Cheers, this mail contains a new index entry, comments to three existing sequences, and 12 pre-numbered new sequences plus one complete re-edit. ------------------------------------------------------------------------ A NEW INDEX ENTRY, to the page http://www.research.att.com/~njas/sequences/Sindx_Lu.html: Lukasiewicz words, sequences related to (start): Lukasiewicz words: A071152, A071153, A071160 (I will add the words for unary-binary trees later (containing only the digits 0, 1 and 2), when I get my Motzkin-algorithm working...) ------------------------------------------------------------------------ ADDITION TO A057514 (with %Y-line replaced and %H-line added): %I A057514 %S A057514 0,1,2,1,3,2,2,2,1,4,3,3,3,2,3,2,3,3,2,2,2,2,1,5,4,4,4,3,4,3,4,4,3,3,3, %H A057514 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %Y A057514 Cf. A057515. For Maple procedure GrayCode see A055095. a(n)-1 gives the number of zeros in A071153(n) (for n>=1). ADDITION TO A057515 (with %Y-line replaced and %H-line added): %I A057515 %S A057515 0,1,2,1,3,2,2,1,1,4,3,3,2,2,3,2,2,1,1,2,1,1,1,5,4,4,3,3,4,3,3,2,2,3,2, %H A057515 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %Y A057515 Cf. A057514, A057516, A057517, A057501 (for procedure binexp2pars). a(n) gives the first digit of A071153(n). ADDITION TO A063171 (with %Y-line replaced and %H-line added): %I A063171 %S A063171 10,1010,1100,101010,101100,110010,110100,111000,10101010,10101100, %H A063171 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %Y A063171 A063171(n) = A071152(n)/2. A014486 gives these terms as converted from decimal to binary system. Cf. also A071153. ------------------------------------------------------------------------ The 12 PRE_NUMBERED new sequences A071152 - A071163 follow: %I A071152 %S A071152 0,20,2020,2200,202020,202200,220020,220200,222000,20202020,20202200,20220020,20220200,20222000, %T A071152 22002020,22002200,22020020,22020200,22022000,22200020,22200200,22202000,22220000,2020202020, %U A071152 2020202200,2020220020,2020220200,2020222000,2022002020,2022002200,2022020020,2022020200,2022022000 %N A071152 Lukasiewicz words for the rooted plane binary trees (interpretation d in Stanley's exercise 19) with the last leaf implicit, i.e. these words are given without the last trailing zero, except for the null tree which is encoded as 0. %H A071152 Index entries for sequences related to Lukasiewicz words %H A071152 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %H A071152 R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997. %H A071152 R. P. Stanley, Exercises on Catalan and Related Numbers %R A071152 %O A071152 0,2 %K A071152 new,nonn %Y A071152 A071152(n) = 2*A063171(n) = A071153(A057123(n)). Cf. also A071153, A071154, A014486. %A A071152 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071152 %p A071152 %I A071153 %S A071153 0,1,20,11,300,201,210,120,111,4000,3001,3010,2020,2011,3100,2101,2200,1300,1201,2110,1210, %T A071153 1120,1111,50000,40001,40010,30020,30011,40100,30101,30200,20300,20201,30110,20210,20120,20111, %U A071153 41000,31001,31010,21020,21011,32000,22001,23000,14000,13001,22010,13010,12020,12011,31100 %N A071153 Lukasiewicz word for each rooted plane tree (interpretation e in Stanley's exercise 19) encoded by A014486 (or A063171), with the last leaf implicit, i.e. these words are given without the last trailing zero, except for the null tree which is encoded as 0. %C A071153 Note: this finite decimal representation works only upto the 6917th term, as the 6918th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071154 shows the initial portion of this sequence sorted. %e A071153 The 11th term of A063171 is 10110010, corresponding to parenthesization ()(())(), thus its Lukasiewicz word is 3010. The 18th term of A063171 is 11011000, corresponding to parenthesization (()(())), thus its Lukasiewicz word is 1201. I.e. in the latter example there is one list on the top-level, which in turn contains two sublists, of which the first is zero elements long, and the second is a sublist containing one empty sublist (the last zero is omitted). %R A071153 %H A071153 Index entries for sequences related to Lukasiewicz words %H A071153 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %H A071153 R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997. %H A071153 R. P. Stanley, Exercises on Catalan and Related Numbers %O A071153 0,3 %K