A123337 -id:A123337 - cp's OEIS frontend

A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4.

1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 1, 12, 18, 12, 6, 4, 12, 12, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Jonathan Vos Post, Oct 31 2006

Through n = 15, a(n) = number of ordered ways to write n as the sum of 4 squares. For n > 15, we must exclude sums which include 4^2, 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 16, 24, 25, 29, 30, 32, 33, 34, 35 and all n > 36. Without the restriction on the size of squares, all natural numbers can be written as the sum of 4 squares, as Lagrange proved in 1750. This sequence is to 4 as A123337 Number of ordered ways to write n as the sum of 5 squares less than 5, is to 5.

			a(0) = 1 because of the unique sum 0 = 0^2 + 0^2 + 0^2 + 0^2.
a(1) = 4 because of the 4 permutations 1 = 0^2 + 0^2 + 0^2 + 1^2 = 0^2 + 0^2 + 1^2 + 0^2 = 0^2 + 1^2 + 0^2 + 0^2 = 1^2 + 0^2 + 0^2 + 0^2.
a(4) = 5 because of 4 = 1^2 + 1^2 + 1^2 + 1^2 plus the 4 permutations of 4 = 0^2 + 0^2 + 0^2 + 2^2.
a(16) = 1 because 16 = 2^2 + 2^2 + 2^2 + 2^2.

Cf. A000118, A014110, A123337.

a[n_] := Total[ Length /@ Permutations /@ IntegerPartitions[n, {4}, Range[0, 3]^2]]; a /@ Range[0, 72] (* Giovanni Resta, Jun 13 2016 *)

a(n) = Card{(a,b,c,d) such that 0<=a,b,c,d<4 and a^2 + b^2 + c^2 + d^2 = n}.

Corrected typo in third example Dave Zobel (dzobel(AT)alumni.caltech.edu), Mar 07 2009

a(16) and related example corrected by Giovanni Resta, Jun 13 2016

A124054 Array(d,n) = number of ordered ways to write n as the sum of d squares less than d, read by rows, through last nonzero value per row.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 0, 3, 3, 0, 0, 1, 1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 1, 12, 18, 12, 6, 4, 12, 12, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 1, 1, 5, 10, 10, 10, 21, 30, 20, 15, 35, 50, 40, 30, 45, 70, 60, 30, 55, 100, 80, 56

Offset: 1

Jonathan Vos Post, Nov 03 2006

Rows terminate with last nonzero element. Row length of row n = A098547 n^3+n^2+1. Row 4 = A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4. Row 5 = A123337 Number of ordered ways to write n as the sum of 5 squares less than 5. Column 0 = A000012 The simplest sequence of positive numbers: the all 1's sequence. Column 1 = A000027 The natural numbers. Column 2 = A000217(n-2) = Triangular numbers C(n-1,2) = n(n-1)/2. Column 3 = A000292(n-2) Tetrahedral numbers = C(n,3).

			A(1,n) = 1 because the unique ordered way to write 1 as the sum of 0 squares less than 0 is the null set {}.
a(2,n) = 1, 2, 1 = Card{0=0^2+0^2}; Card{1=0^2+1^2,1=1^2+0^2}; Card{2=1^2+1^2}.
a(3,n) = 1, 3, 3, 1, 3, 6, 3, 0, 3, 3, 0, 0, 1.
a(4,n) = 1, 4, 6, 4, 5, 12, 12, 4,  6, 16, 18, ... = A123999.
a(5,n) = 1, 5, 10, 10, 10, 21, 30, 20, 15, 35, ... = A123337.
a(6,n) = 1, 6, 15, 20, 21, 36, 61, 60, 45, 72, ...
a(7,n) = 1, 7, 21, 35, 42, 63, 112, 141, 126, 154, ...
a(8,n) = 1, 8, 28, 56, 78, 112, 196, 288, 309, 344, ...
a(9,n) = 1, 9, 36, 84, 135, 198, 336, 540, 675, 766, ...
a(10,n) = 1, 10, 45, 120, 220, 342, 570, 960, 1350, 1640, ...

Cf. A000012, A000027, A000217, A000292, A098547, A123337, A123999.

cntper[v_] := Length[v]!/Times @@ ((Last /@ Tally[v])!); sqq[d_, n_] := Total[ cntper /@ IntegerPartitions[n, {d}, Range[0, d - 1]^2]]; Flatten[ Table[ sqq[d, #] & /@ Range[0, d (d - 1)^2], {d, 1, 6}]] (* Giovanni Resta, Jun 16 2016 *)

A(d,n) for fixed d = Row d = Card{(c_1,c_2,...,c_d) such that 0<=c_i

Data corrected by Giovanni Resta, Jun 16 2016

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A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4.

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