A059825 -id:A059825 - cp's OEIS frontend

A059819 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^2 *Product_{i=1..t} (1-x^i) ).

0, 1, 3, 5, 9, 11, 18, 19, 28, 30, 40, 39, 57, 51, 68, 68, 86, 77, 107, 91, 123, 114, 138, 121, 172, 140, 178, 166, 205, 171, 240, 189, 251, 224, 266, 230, 322, 245, 314, 286, 356, 283, 396, 303, 403, 361, 416, 343, 497, 368, 479, 424, 515, 407

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), here (k=2), A059820 (k=3), ..., A059825 (k=8).

Cf. A000203, A055507.

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=2

a(n) = (sigma(n)+tau(n)+Sum_{k=0..n} tau(k)*tau(n-k))/2.

G.f.: (F(x)+G(x)^2)/2, where F(x) = Sum_{k>0} (k+1)*x^k/(1-x^k) and G(x) = Sum_{k>0} x^k/(1-x^k). - Vladeta Jovovic, Feb 12 2004

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 4, 9, 19, 30, 52, 70, 107, 136, 191, 226, 314, 352, 463, 523, 664, 717, 919, 964, 1205, 1282, 1546, 1603, 1992, 2009, 2414, 2504, 2958, 2974, 3606, 3553, 4223, 4273, 4936, 4912, 5885, 5685, 6634, 6654, 7664, 7454, 8822, 8454, 9845

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), A059821(k=4), A059822 (k=5), A059823 (k=6), A059824 (k=7), A059825 (k=8).
Cf. A002133, A065608, A191822, A191832.
Cf. A000203, A001157, A055507, A191829 (Andrews's D_{0,0,0}(n)), A191831 (Andrews's D_{0,1}(n)).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=3

D(x, y, n) = sum(k=1, n-1, sigma(k, x)*sigma(n-k, y));
D000(n) = sum(k=1, n-1, sigma(k, 0)*D(0, 0, n-k));
a(n) = if(n==0, 0, (3*D(0, 0, n)+3*D(0, 1, n)+D000(n)+2*sigma(n, 0)+3*sigma(n)+sigma(n, 2))/6); \\ Seiichi Manyama, Jul 26 2024

a(n) = ( 3*A055507(n-1) + 3*A191831(n) + A191829(n) + 2*sigma_0(n) + 3*sigma(n) + sigma_2(n) )/6. - Seiichi Manyama, Jul 26 2024

A059832 A ternary tribonacci triangle: form the triangle as follows: start with 3 single values: 1, 2, 3. Each succeeding row is a concatenation of the previous 3 rows.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3

Offset: 0

Jason Earls, Feb 25 2001

Alternatively, define a morphism f: 1 -> 2, 2 -> 3, 3 -> 1,2,3; let S(0)=1, S(k) = f(S(k-1)) for k>0; then sequence is the concatenation S(0) S(1) S(2) S(3) ...

			Rows 0, 1, 2, ..., 8, ... of the triangle are:
0, [1]
1, [2]
2, [3]
3, [1, 2, 3]
4, [2, 3, 1, 2, 3]
5, [3, 1, 2, 3, 2, 3, 1, 2, 3]
6, [1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3]
7, [2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3]
8, [3, 1, 2, 3, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 2, 3]
...

C. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 273.

N. J. A. Sloane, Table of n, a(n) for n = 0..30121 (Roes 0 through 17, flattened.)
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Cf. A059835. Row sums A001590, row lengths A000213.

Rows 0,3,6,9,12,... converge to A305389, rows 1,4,7,10,... converge to A305390, and rows 2,5,8,11,... converge to A305391.

# To get successive rows of A059832
S:=Array(0..100);
S[0]:=[1];
S[1]:=[2];
S[2]:=[3];
for n from 3 to 12 do
S[n]:=[op(S[n-3]),op(S[n-2]), op(S[n-1])];
lprint(S[n]);
od: # N. J. A. Sloane, Jul 04 2018

a(n) = A059825(n) + 1. - Sean A. Irvine, Oct 11 2022

More terms from Larry Reeves (larryr(AT)acm.org), Feb 26 2001

Entry revised by N. J. A. Sloane, Jun 21 2018

A059823 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^6 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 7, 27, 83, 202, 455, 889, 1682, 2892, 4894, 7694, 12090, 17822, 26411, 37206, 52730, 71447, 97984, 128714, 171421, 220064, 285963, 359204, 458506, 565347, 708665, 862163, 1064302, 1276474, 1558090, 1845874, 2226044, 2614188

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=6

A059821 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^4 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 5, 14, 34, 64, 121, 190, 311, 446, 666, 887, 1266, 1599, 2169, 2679, 3504, 4178, 5383, 6253, 7858, 9060, 11114, 12560, 15390, 17076, 20512, 22788, 26993, 29494, 34988, 37750, 44213, 47857, 55281, 59196, 68810, 72754, 83518, 88947

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=4

A059822 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^5 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 6, 20, 55, 119, 246, 435, 766, 1211, 1926, 2807, 4193, 5766, 8161, 10821, 14711, 18820, 24925, 31009, 39984, 48895, 61609, 73844, 91905, 108264, 132400, 154641, 186462, 214772, 257118, 292749, 346430, 392499, 459424, 515579

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=5

A059824 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>=1} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^7 * Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 8, 35, 119, 321, 784, 1672, 3389, 6280, 11285, 18971, 31383, 49162, 76322, 113494, 167785, 239086, 340355, 468636, 646058, 865724, 1161936, 1520105, 1997015, 2559758, 3297599, 4157592, 5266644, 6537922, 8168293, 10003615

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=7

cp's OEIS Frontend

A059819 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^2 *Product_{i=1..t} (1-x^i) ).

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A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

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A059832 A ternary tribonacci triangle: form the triangle as follows: start with 3 single values: 1, 2, 3. Each succeeding row is a concatenation of the previous 3 rows.

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A059823 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^6 *product_{i=1..t} (1-x^i) ).

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A059821 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^4 *product_{i=1..t} (1-x^i) ).

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A059822 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^5 *product_{i=1..t} (1-x^i) ).

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A059824 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>=1} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^7 * Product_{i=1..t} (1-x^i) ).

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A059819 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^2 *Product_{i=1..t} (1-x^i) ).

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

A059823 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^6 *product_{i=1..t} (1-x^i) ).

A059821 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^4 *product_{i=1..t} (1-x^i) ).

A059822 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^5 *product_{i=1..t} (1-x^i) ).

A059824 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>=1} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^7 * Product_{i=1..t} (1-x^i) ).