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

1,2

References

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

Links

Table of n, a(n) for n=1..39.

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Example

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].

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A253225 A039530 A288408 * A005112 A355727 A062637

Adjacent sequences: A095308 A095309 A095310 * A095312 A095313 A095314

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 02 2004

Extensions

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

Status

approved