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A095311 47-gonal numbers. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

LINKS

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

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)

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, etc. (all polygonal sequences).

Sequence in context: A253225 A039530 A288408 * A005112 A062637 A212374

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

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Last modified November 17 21:02 EST 2018. Contains 317278 sequences. (Running on oeis4.)