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%I #56 Dec 15 2024 02:27:27
%S 0,1,28,81,160,265,396,553,736,945,1180,1441,1728,2041,2380,2745,3136,
%T 3553,3996,4465,4960,5481,6028,6601,7200,7825,8476,9153,9856,10585,
%U 11340,12121,12928,13761,14620,15505,16416,17353,18316,19305,20320,21361,22428,23521
%N 28-gonal numbers: a(n) = n*(13*n - 12).
%C The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by _Pierre Gayet_]
%C These are also the star 14-gonal numbers: a(n) = A051866(n) + 14*A000217(n-1). _Luciano Ancora_, Apr 04 2015
%H Daniel Mondot, <a href="/A161935/b161935.txt">Table of n, a(n) for n = 0..1000</a>
%H Pierre Gayet, <a href="/A162316/a162316.gif">Note et Compte rendu</a> (gif version).
%H Pierre Gayet, <a href="/A162316/a162316.pdf">Note et Compte Rendu</a> (pdf version).
%H Pierre Gayet, <a href="/A162316/a162316_1.txt">98 séquences générées ... par la formule générale indiquée</a>.
%H Claude Monet, <a href="http://lycees.ac-rouen.fr/bruyeres/jardin/Nymphea.html">Nymphéas</a>.
%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+1) = a(n) + 26*n + 1. - _Vincenzo Librandi_, Nov 30 2010
%F a(n) = A000217(n) + 25*A000217(n-1). - _Luciano Ancora_, Apr 04 2015
%F Product_{n>=2} (1 - 1/a(n)) = 13/14. - _Amiram Eldar_, Jan 22 2021
%F E.g.f.: exp(x)*(x + 13*x^2). - _Nikolaos Pantelidis_, Feb 05 2023
%F From _Elmo R. Oliveira_, Dec 14 2024: (Start)
%F G.f.: x*(1 + 25*x)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)
%e G.f. = x + 28*x^2 + 81*x^3 + 160*x^4 + 265*x^5 + 396*x^6 + 553*x^7 + ...
%t lst={}; Do[a=13*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
%t Table[n*(13*n - 12), {n, 0, 100}] (* _Robert Price_, Oct 11 2018 *)
%o (Magma) [ (n+1)*(13*n+1): n in[0..50] ];
%o (PARI) {a(n) = n*(13*n - 12)}; /* _Michael Somos_, Dec 07 2016 */
%Y Cf. A000217, A051866, A161532, A161549, A161587, A161617, A162316.
%K easy,nonn,changed
%O 0,3
%A _Pierre Gayet_, Jun 22 2009
%E Edited by _N. J. A. Sloane_, Dec 07 2016 at the suggestion of _Daniel Sterman_.
%E Definition simplified by _Omar E. Pol_, Aug 10 2018