A071153 new,nonn,fini %A A071153 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %Y A071153 For n >= 1, the number of zeros in the term a(n) is given by A057514(n)-1. The first digit of each term is given by A057515. Cf. also A014486, A071152, A071154. Corresponding factorial walk encoding: A071155 (A071157, A071159). %D A071153 %p A071153 %I A071154 %S A071154 1,11,20,111,120,201,210,300,1111,1120,1201,1210,1300,2011,2020,2101,2110,2200,3001,3010, %T A071154 3100,4000,11111,11120,11201,11210,11300,12011,12020,12101,12110,12200,13001,13010,13100,14000, %U A071154 20111,20120,20201,20210,20300,21011,21020,21101,21110,21200,22001,22010,22100,23000,30011 %N A071154 Totally balanced decimal numbers: if we assign the weight w(d) = d-1 for each digit d (i.e. w(0) = -1, w(1) = 0, ..., w(9) = 8), and then read the digits of the term from left to right, the partial sum of weights goes never negative, and the total weighted sum is zero. %C A071154 The initial portion of this sequence (upto the 6917th term) is equal to A071153 (Lukasiewicz words for rooted plane trees) sorted to ascending order by the numerical value. %R A071154 %H A071154 Index entries for sequences related to Lukasiewicz words %O A071154 1,3 %Y A071154 A071153. Subset of A061384. Superset of A071161. Cf. also totally balanced binary numbers: A014486. %K A071154 new,nonn,base %A A071154 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071154 %p A071154 %I A071155 %S A071155 0,1,3,5,9,15,11,17,23,33,57,39,63,87,35,59,41,65,89,47,71,95,119,153,273,177,297,417,159,279, %T A071155 183,303,423,207,327,447,567,155,275,179,299,419,161,281,185,305,425,209,329,449,569,167,287, %U A071155 191,311,431,215,335,455,575,239,359,479,599,719,873,1593,993,1713,2433,897,1617,1017,1737 %N A071155 The Catalan factorial walks (for each rooted plane tree encoded by A014486) encoded as zero-free numbers in factorial base (A007623). %R A071155 %H A071155 C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou, et. al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55. %H A071155 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %O A071155 0,3 %K A071155 new,nonn %A A071155 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071155 %Y A071155 Same sequence sorted: A071156, expanded in the factorial number system: A071157. Corresponding Lukasiewicz words: A071153. %I A071156 %S A071156 0,1,3,5,9,11,15,17,23,33,35,39,41,47,57,59,63,65,71,87,89,95,119,153,155,159,161,167,177,179, %T A071156 183,185,191,207,209,215,239,273,275,279,281,287,297,299,303,305,311,327,329,335,359,417,419, %U A071156 423,425,431,447,449,455,479,567,569,575,599,719,873,875,879,881,887,897,899,903,905,911,927 %N A071156 Apart from the initial term (0), lists all integers whose factorial expansion ends with 1 (i.e. are odd numbers), do not contain a digit zero, and each successive digit to the left is at most one greater than the preceding digit. %C A071156 A071155 sorted. Catalan numbers A000108(n) gives the number of terms whose factorial expansion contain n digits. %R A071156 %O A071156 0,3 %K A071156 new,nonn %A A071156 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071156 %Y A071156 This sequence expanded in the factorial number system: A071158. %p A071156 %I A071157 %S A071157 0,1,11,21,111,211,121,221,321,1111,2111,1211,2211,3211,1121,2121,1221,2221,3221,1321,2321, %T A071157 3321,4321,11111,21111,12111,22111,32111,11211,21211,12211,22211,32211,13211,23211,33211,43211, %U A071157 11121,21121,12121,22121,32121,11221,21221,12221,22221,32221,13221,23221,33221,43221,11321 %N A071157 The zero-free, right-to-left factorial walk encoding for each rooted plane tree encoded by A014486. Sequence A071155 shown with factorial expansion (A007623). %C A071157 Apart from the initial term (0, which encodes the null tree), if we scan the digits from the right (the least significant digit which is always 1) to the left (the most significant), then each successive digit to the left is at most one greater than the previous and never less than one. %C A071157 Note: this finite decimal representation works only upto the 23712nd term, as the 23713rd such walk is already (10,9,8,7,6,5,4,3,2,1). The sequence A071158 shows the initial portion of this sequence sorted. %R A071157 %H A071157 C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou, et. al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55. %H A071157 