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%I #60 Feb 05 2023 19:48:32
%S 0,1,22,63,124,205,306,427,568,729,910,1111,1332,1573,1834,2115,2416,
%T 2737,3078,3439,3820,4221,4642,5083,5544,6025,6526,7047,7588,8149,
%U 8730,9331,9952,10593,11254,11935,12636,13357,14098,14859
%N 22-gonal numbers: a(n) = n*(10*n-9).
%C Sequence found by reading the line from 0, in the direction 0, 22,... and the parallel line from 1, in the direction 1, 63,..., in the square spiral whose vertices are the generalized 22-gonal numbers. - _Omar E. Pol_, Jul 18 2012
%C Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 22,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - _Omar E. Pol_, Jul 29 2012
%C This is also a star hendecagonal number: a(n) = A051682(n) + 11*A000217(n-1). - _Luciano Ancora_, Mar 30 2015
%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="/A051874/b051874.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) = 2*a(n-1)-a(n-2)+20 with n>1, a(0)=0, a(1)=1. - _Zerinvary Lajos_, Feb 18 2008
%F a(n) = 20*n+a(n-1)-19 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 06 2010
%F G.f.: x*(1+19*x)/(1-x)^3. - _Bruno Berselli_, Feb 04 2011
%F a(20*a(n)+191*n+1) = a(20*a(n)+191*n) + a(20*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F Product_{n>=2} (1 - 1/a(n)) = 10/11. - _Amiram Eldar_, Jan 22 2021
%F E.g.f.: exp(x)*(x + 10*x^2). - _Nikolaos Pantelidis_, Feb 05 2023
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+20 od: seq(a[n], n=0..39); # _Zerinvary Lajos_, Feb 18 2008
%t Table[n (10 n -9), {n, 0, 40}] (* _Harvey P. Dale_, Sep 19 2011 *)
%t CoefficientList[Series[x (1 + 19 x) / (1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)
%o (PARI) a(n)=n*(10*n-9) \\ _Charles R Greathouse IV_, Jan 24 2014
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 15 1999
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