A095311 47-gonal numbers.

1, 47, 138, 274, 455, 681, 952, 1268, 1629, 2035, 2486, 2982, 3523, 4109, 4740, 5416, 6137, 6903, 7714, 8570, 9471, 10417, 11408, 12444, 13525, 14651, 15822, 17038, 18299, 19605, 20956, 22352, 23793, 25279, 26810, 28386, 30007, 31673, 33384

Offset: 1

Author

Gary W. Adamson, Jun 02 2004

Keywords

• nonn
• easy

Examples

			a(6) = 681 = 3*a(5) - 3*a(4) + a(3) = 3*455 - 3*274 + 138.
a(37) = 30007 since M^37 * [1 0 0] = [1 37 30007].
		

References

• Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, pp. 185-194.

Links

• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A081422, A000326, A000384, A000566, A000567, ... (all polygonal sequences).

Programs

Magma

I:=[1,47,138]; [n le 3 select I[n]  else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015

Mathematica

a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {1, 45, 1}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 40}] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{3, -3, 1}, {1, 47, 138}, 40] (* Vincenzo Librandi, Jul 25 2015 *)
PolygonalNumber[47,Range[40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2016 *)

PARI

a(n)=([0,1,0; 0,0,1; 1,-3,3]^(n-1)*[1;47;138])[1,1] \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n+3) = 3*a(n+2) - 3*a(n+1) - a(n); a(1) = 1, a(2) = 47, a(3) = 138.

Let M = the 3 X 3 matrix [1 0 0 / 1 1 0 / 1 45 1]. Then M^n * [1 0 0] = [1 n a(n)].

From Colin Barker, Jul 27 2013: (Start)

a(n) = (n*(45*n-43))/2.

G.f.: -x*(44*x+1) / (x-1)^3. (End)

E.g.f.: exp(x)*(x + 45*x^2/2). - Nikolaos Pantelidis, Feb 10 2023

Extensions

Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 05 2004