1,3

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(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}.

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, ... = A123999.

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

a(6,n) = 1, 6, 15, 20, 21, 20, 61, ...

a(7,n) = 1, 7, 21, 35, ...

a(8,n) = 1, 8, 28, 56, ...

a(9,n) = 1, 9, 36, 84, ...

a(10,n) = 1, 10, 45, 120, ...

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

Sequence in context: A114162 A162981 A029264 this_sequence A082870 A026009 A137171

Adjacent sequences: A124051 A124052 A124053 this_sequence A124055 A124056 A124057

easy,nonn,tabl

Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 03 2006