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