A302998 -id:A302998 - cp's OEIS frontend

A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

1, 2, 6, 29, 165, 1203, 9763, 83877, 793049, 7903501, 83570177, 933697153, 10905583809, 133352809334, 1695473999478, 22354920990148, 305096197935075, 4296142551821184, 62336908825014452, 930284705538262688, 14255992611680074754, 224065160215526683317, 3607018540134004189466

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n^2.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Main diagonal of A302998.

Cf. A000603, A000604, A010052, A055403, A055404, A055405, A055406, A055407, A055408, A055409, A298329, A302861, A302862.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n^2}], {n, 0, 22}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n^2}], {n, 0, 22}]

A055417 Number of points in N^n of norm <= 2.

Original entry on oeis.org

1, 3, 6, 11, 20, 36, 63, 106, 171, 265, 396, 573, 806, 1106, 1485, 1956, 2533, 3231, 4066, 5055, 6216, 7568, 9131, 10926, 12975, 15301, 17928, 20881, 24186, 27870, 31961, 36488, 41481, 46971, 52990, 59571, 66748, 74556, 83031, 92210, 102131, 112833, 124356

Offset: 0

David W. Wilson

Binomial transform of [1, 0, 2, -1, 2, -1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Mar 12 2009

			{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), (0, 1, 1), (0, 2, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1), (2, 0, 0)} are all the points in N^3 of norm <= 2 so a(3)=11.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row n=2 of A302998.

CoefficientList[Series[(-z^3 - z^2 + 2*z - 1)/(z - 1)^5, {z, 0, 100}], z] (* and *) Table[(n^4 - 6*n^3 + 23 n^2 + 6*n)/24, {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{1,3,6,11,20},50] (* Harvey P. Dale, Aug 30 2025 *)

PARI
```
a(n)=(n^3-3*n^2+14*n+24)*(n+1)/24
```

a(n) = (n^3 - 3*n^2 + 14*n + 24)*(n+1)/24. Proof: The coordinates of such a point are a permutation of one of the vectors (0, ..., 0), (0, ..., 0, 1), (0, ..., 0, 2), (0, ..., 0, 1, 1), (0, ..., 0, 1, 1, 1), or (0, ..., 0, 1, 1, 1, 1), so the number of points is 1 + n + n + binomial(n,2) + binomial(n,3) + binomial(n,4). - Formula conjectured by Frank Ellermann, Mar 16 2002 and explained by Michael Somos, Apr 25 2003

G.f.: (1-2*x+x^2+x^3)/(1-x)^5. - Michael Somos, Apr 25 2003

A055418 Number of points in N^n of norm <= 3.

Original entry on oeis.org

1, 4, 11, 29, 70, 157, 337, 702, 1420, 2780, 5258, 9615, 17043, 29381, 49430, 81404, 131563, 209084, 327237, 504945, 768820, 1155781, 1716375, 2518938, 3654750, 5244356, 7445244, 10461091, 14552809, 20051645, 27374612, 37042552, 49701157

Offset: 0

David W. Wilson

			There are exactly 19 coordinate configurations (up to permutation) with up to 9 nonzero positive coordinates that can produce a vector of norm <= 3:
{..., 0, 0, 0, 0, 0, 0, 0, 0, 0}   0
{..., 0, 0, 0, 0, 0, 0, 0, 0, 1}   1
{..., 0, 0, 0, 0, 0, 0, 0, 0, 2}   2
{..., 0, 0, 0, 0, 0, 0, 0, 0, 3}   3
{..., 0, 0, 0, 0, 0, 0, 0, 1, 1}   sqrt(2)
{..., 0, 0, 0, 0, 0, 0, 0, 1, 2}   sqrt(5)
{..., 0, 0, 0, 0, 0, 0, 0, 2, 2}   2 sqrt(2)
{..., 0, 0, 0, 0, 0, 0, 1, 1, 1}   sqrt(3)
{..., 0, 0, 0, 0, 0, 0, 1, 1, 2}   sqrt(2) sqrt(3)
{..., 0, 0, 0, 0, 0, 0, 1, 2, 2}   3
{..., 0, 0, 0, 0, 0, 1, 1, 1, 1}   2
{..., 0, 0, 0, 0, 0, 1, 1, 1, 2}   sqrt(7)
{..., 0, 0, 0, 0, 1, 1, 1, 1, 1}   sqrt(5)
{..., 0, 0, 0, 0, 1, 1, 1, 1, 2}   2 sqrt(2)
{..., 0, 0, 0, 1, 1, 1, 1, 1, 1}   sqrt(6)
{..., 0, 0, 0, 1, 1, 1, 1, 1, 2}   3
{..., 0, 0, 1, 1, 1, 1, 1, 1, 1}   sqrt(7)
{..., 0, 1, 1, 1, 1, 1, 1, 1, 1}   2 sqrt(2)
{..., 1, 1, 1, 1, 1, 1, 1, 1, 1}   3
To produce the formula for a(n), it is sufficient to sum the number of permutations of these configurations in a vector of arbitrary length n.
This gives in the same order:
a(n) = 1 + n + n + n + binomial(n, 2) + n*(n - 1) + binomial(n, 2) + binomial(n, 3) + n*binomial(n-1, 2) + n*binomial(n-1, 2) + binomial(n, 4) + n*binomial(n-1, 3) + binomial(n, 5) + n*binomial(n-1, 4) + binomial(n, 6) + n*binomial(n-1, 5) + binomial(n, 7) + binomial(n, 8) + binomial(n, 9).
This is a polynomial of degree 9 in n.
a(n) = (1 + n) (9! + n (452016 + n (-224244 + n (152108 + n (-17351 + n (-16 + n (394 + (-28 + n) n)))))))/(9!).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=3 of A302998.

