%I #35 Jul 12 2024 18:45:32

%S 0,1,26,75,148,245,366,511,680,873,1090,1331,1596,1885,2198,2535,2896,

%T 3281,3690,4123,4580,5061,5566,6095,6648,7225,7826,8451,9100,9773,

%U 10470,11191,11936,12705,13498,14315,15156,16021,16910,17823,18760

%N 26-gonal numbers: a(n) = n*(12*n-11).

%C See comments in A255184.

%C Also star 13-gonal number: a(n) = A051865(n) + 13*A000217(n-1).

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (24th row of the table).

%H Luciano Ancora, <a href="/A255185/b255185.txt">Table of n, a(n) for n = 0..1000</a>

%H Luciano Ancora, <a href="/A256645/a256645_1.pdf">Polygonal and Pyramidal numbers</a>, Section 1.

%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 G.f.: x*(1 + 23*x)/(1 - x)^3.

%F a(n) = A000217(n) + 23*A000217(n-1).

%F Product_{n>=2} (1 - 1/a(n)) = 12/13. - _Amiram Eldar_, Jan 22 2021

%F E.g.f.: exp(x)*(x + 12*x^2). - _Nikolaos Pantelidis_, Feb 05 2023

%t Table[n (12 n - 11), {n, 50}]

%t PolygonalNumber[26,Range[0,50]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{0,1,26},50] (* _Harvey P. Dale_, Feb 02 2017 *)

%o (PARI) a(n)=n*(12*n-11) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Magma) [n*(12*n-11): n in [0..50]]; // _G. C. Greubel_, Jul 12 2024

%o (SageMath) [n*(12*n-11) for n in range(51)] # _G. C. Greubel_, Jul 12 2024

%Y Cf. similar sequences listed in A255184.

%K nonn,easy

%O 0,3

%A _Luciano Ancora_, Apr 04 2015