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