A141666 - OEIS

%I #11 Feb 16 2025 08:33:08

%S 1,4,4,4,16,4,0,16,16,0,4,0,16,0,4,8,16,0,0,16,8,0,32,16,0,16,32,0,0,

%T 0,32,0,0,32,0,0,4,0,0,0,16,0,0,0,4,4,16,0,0,32,32,0,0,16,4,8,16,16,0,

%U 0,64,0,0,16,16,8

%N A symmetrical triangle of coefficients based on A004018 (or number of ways of writing n as a sum of 2 squares): t(n,m) = r2(n-m+1)*r2(m+1).

%C Row sums are {1, 8, 24, 32, 24, 48, 96, 64, 24, 104, 144}.

%D G. E. Andrews, Number Theory, 1971, Dover Publications New York, p. 44, p. 201-207.

%H G. C. Greubel, <a href="/A141666/b141666.txt">Rows n = 0..100 of triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SumofSquaresFunction.html">Sum of Squares Function</a>

%F t(n,m) = r2(n-m+1)*r2(m+1).

%e Triangle begins

%e   {1},

%e   {4,  4},

%e   {4, 16,  4},

%e   {0, 16, 16,  0},

%e   {4,  0, 16,  0,  4},

%e   {8, 16,  0,  0, 16,  8},

%e   {0, 32, 16,  0, 16, 32,  0},

%e   {0,  0, 32,  0,  0, 32,  0,  0},

%e   {4,  0,  0,  0, 16,  0,  0,  0,  4},

%e   {4, 16,  0,  0, 32, 32,  0,  0, 16,  4},

%e   {8, 16, 16,  0,  0, 64,  0,  0, 16, 16,  8}

%t Clear[a]; a = CoefficientList[Series[1 + 4*Sum[(-1)^(1 + n)/(-1 + x^(1 - 2*n)), {n, 100}], {x, 0, 100}], x]; Table[Table[a[[n - m + 1]]*a[[m + 1]], {m, 0, n}], {n, 0, 10}]//Flatten

%Y Cf. A004018.

%K nonn,tabl,changed

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 05 2008