A100583 Number of triangles in an n X n grid of squares with diagonals.

0, 8, 44, 124, 268, 492, 816, 1256, 1832, 2560, 3460, 4548, 5844, 7364, 9128, 11152, 13456, 16056, 18972, 22220, 25820, 29788, 34144, 38904, 44088, 49712, 55796, 62356, 69412, 76980, 85080, 93728, 102944, 112744, 123148, 134172, 145836

Offset

0,2

Links

Table of n, a(n) for n=0..36.

Author?, WisFaq (Dutch)

Dave Richeson, Counting triangles on a tin ceiling (solution, take 2) (2011)

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = (12*n^3+18*n^2+4*n+(-1)^n-1)/4. (For a proof see the Richeson link.)

a(n) = 4*Sum{i=1 to n}(i^2 + (n+1-i)*(n+1-round(i/2))).

G.f.: 4*x*(x+2)*(2*x+1) / ((x-1)^4*(x+1)). - Colin Barker, Aug 19 2014

Prog

(PARI) a(n)=3*n^3+9*n^2\2+n \\ Charles R Greathouse IV, Aug 19 2014

Crossrefs

Sequence in context: A290787 A188148 A316466 * A261996 A036464 A000938

Adjacent sequences: A100580 A100581 A100582 * A100584 A100585 A100586

Keyword

nonn,easy

Author

Floor van Lamoen, Nov 30 2004

Status

approved