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

0,3

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).

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

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 *)

Prog

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

Crossrefs

First differences of A082289.

Cf. A000447, A002412, A051895.

Sequence in context: A166381 A083723 A294866 * A321123 A145066 A014148

Adjacent sequences: A045944 A045945 A045946 * A045948 A045949 A045950

Keyword

nonn,easy,nice

Author

R. K. Guy

Status

approved