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A051865 13-gonal (or tridecagonal) numbers: a(n) = n*(11*n - 9)/2.

%I A051865 #71 Feb 06 2023 08:04:12
%S A051865 0,1,13,36,70,115,171,238,316,405,505,616,738,871,1015,1170,1336,1513,
%T A051865 1701,1900,2110,2331,2563,2806,3060,3325,3601,3888,4186,4495,4815,
%U A051865 5146,5488,5841,6205,6580,6966,7363,7771,8190,8620,9061,9513
%N A051865 13-gonal (or tridecagonal) numbers: a(n) = n*(11*n - 9)/2.
%C A051865 Sequence found by reading the line from 0, in the direction 0, 13, ... and the parallel line from 1, in the direction 1, 36, ..., in the square spiral whose vertices are the generalized 13-gonal numbers A195313. - _Omar E. Pol_, Jul 18 2012
%D A051865 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
%D A051865 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%H A051865 T. D. Noe, <a href="/A051865/b051865.txt">Table of n, a(n) for n = 0..1000</a>
%H A051865 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H A051865 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A051865 a(n) = 11*n + a(n-1) - 10 with n > 0, a(0) = 0. - _Vincenzo Librandi_, Aug 06 2010
%F A051865 G.f.: x*(1+10*x)/(1-x)^3. - _Bruno Berselli_, Feb 04 2011
%F A051865 a(11*a(n) + 56*n + 1) = a(11*a(n) + 56*n) + a(11*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F A051865 Product_{n>=2} (1 - 1/a(n)) = 11/13. - _Amiram Eldar_, Jan 21 2021
%F A051865 E.g.f.: exp(x)*(x + 11*x^2/2). - _Nikolaos Pantelidis_, Feb 06 2023
%t A051865 CoefficientList[Series[x (1 + 10 x) / (1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)
%t A051865 LinearRecurrence[{3,-3,1},{0,1,13},50] (* _Harvey P. Dale_, Jul 12 2014 *)
%t A051865 Table[n*(11*n - 9)/2, {n, 0, 100}] (* _Robert Price_, Oct 11 2018 *)
%o A051865 (PARI) a(n)=(11*n^2-9*n)/2 \\ _Charles R Greathouse IV_, May 27 2011
%K A051865 nonn,easy
%O A051865 0,3
%A A051865 _N. J. A. Sloane_, Dec 15 1999