A255212 -id:A255212 - cp's OEIS frontend

A016727 Number of inequivalent solutions to x^2+y^2+z^2 = n^2.

1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 3, 2, 5, 1, 4, 4, 4, 2, 7, 3, 4, 2, 5, 3, 9, 2, 5, 5, 5, 1, 11, 4, 7, 4, 6, 4, 10, 2, 7, 7, 7, 3, 13, 4, 7, 2, 9, 5, 14, 3, 8, 9, 10, 2, 16, 5, 9, 5, 9, 5, 21, 1, 11, 11, 10, 4, 17, 7, 10, 4, 11, 6, 18, 4, 16, 10, 11, 2, 23, 7, 12, 7, 14, 7, 20, 3

Offset: 0

csvcjld(AT)nomvst.lsumc.edu

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
C. D. Olds, On the representations N_3(n^2), Bull. Am. Math. Soc. 47 (6) (1941) 499-503

Cf. A065458.

Column k=3 of A255212.

Table[Length[PowersRepresentations[n^2, 3, 2]], {n, 0, 100}]

a(n) = A000164(n^2). - R. J. Mathar, Feb 12 2017

A063014 Number of solutions to n^2 = b^2 + c^2 (with c >= b >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 5, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2

Offset: 0

Henry Bottomley, Jul 26 2001

nonn

			a(0)=1 since 0^2 = 0^2 + 0^2;
a(5)=2 since 5^2 = 0^2 + 5^2 = 3^2 + 4^2;
a(25)=3 since 25^2 = 0^2 + 25^2 = 7^2 + 24^2 = 15^2 + 20^2.

Felix Huber, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000161, A046080.

Column k=2 of A255212.

with(NumberTheory):
A063014 := n -> nops(SumOfSquares(n^2));
seq(A063014(n), n = 0 .. 100); # Felix Huber, Jun 01 2024

a(0) = 1; a(n) = A046080(n) + 1 for n > 0. [amended by Georg Fischer, Jan 25 2020]

a(n) = A000161(n^2). - Christian Krause, Dec 08 2022

A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 4, 2, 0, 1, 8, 6, 4, 2, 0, 1, 10, 24, 30, 4, 2, 0, 1, 12, 90, 104, 6, 12, 2, 0, 1, 14, 252, 250, 24, 30, 4, 2, 0, 1, 16, 574, 876, 730, 248, 30, 4, 2, 0, 1, 18, 1136, 3542, 4092, 1210, 312, 54, 4, 2, 0, 1, 20, 2034, 12112, 18494, 7812, 2250, 456, 6, 4, 2, 0

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  0,  2,   4,   6,    8,    10,  ...
  0,  2,   4,   6,   24,    90,  ...
  0,  2,   4,  30,  104,   250,  ...
  0,  2,   4,   6,   24,   730,  ...
  0,  2,  12,  30,  248,  1210,  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.
Main diagonal gives A232173.
Cf. A000122, A122141, A255212, A286815.

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
A:= (n, k)-> b(n^2, k):
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 10 2023

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.

A105152 Number of sum of squares representations of n^2 in n dimensions disregarding order and sign.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 9, 13, 25, 52, 94, 195, 354, 691, 1341, 2514, 4746, 8984, 16639, 31031, 57421, 105091, 192266, 349765, 632223, 1138114, 2043082, 3642712, 6474196, 11462149, 20189285, 35427948, 61987644, 107928280, 187336851, 324080969, 558451251, 959214488

Offset: 0

Wouter Meeussen, Apr 10 2005

nonn

			a(5)=4 since we can write 5^2 as 0^2+0^2+0^2+0^2+5^2, 0^2+0^2+0^2+3^2+4^2, 0^2+1^2+2^2+2^2+4^2, 2^2+2^2+2^2+2^2+3^2.

David A. Corneth, Table of n, a(n) for n = 0..500 (first 201 terms from Alois P. Heinz)

Main diagonal of A255212.

<< NumberTheory`NumberTheoryFunctions`; Table[Length@SumOfSquaresRepresentations[k, k^2], {k, 16}]

a(18)-a(20) from Robert G. Wilson v, Apr 12 2005

a(0), a(21)-a(37) from Alois P. Heinz, Feb 16 2015

A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 2, 6, 7, 6, 4, 8, 10, 14, 2, 11, 14, 13, 7, 23, 15, 17, 4, 24, 21, 31, 10, 25, 37, 28, 2, 46, 29, 49, 14, 38, 35, 61, 7, 45, 62, 49, 15, 93, 47, 57, 4, 72, 67, 97, 21, 71, 84, 101, 10, 119, 70, 86, 37, 92, 79, 165, 2, 138, 127, 109, 29, 168, 140, 121, 14

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=3 because 25 produces {0,0,0,5}, {0,0,3,4}, {1,2,2,4}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A063014, A016727, A065459.

Column k=4 of A255212.

