|
%I #40 Feb 10 2023 09:11:38
%S 1,47,138,274,455,681,952,1268,1629,2035,2486,2982,3523,4109,4740,
%T 5416,6137,6903,7714,8570,9471,10417,11408,12444,13525,14651,15822,
%U 17038,18299,19605,20956,22352,23793,25279,26810,28386,30007,31673,33384
%N 47-gonal numbers.
%D Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, pp. 185-194.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n+3) = 3*a(n+2) - 3*a(n+1) - a(n); a(1) = 1, a(2) = 47, a(3) = 138.
%F 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)].
%F From _Colin Barker_, Jul 27 2013: (Start)
%F a(n) = (n*(45*n-43))/2.
%F G.f.: -x*(44*x+1) / (x-1)^3. (End)
%F E.g.f.: exp(x)*(x + 45*x^2/2). - _Nikolaos Pantelidis_, Feb 10 2023
%e a(6) = 681 = 3*a(5) - 3*a(4) + a(3) = 3*455 - 3*274 + 138.
%e a(37) = 30007 since M^37 * [1 0 0] = [1 37 30007].
%t 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 *)
%t LinearRecurrence[{3, -3, 1}, {1, 47, 138}, 40] (* _Vincenzo Librandi_, Jul 25 2015 *)
%t PolygonalNumber[47,Range[40]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Dec 01 2016 *)
%o (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
%o (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
%Y Cf. A081422, A000326, A000384, A000566, A000567, ... (all polygonal sequences).
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Jun 02 2004
%E Edited by _N. J. A. Sloane_ and _Robert G. Wilson v_, Jun 05 2004
|
