A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

0, 0, 2, 7, 17, 33, 57, 90, 134, 190, 260, 345, 447, 567, 707, 868, 1052, 1260, 1494, 1755, 2045, 2365, 2717, 3102, 3522, 3978, 4472, 5005, 5579, 6195, 6855, 7560, 8312, 9112, 9962, 10863, 11817, 12825, 13889, 15010, 16190, 17430, 18732, 20097, 21527

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy
• nice

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

First differences of A082289.

Cf. A000447, A002412, A051895.

Programs

Magma

[Floor((4*n^3+2*n^2-4*n)/16): n in [0..50]]; // Vincenzo Librandi, Aug 29 2011

Mathematica

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 17}, 45] (* Jean-François Alcover, Dec 12 2016 *)
CoefficientList[Series[(2x^2+x^3)/((1-x)^3(1-x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 26 2021 *)

PARI

a(n)=(4*n^3+2*n^2-4*n)\16

Formula

G.f.: (2*x^2+x^3)/((1-x)^3*(1-x^2)). - Michael Somos

a(n) = (1/16)*(2*n*(2*n^2+n-2)+(-1)^n-1). - Bruno Berselli, Aug 29 2011

a(2*n) = A000447(n)+A002412(n); a(2*n+1) = A051895(n). - J. M. Bergot, Apr 12 2018

E.g.f.: (x*(1 + 7*x + 2*x^2)*cosh(x) - (1 - x - 7*x^2 - 2*x^3)*sinh(x))/8. - Stefano Spezia, Aug 22 2023