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%I #82 Jun 09 2021 02:24:25
%S 0,1,14,39,76,125,186,259,344,441,550,671,804,949,1106,1275,1456,1649,
%T 1854,2071,2300,2541,2794,3059,3336,3625,3926,4239,4564,4901,5250,
%U 5611,5984,6369,6766,7175,7596,8029,8474,8931,9400,9881,10374
%N 14-gonal (or tetradecagonal) numbers: a(n) = n*(6*n-5).
%C Sequence found by reading the line from 0, in the direction 0, 14, ... and the parallel line from 1, in the direction 1, 39, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818. Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 14, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Jul 18 2012
%C After 0, partial sums of A017533. - _Bruno Berselli_, Sep 11 2013
%C This is also a star heptagonal number: a(n) = A000566(n) + 7*A000217(n-1). - _Luciano Ancora_, Mar 30 2015
%C Starting with offset 1, this is the binomial transform of (1, 13, 12, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 29 2015
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
%D Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing, 2012, page 6.
%H T. D. Noe, <a href="/A051866/b051866.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 G.f.: x*(1+11*x)/(1-x)^3. - _Bruno Berselli_, Feb 04 2011
%F a(n) = 12*n + a(n-1) - 11, with n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = A033568(n) - 1. - _Omar E. Pol_, Jul 18 2012
%F a(12*a(n)+67*n+1) = a(12*a(n) + 67*n) + a(12*n + 1). - _Vladimir Shevelev_, Jan 24 2014
%F From _Amiram Eldar_, Oct 20 2020: (Start)
%F Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + log(432))/10.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*sqrt(3)*arccoth(sqrt(3)) - log(2))/5. (End)
%F Product_{n>=2} (1 - 1/a(n)) = 6/7. - _Amiram Eldar_, Jan 21 2021
%F E.g.f.: exp(x)*x*(1 + 6*x). - _Stefano Spezia_, Jun 08 2021
%p A051866 := proc(n) n*(6*n-5) ; end proc: seq(A051866(n),n=0..30) ; # _R. J. Mathar_, Feb 05 2011
%t Table[n*(6*n - 5), {n, 0, 100}] (* _Robert Price_, Oct 11 2018 *)
%o (PARI) a(n)=n*(6*n-5); \\ _Joerg Arndt_, Feb 01 2014
%Y Cf. A000217, A000566, A001318, A017533, A033568, A195818.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 15 1999
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