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A341441

Total number of triangles visible in a regular (2n+1)-gon with all diagonals drawn.

1, 35, 287, 1302, 4257, 11297, 25935, 53516, 101745, 181279, 306383, 495650, 772785, 1167453, 1716191, 2463384, 3462305, 4776219, 6479551, 8659118, 11415425, 14864025, 19136943, 24384164, 30775185, 38500631, 47773935, 58833082, 71942417, 87394517, 105512127

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

OFFSET

1,2

COMMENTS

For n=1, an equilateral triangle, there is no diagonal, and thus the polygon itself is the only triangle.

LINKS

Table of n, a(n) for n=1..31.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

FORMULA

a(n) = n*(2*n+1)*(2*n-1)*(2*n^3+21*n^2-2*n+9)/90.

G.f.: x*(x^5+20*x^4+7*x^3-63*x^2-28*x-1)/(x-1)^7. - Alois P. Heinz, Feb 11 2021

E.g.f.: exp(x)*x*(90 + 1485*x + 2775*x^2 + 1350*x^3 + 204*x^4 + 8*x^5)/90. - Stefano Spezia, Feb 12 2021