A010854 -id:A010854 - cp's OEIS frontend

A181404 Total number of positive integers below 10^n requiring 8 positive cubes in their representation as sum of cubes.

0, 3, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Martin Renner, Jan 28 2011

Also continued fraction expansion of (9+sqrt(229))/74. - Bruno Berselli, Sep 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A018889, A010854.

Cf. A002283, A061439, A130130, A181375, A181377, A181379, A181381, A181400, A181402.

PadRight[{0, 3}, 100, 15] (* Paolo Xausa, May 24 2024 *)

a(n)=if(n>2,15,3*n-3) \\ Charles R Greathouse IV, Oct 07 2015

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + a(n) + A130130(n) = A002283(n).
a(n) = 15 for n > 2. - Charles R Greathouse IV, Sep 09 2011
G.f.: 3*x^2*(1+4*x)/(1-x). - Bruno Berselli, Sep 09 2011
E.g.f.: 3*(5*(exp(x) - 1 - x) - 2*x^2). - Stefano Spezia, May 21 2024

a(5)-a(7) from Lars Blomberg, May 04 2011

A023013 Number of partitions of n into parts of 15 kinds.

Original entry on oeis.org

1, 15, 135, 920, 5220, 25893, 115700, 475065, 1817910, 6551390, 22414314, 73265580, 229972855, 696109950, 2039031360, 5796944357, 16036186005, 43259046975, 114012183695, 294067720380, 743368453326, 1844121021245, 4494803760045

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010854. - Alois P. Heinz, Oct 17 2008

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

15th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*15, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

CoefficientList[Series[1/QPochhammer[x]^15, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

Vec(1/eta(x)^15 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

a(0) = 1, a(n) = (15/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

a(n) ~ m^((m+1)/4) * exp(Pi*sqrt(2*m*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)) * (1 - ((9+Pi^2)*m^2+36*m+27) / (24*Pi*sqrt(6*m*n))), set m = 15. - Vaclav Kotesovec, Jun 28 2025

A166126 Decimal expansion of 1/(imaginary part of (15+2*I)^(1/2))^2.

Original entry on oeis.org

1, 5, 0, 6, 6, 3, 7, 2, 9, 7, 5, 2, 1, 0, 7, 7, 7, 9, 6, 3, 5, 9, 5, 9, 3, 1, 0, 2, 4, 6, 7, 0, 5, 3, 2, 6, 0, 5, 8, 6, 2, 4, 3, 7, 7, 4, 1, 9, 2, 5, 9, 8, 5, 0, 9, 1, 1, 4, 3, 4, 5, 1, 4, 9, 6, 4, 9, 1, 4, 0, 5, 5, 5, 1, 7, 5, 8, 5, 0, 8, 3, 8, 6, 2, 3, 1, 1, 4, 7, 6, 6, 8, 8, 7, 4, 3, 7, 9, 8, 5, 5, 5, 6, 9, 3

Offset: 2

Klaus Brockhaus, Oct 07 2009

lim_{n -> infinity} b(n)/b(n-1) = 1/imag((15+2*I)^(1/2))^2 for b = A154597.
Contribution from Klaus Brockhaus, May 28 2010: (Start)
Also decimal expansion of (15+sqrt(229))/2.
Continued fraction expansion of (15+sqrt(229))/2 is A010854. (End)

			1/imag((15+2*I)^(1/2))^2 = 15.06637297521077796....

A154597, A166125 (decimal expansion of sqrt(229)).

Cf. A010854 (all 15's sequence). [From Klaus Brockhaus, May 28 2010]

A343052 Table read by ascending antidiagonals: T(k, n) is the minimum vertex sum in a perimeter-magic k-gon of order n.

Original entry on oeis.org

6, 12, 6, 15, 10, 6, 24, 15, 12, 6, 28, 21, 15, 10, 6, 40, 28, 24, 15, 12, 6, 45, 36, 28, 21, 15, 10, 6, 60, 45, 40, 28, 24, 15, 12, 6, 66, 55, 45, 36, 28, 21, 15, 10, 6, 84, 66, 60, 45, 40, 28, 24, 15, 12, 6, 91, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 112, 91, 84, 66, 60, 45, 40, 28, 24, 15, 12, 6

Offset: 3

Stefano Spezia, Apr 03 2021

			The table begins:
k\n|   3   4   5   6   7 ...
---+--------------------
3  |   6   6   6   6   6 ...
4  |  12  10  12  10  12 ...
5  |  15  15  15  15  15 ...
6  |  24  21  24  21  24 ...
7  |  28  28  28  28  28 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 5 and 7).

Cf. A000217 (n = 4), A010722 (k = 3), A010854 (k = 5), A010867 (k = 7), A265225, A343053 (maximum).

T[k_,n_]:=k(1+k+Mod[n,2](1-Mod[k,2]))/2; Table[T[k+3-n,n],{k,3,14},{n,3,k}]//Flatten

O.g.f.: x*(1 + x^2 + y + x*(2 + 3*y))/((1 - x)^3*(1 + x)^2*(1 - y^2)).
E.g.f.: x*((5 + 2*x)*cosh(x + y) - cosh(x - y) + 2*(2 + x)*sinh(x + y))/4.
T(k, n) = k*(1 + k + (n mod 2)*(1 - (k mod 2)))/2.
T(k, 3) = A265225(k-1) (conjectured).

cp's OEIS Frontend

A181404 Total number of positive integers below 10^n requiring 8 positive cubes in their representation as sum of cubes.

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A023013 Number of partitions of n into parts of 15 kinds.

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A166126 Decimal expansion of 1/(imaginary part of (15+2*I)^(1/2))^2.

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A343052 Table read by ascending antidiagonals: T(k, n) is the minimum vertex sum in a perimeter-magic k-gon of order n.

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Mathematica

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