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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 3 09:50 EST 2022. Contains 358517 sequences. (Running on oeis4.)