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %O A071157 0,3 %K A071157 new,nonn,fini %A A071157 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071157 %Y A071157 A071157(n) = A007623(A071155(n)). Corresponding Lukasiewicz words: A071153. Essentially the same as A071159 but with digits reversed. %p A071157 %I A071158 %S A071158 1,11,21,111,121,211,221,321,1111,1121,1211,1221,1321,2111,2121,2211,2221,2321,3211,3221, %T A071158 3321,4321,11111,11121,11211,11221,11321,12111,12121,12211,12221,12321,13211,13221,13321,14321, %U A071158 21111,21121,21211,21221,21321,22111,22121,22211,22221,22321,23211,23221,23321,24321,32111 %N A071158 Integers whose decimal expansion end with 1, do not contain zeroes, and each successive digit to the left is at most one greater than the previous. %C A071158 The initial portion of this sequence equals to A071157 sorted. %R A071158 %O A071158 1,3 %K A071158 new,nonn,base %A A071158 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071158 %Y A071158 A071158(n) = A007623(A071156(n)). Cf. also A071159. %p A071158 %I A071159 %S A071159 1,11,12,111,112,121,122,123,1111,1112,1121,1122,1123,1211,1212,1221,1222,1223,1231,1232, %T A071159 1233,1234,11111,11112,11121,11122,11123,11211,11212,11221,11222,11223,11231,11232,11233,11234, %U A071159 12111,12112,12121,12122,12123,12211,12212,12221,12222,12223,12231,12232,12233,12234,12311 %N A071159 Integers whose decimal expansion start with 1, do not contain zeroes, and each successive digit to the right is at most one greater than the previous. %R A071159 %O A071159 1,3 %K A071159 new,nonn,base %A A071159 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %Y A071159 Essentially the same as A071157 but with digits reversed. Corresponding Lukasiewicz words: A071153. %D A071159 %p A071159 %I A071160 %S A071160 0,1,20,11,300,201,120,111,4000,3001,2020,2011,1300,1201,1120,1111,50000,40001,30020,30011, %T A071160 20300,20201,20120,20111,14000,13001,12020,12011,11300,11201,11120,11111,600000,500001,400020, %U A071160 400011,300300,300201,300120,300111,204000,203001,202020,202011,201300,201201,201120,201111 %N A071160 Lukasiewicz words that are also valid asynchronic siteswap juggling patterns. %F A071160 Construction: starting from the most significant (the leftmost) bit, replace each 1-bit in the binary expansion of n with the distance to the next 1-bit to the right, allowing a cyclic wrap-over from the least-significant 1-bit to the most significant 1-bit. I.e. from 22 = 10110 in binary we get 20120, the 22nd term of this sequence. %C A071160 Note: this finite decimal representation works only upto the 511th term, as the 512th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071161 shows the initial portion of this sequence sorted. %H A071160 Peter J. Beek and Arthur Lewbel, The Science of Juggling, Scientific American, Nov, 1995, Vol. 273, Number 5, pp. 92-97. %H A071160 Joe Buhler and Ron Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519. %H A071160 Juggling Information Service: Site Swap FAQs %H A071160 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %H A071160 R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997. %H A071160 Index entries for sequences related to Lukasiewicz words %R A071160 %O A071160 0,3 %K A071160 new,nonn,fini %A A071160 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %Y A071160 A071160(n) = A071161(A054429(n)). Subset of A071153. A071162, A071163, Cf. also A060495, A060498, A065177. %D A071160 %p A071160 %I A071161 %S A071161 0,1,11,20,111,120,201,300,1111,1120,1201,1300,2011,2020,3001,4000,11111,11120,11201,11300, %T A071161 12011,12020,13001,14000,20111,20120,20201,20300,30011,30020,40001,50000,111111,111120,111201, %U A071161 111300,112011,112020,113001,114000,120111,120120,120201,120300,130011,130020,140001,150000 %N A071161 Integers whose decimal expansion satisfies the condition that if we read each term from the left to right (the most significant to the least significant digit) then each non-zero digit gives a distance to the next non-zero digit to right (with a cyclic wrap-over from the least-significant to the most significant non-zero digit). %R A071161 %O A071161 0,3 %K A071161 new,nonn,base %A A071161 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071161 %Y A071161 Subset of A071154. The initial portion (upto the 511th term) of this sequence satisfies A071161(n) = A071160(A054429(n)). %p