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%I #60 Mar 19 2023 15:32:42
%S 0,1,21,60,118,195,291,406,540,693,865,1056,1266,1495,1743,2010,2296,
%T 2601,2925,3268,3630,4011,4411,4830,5268,5725,6201,6696,7210,7743,
%U 8295,8866,9456,10065,10693,11340,12006,12691,13395,14118
%N 21-gonal numbers: a(n) = n*(19n - 17)/2.
%C Sequence found by reading the line from 0, in the direction 0, 21, ... and the parallel line from 1, in the direction 1, 60, ..., in the square spiral whose vertices are the generalized 21-gonal numbers. - _Omar E. Pol_, Jul 18 2012
%C Partial sums of A215144. - _Leo Tavares_, Mar 17 2023
%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 Vincenzo Librandi, <a href="/A051873/b051873.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+18*x)/(1-x)^3. - _Bruno Berselli_, Feb 04 2011
%F a(n) = 19*n+a(n-1)-18 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = A226490(n) - n. - _Bruno Berselli_, Jun 11 2013
%F a(19*a(n)+172*n+1) = a(19*a(n)+172*n) + a(19*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F Product_{n>=2} (1 - 1/a(n)) = 19/21. - _Amiram Eldar_, Jan 22 2021
%F E.g.f.: exp(x)*(x + 19*x^2/2). - _Nikolaos Pantelidis_, Feb 06 2023
%p A051873 := proc(n) n*(19*n-17)/2 ;end proc: seq(A051873(n),n=0..30) ; # _R. J. Mathar_, Feb 05 2011
%t PolygonalNumber[21,Range[0,40]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 22 2016 *)
%t Table[n*(19*n - 17)/2, {n, 0, 100}] (* _Robert Price_, Oct 11 2018 *)
%o (PARI) n*(19*n-17)/2 \\ _Charles R Greathouse IV_, Jan 24 2014
%Y Cf. A215144, A226490.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 15 1999
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