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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
OFFSET
1,2
REFERENCES
Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, pp. 185-194.
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
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