Cheers,
12 NEW PRENUMBERED SEQUENCES, A112083-A112086 & A112132-A112139.


%I A112083
%S A112083 2,4,12,35,155,239,779,2855,5279,9095,15695,59135,350699,183395,
%T A112083 1352339,1477295,1077959,6922920,3038555,12705840,14199120
%N A112083 Column 2 of A112060.
%K A112083 nonn
%Y A112083 Also row 2 of A112061. C.f. A112084.
%O A112083 1,1
%A A112083 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112084
%S A112084 5,9,25,71,311,479,1559,5711,10559,18191,31391,118271,701399,366791,
%T A112084 2704679,2954591,2155919,13845841,6077111,25411681,28398241
%N A112084 Column 2 of A112070.
%C A112084 Note the subsequence equal to the portion of A045535 / A062241, and also the non-monotone drops present, like the one from a(13)=701399 to a(14)=366791.
%C A112084 An independent recomputation with another software, e.g. Mathematica, would be welcome.
%Y A112084 Row 2 of A112071. a(n) = A005408(A112083(n)).
%K A112084 nonn,more
%O A112084 1,1
%A A112084 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112085
%S A112085 1,2,2,1,4,1,2,2,1,2,2,1,4,1,2,2,1,2,2,1,2,2,1,4,1,2,2,1,2,2,1,4,1,2,
%T A112085 2,1,2,2,1,2,2,1,4,1,2,2,1,2,2,1,4,1,2,2,1,4,1,2,2,1,4,1,2,2,1,2,2,1,
%U A112085 4,1,2,2,1,4,1,2,2,1,4,1,2,2,1,2,2,1,4,1,2,2,1,2,2,1,2,2,1,4,1,2,2,1
%N A112085 First differences of A112055.
%F A112085 a(n) = A112055(n+1)-A112055(n).
%C A112085 Will other terms than 1, 2 or 4 ever appear? Is there genuine periodicity?
%K A112085 nonn
%O A112085 1,2
%A A112085 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112086
%S A112086 2,4,6,16,72,420,3240
%N A112086 a(n) = the period of the first differences of the nth row of A112060 (or A112070), or 0 if that row has not a periodic first difference.
%C A112086 These values have been computed empirically. An independent recomputation or a mathematical proof would be welcome. The initial terms factored: 2, 2*2, 2*3, 2*2*2*3*3, 2*2*7*3*5, 2*2*2*3*3*3*3*5, ...
%Y A112086 These are the periods of A010684, A112132, A112133, A112134, A112135, A112136, A112137, etc. (For A112138 & A112139 not computed yet). If we sum the period length prefixes of these sequences, as: Sum_{i=1..a(1)} A010684(i), Sum_{i=1..a(2)} A112132(i), Sum_{i=1..a(3)} A112133(i), etc., we get the sequence 4, 12, 60, 420, 4620, 60060, 1021020, ... (c.f. A097250) and when doubled, it yields: 8, 24, 120, 840, 9240, 120120, 2042040, ... (c.f. A066631 and A102476).
%K A112086 nonn,more
%O A112086 1,1
%A A112086 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112132
%S A112132 1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,
%T A112132 1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,
%U A112132 1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3,1,7,1,3
%N A112132 First differences of A112062.
%Y A112132 Period=4. Also half of the first differences of A112072. C.f. A112086.
%K A112132 nonn
%O A112132 1,2
%A A112132 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112133
%S A112133 1,11,13,11,1,23,1,11,13,11,1,23,1,11,13,11,1,23,1,11,13,11,1,23,1,11,
%T A112133 13,11,1,23,1,11,13,11,1,23,1,11,13,11,1,23,1,11,13,11,1,23,1,11,13,
%U A112133 11,1,23,1,11,13,11,1,23,1,11,13,11,1,23,1,11,13,11,1,23,1,11,13,11,1
%N A112133 First differences of A112063.
%Y A112133 Period=6. Also half of the first differences of A112073. C.f. A112086.
%K A112133 nonn
%O A112133 1,2
%A A112133 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112134
%S A112134 11,24,36,24,1,59,25,11,25,59,1,24,36,24,11,49,11,24,36,24,1,59,25,11,
%T A112134 25,59,1,24,36,24,11,49,11,24,36,24,1,59,25,11,25,59,1,24,36,24,11,49,
%U A112134 11,24,36,24,1,59,25,11,25,59,1,24,36,24,11,49,11,24,36,24,1,59,25,11
%N A112134 First differences of A112064.
%Y A112134 Period=16. Also half of the first differences of A112074. C.f. A112086.
%K A112134 nonn
%O A112134 1,1
%A A112134 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112135
%S A112135 95,120,60,24,60,85,60,36,95,60,145,84,11,25,59,36,84,60,1,155,25,84,
%T A112135 71,24,60,1,144,95,25,71,24,61,84,71,25,59,25,71,84,61,24,71,25,95,
%U A112135 144,1,60,24,71,84,25,155,1,60,84,36,59,25,11,84,145,60,95,36,60,85
%N A112135 First differences of A112065.
%Y A112135 Period=72. Also half of the first differences of A112075. C.f. A112086.
%K A112135 nonn
%O A112135 1,1
%A A112135 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112136
%S A112136 155,336,84,180,336,145,24,420,71,120,144,61,215,84,300,25,95,205,215,
%T A112136 216,180,84,1,60,95,204,300,36,145,144,36,59,25,156,504,251,169,60,95,
%U A112136 36,205,95,25,95,205,299,61,264,336,95,84,216,265,60,204,95,60,156
%N A112136 First differences of A112066.
%Y A112136 Period=420. Also half of the first differences of A112076. C.f. A112086.
%K A112136 nonn
%O A112136 1,1
%A A112136 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112137
%S A112137 635,720,36,1680,60,769,311,25,155,924,1,1404,131,289,95,120,205,611,
%T A112137 84,660,361,84,575,85,396,215,205,95,624,61,239,25,431,480,780,60,265,
%U A112137 479,120,961,24,335,685,95,565,336,155,600,324,181,755,169,516,95
%N A112137 First differences of A112067.
%Y A112137 Period=3240. Also half of the first differences of A112077. C.f. A112086.
%K A112137 nonn
%O A112137 1,1
%A A112137 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112138
%S A112138 2675,1344,420,576,204,1320,756,1080,445,815,924,3120,481,1895,84,205,
%T A112138 635,1140,265,180,1271,564,780,961,204,95,1140,516,229,456,924,660,
%U A112138 911,265,659,180,925,480,611,360,745,1140,864,155,456,205,1115,169
%N A112138 First differences of A112068.
%Y A112138 Period=?. Also half of the first differences of A112078. C.f. A112086.
%K A112138 nonn
%O A112138 1,1
%A A112138 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005

%I A112139
%S A112139 5015,2616,9240,2844,5496,1140,325,5639,2220,6636,1465,420,719,420,
%T A112139 1560,265,2675,2041,5939,1621,3035,2664,336,769,3791,1980,949,4211,
%U A112139 2329,755,2941,839,145,635,2665,71,420,204,216,685,1824,3060,420,731
%N A112139 First differences of A112069.
%Y A112139 Period=?. Also half of the first differences of A112079. C.f. A112086.
%K A112139 nonn
%O A112139 1,1
%A A112139 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 28 2005




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Yours,

Antti Karttunen