A045848 -id:A045848 - cp's OEIS frontend

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 1, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 2, 0, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 6, 2, 0, 0

Offset: 0

N. J. A. Sloane

			Rows are
1,0,0,..;
1,1,0,0,1,0..;
1,2,1,0,2,2,..;
1,3,3,1,...

Seiichi Manyama, Ascending antidiagonals n = 0..139, flattened
H. Wilf, A combinatorial determinant, arXiv:math/9809120 [math.CO], 1998.

Rows k=0-10 give: A000007, A010052, A000925, A002102, A014110, A038671, A045848, A045849, A045850, A045851, A045852.
Diagonal gives A287617.
Antidiagonal sums give A302018.

i-th row is expansion of (1+x+x^4+x^9+...)^i.

More terms from Erich Friedman

A340905 Number of ways to write n as an ordered sum of 6 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 15, 0, 6, 20, 0, 30, 15, 0, 60, 12, 15, 60, 31, 60, 30, 60, 90, 36, 86, 60, 120, 120, 15, 180, 141, 60, 165, 140, 180, 186, 120, 180, 285, 156, 126, 360, 255, 216, 270, 260, 390, 240, 262, 420, 426, 360, 210, 540, 530, 216, 540, 540, 480, 600, 300, 600, 825, 312, 576, 840

Offset: 6

Ilya Gutkovskiy, Jan 31 2021

Alois P. Heinz, Table of n, a(n) for n = 6..20000
Index entries for sequences related to sums of squares

Cf. A000141, A000290, A010052, A025430, A045848, A063691, A063725, A063730, A340481, A340906, A340915, A340946, A340947.

Column k=6 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..71);  # Alois P. Heinz, Jan 31 2021

nmax = 71; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (theta_3(x) - 1)^6 / 64, where theta_3() is the Jacobi theta function.

A340988 Number of partitions of n into 6 distinct nonzero squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 3, 2, 0, 1, 2, 1, 1, 1, 0, 3, 0, 0, 4, 1, 0, 1

Offset: 91

Ilya Gutkovskiy, Feb 02 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000141, A000290, A010052, A025430, A025441, A025442, A025443, A025444, A045848, A224981, A340905, A340998, A340999, A341000, A341001.

Column k=6 of A341040.

A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

Original entry on oeis.org

1, 7, 22, 42, 63, 99, 160, 220, 265, 337, 457, 577, 672, 792, 978, 1178, 1319, 1469, 1739, 2039, 2255, 2507, 2882, 3242, 3513, 3819, 4269, 4769, 5159, 5555, 6181, 6841, 7246, 7666, 8401, 9181, 9763, 10363, 11188, 12108, 12828, 13434, 14394, 15534, 16359, 17211, 18477, 19677

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045848.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045848, A055405, A055412, A175361, A224212, A224213, A302862, A341400, A341402.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 6)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..47);  # Alois P. Heinz, Feb 10 2021

nmax = 47; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^6/(64 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^6 / (64 * (1 - x)).

a(n^2) = A055405(n).

A282173 Expansion of (Sum_{k>=0} x^(k(k+1)(2*k+1)/6))^6.

Original entry on oeis.org

1, 6, 15, 20, 15, 12, 31, 60, 60, 30, 21, 60, 90, 60, 21, 50, 120, 120, 50, 36, 135, 210, 135, 30, 60, 186, 186, 60, 15, 120, 217, 150, 75, 120, 240, 246, 180, 180, 210, 216, 150, 180, 200, 180, 150, 200, 300, 240, 165, 180, 390, 390, 180, 60, 180, 372, 225, 110, 135, 330, 351, 270, 300, 360, 435, 300, 375, 360, 300, 210

Offset: 0

Ilya Gutkovskiy, Feb 07 2017

nonn

Number of ways to write n as an ordered sum of 6 square pyramidal numbers (A000330).
Conjecture: a(n) > 0 for all n.
Extended conjecture: every number is the sum of at most 6 square pyramidal numbers.
Generalized conjecture: every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5). - Ilya Gutkovskiy, Feb 10 2017

			a(5) = 12 because we have:
[5, 0, 0, 0, 0, 0]
[0, 5, 0, 0, 0, 0]
[0, 0, 5, 0, 0, 0]
[0, 0, 0, 5, 0, 0]
[0, 0, 0, 0, 5, 0]
[0, 0, 0, 0, 0, 5]
[1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 0, 1]
[1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1]
[1, 0, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1]

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A045848.

nmax = 69; CoefficientList[Series[(Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, nmax}])^6, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.

A280719 Expansion of (Sum_{k>=0} x^(k(2k-1)))^6.

Original entry on oeis.org

1, 6, 15, 20, 15, 6, 7, 30, 60, 60, 30, 6, 15, 60, 90, 66, 45, 60, 80, 90, 66, 50, 120, 180, 135, 60, 15, 60, 186, 210, 141, 126, 120, 126, 165, 180, 241, 300, 210, 90, 90, 180, 270, 270, 210, 212, 270, 270, 200, 210, 366, 450, 390, 270, 135, 210, 375, 360, 396, 420, 300, 330, 375, 380, 510, 480, 336, 450, 510, 390, 330

Offset: 0

Ilya Gutkovskiy, Feb 10 2017

nonn

Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 6 hexagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

			a(6) = 7 because we have:
[6, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0]
[0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000384, A007536, A008440, A045848, A280718, A282248.

nmax = 70; CoefficientList[Series[Sum[x^(k (2 k - 1)), {k, 0, nmax}]^6, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(2*k-1)))^6.

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A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

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A340905 Number of ways to write n as an ordered sum of 6 squares of positive integers.

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A340988 Number of partitions of n into 6 distinct nonzero squares.

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A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

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A282173 Expansion of (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.

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Mathematica

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A280719 Expansion of (Sum_{k>=0} x^(k*(2*k-1)))^6.

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Examples

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Programs

Mathematica

Formula

A282173 Expansion of (Sum_{k>=0} x^(k(k+1)(2*k+1)/6))^6.

A280719 Expansion of (Sum_{k>=0} x^(k(2k-1)))^6.