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A124054
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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.
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0
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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}.
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EXAMPLE
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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, ...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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