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.

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

Author

Jonathan Vos Post, Nov 03 2006

Keywords

• easy
• nonn
• tabf

Comments

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).

Examples

			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, ...
		

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

Extensions

Data corrected by Giovanni Resta, Jun 16 2016

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