login
Centered 47-gonal numbers.
1

%I #24 Feb 05 2024 09:26:51

%S 1,48,142,283,471,706,988,1317,1693,2116,2586,3103,3667,4278,4936,

%T 5641,6393,7192,8038,8931,9871,10858,11892,12973,14101,15276,16498,

%U 17767,19083,20446,21856,23313,24817,26368,27966,29611,31303,33042,34828,36661,38541

%N Centered 47-gonal numbers.

%H Vincenzo Librandi, <a href="/A129428/b129428.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = (1/2)*(2 + 47*n + 47*n^2).

%F From _Colin Barker_, Jul 27 2013: (Start)

%F a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).

%F G.f.: (1 + 45*x + x^2)/(1-x)^3. (End)

%F E.g.f.: (1/2)*(2 + 94*x + 47*x^2)*exp(x). - _G. C. Greubel_, Feb 05 2024

%t LinearRecurrence[{3, -3, 1}, {1, 48, 142}, 70] (* _Vincenzo Librandi_, Sep 09 2016 *)

%o (Magma) [1+(47/2)*n+(47/2)*n^2: n in [0..50]]; // _Vincenzo Librandi_, Sep 09 2016

%o (PARI) a(n)=47*n*(n+1)/2+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%o (SageMath) [1+47*binomial(n+1,2) for n in range(51)] # _G. C. Greubel_, Feb 05 2024

%Y Cf. A095311, A130566, A130859, A130876.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Oct 06 2007, based on a suggestion from an unknown correspondent in 2004.