

A100583


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


1



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
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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)^n1)/4. (For a proof see the Richeson link.)
a(n) = 4*Sum{i=1 to n}(i^2 + (n+1i)*(n+1round(i/2))).
G.f.: 4*x*(x+2)*(2*x+1) / ((x1)^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: A046377 A075816 A188148 * A261996 A036464 A000938
Adjacent sequences: A100580 A100581 A100582 * A100584 A100585 A100586


KEYWORD

nonn,easy


AUTHOR

Floor van Lamoen, Nov 30 2004


STATUS

approved



