A045849 -id:A045849 - 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

A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 7, 35, 0, 42, 35, 0, 105, 28, 21, 140, 49, 105, 105, 106, 210, 84, 182, 210, 217, 287, 105, 420, 378, 126, 497, 392, 420, 532, 350, 630, 714, 434, 546, 980, 742, 609, 980, 896, 1071, 882, 875, 1470, 1239, 1099, 1155, 1722, 1652, 882, 1933, 1995, 1554, 2072, 1505

Offset: 7

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A045849, A063691, A063725, A063730, A340481, A340905, A340915, A340946, A340947.

Column k=7 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, 7):
seq(a(n), n=7..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (theta_3(x) - 1)^7 / 128, where theta_3() is the Jacobi theta function.

A340998 Number of partitions of n into 7 distinct nonzero squares.

Original entry on oeis.org

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

Offset: 140

Ilya Gutkovskiy, Feb 02 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A025441, A025442, A025443, A025444, A045849, A224982, A340906, A340988, A340999, A341000, A341001.

Column k=7 of A341040.

A341402 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 8, 29, 64, 106, 169, 281, 422, 548, 702, 961, 1276, 1556, 1864, 2326, 2893, 3390, 3852, 4545, 5455, 6253, 7002, 8080, 9361, 10453, 11496, 12903, 14618, 16194, 17643, 19589, 22011, 24027, 25714, 28143, 31188, 33792, 36137, 39203, 42920, 46294, 49108, 52580, 57165, 61365

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045849.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045849, A055406, A055413, A224212, A224213, A302862, A341400, A341401.

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, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 10 2021

nmax = 44; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^7/(128 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^7 / (128 * (1 - x)).

a(n^2) = A055406(n).

A282248 Expansion of (Sum_{k>=0} x^(k(5k-3)/2))^7.

Original entry on oeis.org

1, 7, 21, 35, 35, 21, 7, 8, 42, 105, 140, 105, 42, 7, 21, 105, 210, 210, 112, 63, 105, 175, 245, 252, 147, 77, 210, 420, 455, 315, 147, 35, 105, 420, 637, 483, 273, 266, 315, 392, 532, 483, 357, 532, 840, 840, 567, 315, 210, 421, 840, 1050, 777, 462, 497, 707, 882, 917, 735, 525, 889, 1407, 1407, 1050, 770, 525, 630, 1302

Offset: 0

Ilya Gutkovskiy, Feb 09 2017

nonn

Number of ways to write n as an ordered sum of 7 heptagonal numbers (A000566).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 7 heptagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

			a(7) = 8 because we have
[7, 0, 0, 0, 0, 0, 0]
[0, 7, 0, 0, 0, 0, 0]
[0, 0, 7, 0, 0, 0, 0]
[0, 0, 0, 7, 0, 0, 0]
[0, 0, 0, 0, 7, 0, 0]
[0, 0, 0, 0, 0, 7, 0]
[0, 0, 0, 0, 0, 0, 7]
[1, 1, 1, 1, 1, 1, 1]

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

Cf. A000566, A045849, A213523, A226252.

nmax = 67; CoefficientList[Series[Sum[x^(k (5 k - 3)/2), {k, 0, nmax}]^7, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(5*k-3)/2))^7.

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.

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A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

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A340998 Number of partitions of n into 7 distinct nonzero squares.

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A341402 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

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Maple

Mathematica

Formula

A282248 Expansion of (Sum_{k>=0} x^(k*(5*k-3)/2))^7.

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Examples

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Crossrefs

Programs

Mathematica

Formula

A282248 Expansion of (Sum_{k>=0} x^(k(5k-3)/2))^7.