A071161 %I A071162 %S A071162 0,2,10,12,42,44,52,56,170,172,180,184,212,216,232,240,682,684,692,696,724,728,744,752,852, %T A071162 856,872,880,936,944,976,992,2730,2732,2740,2744,2772,2776,2792,2800,2900,2904,2920,2928,2984, %U A071162 2992,3024,3040,3412,3416,3432,3440,3496,3504,3536,3552,3752,3760,3792,3808,3920,3936,4000 %N A071162 Those totally balanced binary sequences which encode rooted plane trees whose Lukasiewicz words are also valid asynchronic siteswap juggling patterns. %C A071162 The terms of the binary length 2n are counted by 2's powers, A000079. %R A071162 %H A071162 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %O A071162 0,2 %K A071162 new,nonn %A A071162 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %Y A071162 A071160 gives the finite decimal representation of the said Lukasiewicz words and A071163 gives the positions of these terms in A014486, whose subset this sequence is. %D A071162 %p A071162 %I A071163 %S A071163 0,1,2,3,4,5,7,8,9,10,12,13,17,18,21,22,23,24,26,27,31,32,35,36,45,46,49,50,58,59,63,64,65, %T A071163 66,68,69,73,74,77,78,87,88,91,92,100,101,105,106,129,130,133,134,142,143,147,148,170,171,175, %U A071163 176,189,190,195,196,197,198,200,201,205,206,209,210,219,220,223,224,232,233,237,238,261,262 %N A071163 Positions of A071162 in A014486. %R A071163 %H A071163 A. Karttunen, Gatomorphisms and other excursions amidst the plane trees & parenthesizations (Includes the complete Scheme program for computing this sequence) %O A071163 0,3 %K A071163 new,nonn %A A071163 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 14 2002 %D A071163 %p A071163 ------------------------------------------------------------------------ ADDITIONAL OBSERVATIONS: Note how the semantics of some of the sequence pairs like A071153 - A071154 (sorted), A071157 - A071158 (sorted) A071160 - A071161 (sorted) differ. The first one of these pairs (marked with the keyword fini) means a _finite_ prefix of the infinite sequence of words, upto the point where they cannot be anymore presented with just decimal digits (0-9). On the other hand, the second one of these pairs, marked with the keyword base, is an infinite list of terms constructed with a similar condition - but using just the decimal representation of n -, so they are actually subsets of the infinite sequences of words given first, but with all the words that are not representable with the decimal digits omitted. If we were strict, the similar notes should be added to the sequences A007623, A049345, A064039, A060495, A060496, A060498 and A060499 which are likewise finite prefixes of infinite sequences of "vectors" not wholly representable with decimal digits only: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A007623 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A049345 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A064039 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A060495 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A060496 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A060498 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A060499 Hmm, and A030299 is the perfect example of this thing. Here's a complete re-edit for it: %I A030299 %S A030299 1,12,21,123,132,213,231,312,321,1234,1243,1324,1342,1423,1432,2134, %T A030299 2143,2314,2341,2413,2431,3124,3142,3214,3241,3412,3421,4123,4132,4213, %U A030299 4231,4312,4321,12345,12354,12435,12453,12534,12543,13245,13254,13425 %N A030299 Permutations of lengths 1, 2, 3, ... arranged lexicographically. %C A030299 Note: this finite decimal representation works only upto the term 987654321, as the next permutation in the infinite list is already [1,2,3,4,5,6,7,8,9,10]. %K A030299 nonn,easy,base,fini %Y A030299 Cf. also A055089, A060117. %O A030299 1,2 %A A030299 njas, Clark Kimberling (ck6@cedar.evansville.edu) Kill the duplicate: %I A064775 %S A064775 1,12,21,123,132,213,231,312,321,1234,1243,1324,1342,1423,1432,2134, %T A064775 2143,2314,2341,2413,2431,3124,3142,3214,3241,3412,3421,4123,4132,4213, %U A064775 4231,4312,4321,12345,12354,12435,12453,12534,12543,13245,13254,13425 %N A064775 Numbers whose digits are permutations of (1,2,3,4,...,i) for some i >= 1. %K A064775 base,nonn %O A064775 1,2 %A A064775 Klaus Strassburger (strass@ddfi.uni-duesseldorf.de), Oct 19 2001 ----------------------------------------------------------------------------- Yours, Antti Karttunen