Cf. A055417 (case for norm <= 2).

Satisfies a degree nine polynomial (see Example section). - Olivier Gérard, Mar 30 2015

G.f.: -(8*x^8-35*x^7+51*x^6-30*x^5-5*x^4+21*x^3-16*x^2+6*x-1) / (x-1)^10. - Colin Barker, Jul 07 2013

A055419 Number of points in N^n of norm <= 4.

Original entry on oeis.org

1, 5, 17, 54, 165, 482, 1319, 3390, 8262, 19250, 43154, 93601, 197381, 406200, 817954, 1614348, 3126068, 5943728, 11103912, 20396957, 36869790, 65637607, 115178814, 199377387, 340708159, 575150851, 959691775, 1583660542

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Row n=4 of A302998.

Empirical g.f.: (11*x^16 -164*x^15 +936*x^14 -2841*x^13 +5310*x^12 -6697*x^11 +6044*x^10 -3984*x^9 +1734*x^8 -75*x^7 -741*x^6 +827*x^5 -539*x^4 +235*x^3 -68*x^2 +12*x -1) / (x -1)^17. - Colin Barker, Jul 07 2013

A055420 Number of points in N^n of norm <= 5.

Original entry on oeis.org

1, 6, 26, 99, 357, 1203, 3819, 11496, 33044, 91067, 241283, 616618, 1525908, 3670422, 8609234, 19739642, 44323092, 97594520, 210952363, 448015366, 935625977, 1922870258, 3891946458, 7763818914, 15275110988, 29660831057

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 26.

Row n=5 of A302998.

A055421 Number of points in N^n of norm <= 6.

Original entry on oeis.org

1, 7, 35, 163, 688, 2673, 9763, 33792, 111155, 349122, 1053323, 3070003, 8681103, 23885604, 64075075, 167836570, 429817326, 1077457554, 2646942189, 6379963205, 15104418023, 35160248084, 80550746390, 181768697529

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 37.

Row n=6 of A302998.

A055422 Number of points in N^n of norm <= 7.

Original entry on oeis.org

1, 8, 45, 239, 1154, 5139, 21374, 83877, 312913, 1115334, 3813001, 12556041, 39993741, 123663399, 372232239, 1093036739, 3136328324, 8805588398, 24218635622, 65321490327, 172947226965, 449923333530, 1151130040924

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 50.

Row n=7 of A302998.

A055423 Number of points in N^n of norm <= 8.

Original entry on oeis.org

1, 9, 58, 344, 1867, 9389, 43774, 191433, 793049, 3134307, 11887373, 43482354, 154000286, 529571812, 1771709727, 5775999764, 18376959165, 57141981155, 173888869424, 518545311255, 1517106917098, 4359360903570, 12314479965763

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 65.

Row n=8 of A302998.

A055424 Number of points in N^n of norm <= 9.

Original entry on oeis.org

1, 10, 73, 486, 2907, 15930, 81586, 394251, 1807754, 7903501, 33107532, 133503791, 520255432, 1965535471, 7218099342, 25820149509, 90120974981, 307356231936, 1025551245320, 3351842646799, 10742435671415

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 82.

Row n=9 of A302998.

A055425 Number of points in N^n of norm <= 10.

Original entry on oeis.org

1, 11, 90, 648, 4272, 25810, 145138, 766518, 3834490, 18286018, 83570177, 367774928, 1564548863, 6452004367, 25847879345, 100777951798, 383035877989, 1421435174008, 5157779243758, 18324309623403, 63818201253153

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 101.

Row n=10 of A302998.

cp's OEIS Frontend

A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

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A055417 Number of points in N^n of norm <= 2.

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A055418 Number of points in N^n of norm <= 3.

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A055419 Number of points in N^n of norm <= 4.

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A055420 Number of points in N^n of norm <= 5.

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A055421 Number of points in N^n of norm <= 6.

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A055422 Number of points in N^n of norm <= 7.

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A055423 Number of points in N^n of norm <= 8.

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A055424 Number of points in N^n of norm <= 9.

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A055425 Number of points in N^n of norm <= 10.

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