N:= 100:
R:= Vector(N,1):
for a from 0 to N do
  for b from a to floor(sqrt(N^2-a^2)) do
     for c from b to floor(sqrt(N^2-a^2-b^2)) do
       q:= a^2 + b^2 + c^2;
       for f in numtheory:-divisors(q) do
          if f^2 + 2*f*c <= q and (f + q/f)::even then
             r:= (f + q/f)/2;
             if r <= N then R[r]:= R[r]+1 fi;
          fi
od od od od:
convert(R,list); # Robert Israel, Feb 16 2015

Length/@Table[SumOfSquaresRepresentations[4, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065459 Number of inequivalent (ordered) solutions to n^2 = sum of 5 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 13, 12, 13, 17, 25, 22, 27, 31, 35, 38, 46, 49, 61, 61, 61, 73, 92, 83, 112, 106, 118, 127, 147, 138, 185, 175, 178, 198, 239, 212, 254, 262, 298, 294, 341, 304, 404, 376, 385, 432, 483, 441, 539, 517, 560, 551, 680, 587, 745, 693, 698

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=4 because 25 produces {0,0,0,0,5}, {0,0,0,3,4}, {0,1,2,2,4}, {2,2,2,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..750

Cf. A063014, A016727.

Column k=5 of A255212.

Length/@Table[SumOfSquaresRepresentations[5, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065461 Number of inequivalent (ordered) solutions to n^2 = sum of 7 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 13, 16, 27, 36, 43, 58, 72, 99, 130, 146, 178, 254, 265, 342, 417, 507, 540, 726, 745, 975, 1092, 1289, 1338, 1845, 1751, 2246, 2447, 2948, 2852, 3932, 3638, 4728, 4868, 5778, 5618, 7659, 6887, 8891, 8887, 10825, 10109, 13712

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(4)=4 because 16 produces {0,0,0,0,0,0,4}, {0,0,0,2,2,2,2}, {0,0,1,1,1,2,3}, {1,1,1,1,2,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A063014, A016727.

Column k=7 of A255212.

Table[Length[PowersRepresentations[n^2,7,2]],{n,0,50}] (* Harvey P. Dale, Sep 08 2020 *)

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

Previous Mathematica program replaced by Harvey P. Dale, Sep 08 2020

A065460 Number of inequivalent (ordered) solutions to n^2 = sum of 6 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 9, 9, 9, 18, 23, 24, 29, 37, 53, 62, 59, 77, 116, 106, 130, 156, 199, 192, 221, 257, 342, 336, 384, 402, 577, 501, 599, 639, 835, 774, 910, 912, 1220, 1113, 1378, 1298, 1703, 1530, 1907, 1862, 2398, 2094, 2471, 2393, 3356, 2765, 3543

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=5 because 25 produces {0,0,0,0,0,5}, {0,0,0,0,3,4}, {0,0,1,2,2,4}, {0,2,2,2,2,3}, {1,1,1,2,3,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A063014, A016727.

Column k=6 of A255212.

Length/@Table[SumOfSquaresRepresentations[6, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065462 Number of inequivalent (ordered) solutions to n^2 = sum of 8 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 18, 25, 36, 51, 73, 90, 133, 169, 223, 295, 380, 452, 603, 763, 903, 1115, 1385, 1668, 2025, 2398, 2811, 3535, 4011, 4683, 5503, 6724, 7316, 8684, 9946, 11844, 12994, 15091, 16712, 20493, 21663, 24975, 27536, 33079, 34654, 39957, 43315

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(4)=5 because 16 produces {0,0,0,0,0,0,0,4}, {0,0,0,0,2,2,2,2}, {0,0,0,1,1,1,2,3}, {0,1,1,1,1,2,2,2}, {1,1,1,1,1,1,1,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A016727, A063014.

Column k=8 of A255212.

Length/@Table[SumOfSquaresRepresentations[8, (k)^2], {k, 36}]

a(0), a(37)-a(47) from Alois P. Heinz, Feb 16 2015

A255213 Number of partitions of n^2 into at most 9 square parts.

Original entry on oeis.org

1, 1, 2, 4, 5, 9, 15, 21, 30, 52, 70, 100, 142, 196, 264, 378, 477, 640, 841, 1082, 1389, 1783, 2203, 2786, 3460, 4290, 5209, 6410, 7810, 9397, 11190, 13501, 16122, 18936, 22374, 26491, 30871, 36211, 41932, 48389, 56703, 65192, 73967, 85947, 98687, 110647

Offset: 0

Alois P. Heinz, Feb 17 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=9 of A255212.

cp's OEIS Frontend

A016727 Number of inequivalent solutions to x^2+y^2+z^2 = n^2.

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A063014 Number of solutions to n^2 = b^2 + c^2 (with c >= b >= 0).

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A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

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A105152 Number of sum of squares representations of n^2 in n dimensions disregarding order and sign.

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A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

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A065459 Number of inequivalent (ordered) solutions to n^2 = sum of 5 squares of integers >= 0.

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A065461 Number of inequivalent (ordered) solutions to n^2 = sum of 7 squares of integers >= 0.

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A065460 Number of inequivalent (ordered) solutions to n^2 = sum of 6 squares of integers >= 0.

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