A130748 Place n points on each of the three sides of a triangle, 3n points in all; a(n) = number of nondegenerate triangles that can be constructed using these points (plus the 3 original vertices) as vertices.

17, 72, 190, 395, 711, 1162, 1772, 2565, 3565, 4796, 6282, 8047, 10115, 12510, 15256, 18377, 21897, 25840, 30230, 35091, 40447, 46322, 52740, 59725, 67301, 75492, 84322, 93815, 103995, 114886, 126512, 138897, 152065, 166040, 180846, 196507, 213047, 230490

Offset

1,1

Comments

Define b(1)=1 and b(n)=a(n-1) for n>1. Then (b(n)) is the principal diagonal of the convolution array A213833. - Clark Kimberling, Jul 04 2012

Links

Table of n, a(n) for n=1..38.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = C(3*(n+1), 3) - 3*C(n+2, 3) where n>0.

a(n) = (n+1)*A002414(n+1) - n*A002414(n). - Bruno Berselli, Dec 11 2012

a(n) = (8n^3 + 15n^2 + 9n + 2)/2 \\ Charles R Greathouse IV, Feb 14 2013

Example

5 points are put on each side of a triangle (n = 5); we then have 18 vertices to construct with: 5 * 3 + 3 originals. The number of total arrangements = combi(18,3) : combi[3(n+1),3]. But these include degenerates along the 3 sides: 7 points on each side, so combi(7,3) on each side : 3 * combi[n+2, 3] combi[18,3] - 3 * combi[7,3] = 816 - 105 = 711.

Maple

A130748:=n->(8*n^3 + 15*n^2 + 9*n + 2)/2; seq(A130748(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Mathematica

Table[(8 n^3 + 15 n^2 + 9 n + 2)/2, {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

Prog

(PARI) a(n)=4*n^3+n*(15*n+9)/2+1 \\ Charles R Greathouse IV, Feb 14 2013

Crossrefs

Cf. A002414, A213833, A220084 (for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).

Cf. A260260 (comment). [Bruno Berselli, Jul 22 2015]

Sequence in context: A214530 A087514 A119625 * A131692 A157864 A229496

Adjacent sequences: A130745 A130746 A130747 * A130749 A130750 A130751

Keyword

nonn,easy

Author

Denis Borris, Jul 12 2007

Extensions

More terms from Wesley Ivan Hurt, Jan 28 2014

Status

approved