A141666 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).

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, 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, 0, 64, 0, 0, 16, 16, 8

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 05 2008

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins
  {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,  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,  0, 64,  0,  0, 16, 16,  8}
		

References

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

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Eric Weisstein's World of Mathematics, Sum of Squares Function

Crossrefs

Cf. A004018.

Programs

Mathematica

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

Formula

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