A124054 - OEIS

%I #11 Jun 16 2016 16:59:10

%S 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,

%T 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,

%U 30,20,15,35,50,40,30,45,70,60,30,55,100,80,56

%N 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.

%C 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).

%F A(d,n) for fixed d = Row d = Card{(c_1,c_2,...,c_d) such that 0<=c_i<d and (c_1)^2 + (c_2)^2 + ... + (c_d)^2 = n}.

%e 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 {}.

%e 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}.

%e a(3,n) = 1, 3, 3, 1, 3, 6, 3, 0, 3, 3, 0, 0, 1.

%e a(4,n) = 1, 4, 6, 4, 5, 12, 12, 4,  6, 16, 18, ... = A123999.

%e a(5,n) = 1, 5, 10, 10, 10, 21, 30, 20, 15, 35, ... = A123337.

%e a(6,n) = 1, 6, 15, 20, 21, 36, 61, 60, 45, 72, ...

%e a(7,n) = 1, 7, 21, 35, 42, 63, 112, 141, 126, 154, ...

%e a(8,n) = 1, 8, 28, 56, 78, 112, 196, 288, 309, 344, ...

%e a(9,n) = 1, 9, 36, 84, 135, 198, 336, 540, 675, 766, ...

%e a(10,n) = 1, 10, 45, 120, 220, 342, 570, 960, 1350, 1640, ...

%t 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 *)

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

%K easy,nonn,tabf

%O 1,3

%A _Jonathan Vos Post_, Nov 03 2006

%E Data corrected by _Giovanni Resta_, Jun 16 2016