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A010021 a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.

%I A010021 #56 Jul 08 2025 01:27:58
%S A010021 1,34,130,290,514,802,1154,1570,2050,2594,3202,3874,4610,5410,6274,
%T A010021 7202,8194,9250,10370,11554,12802,14114,15490,16930,18434,20002,21634,
%U A010021 23330,25090,26914,28802,30754,32770,34850,36994,39202,41474,43810,46210,48674,51202
%N A010021 a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.
%C A010021 From _Omar E. Pol_, Apr 21 2021: (Start)
%C A010021 Sequence found by reading the line segment from 1 to 34 together with the line from 34, in the direction 34, 130, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979.
%C A010021 The spiral begins as follows:
%C A010021        46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
%C A010021         |                                                       |
%C A010021         |                           0                           |
%C A010021         |                           |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
%C A010021         |                           1                           15
%C A010021         |
%C A010021        51
%C A010021 (End)
%H A010021 Bruno Berselli, <a href="/A010021/b010021.txt">Table of n, a(n) for n = 0..1000</a>
%H A010021 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A010021 G.f.: (1+x)*(1+30*x+x^2)/(1-x)^3. [_Bruno Berselli_, Feb 07 2012]
%F A010021 a(n) = A005893(4n) = A008527(2n); a(n+1) = A108100(2n+2). [_Bruno Berselli_, Feb 07 2012]
%F A010021 E.g.f.: (x*(x+1)*32+2)*e^x-1. - _Gopinath A. R._, Feb 14 2012
%F A010021 a(n) = (4n+1)^2+(4n-1)^2 for n>0. [_Bruno Berselli_, Jun 24 2014]
%F A010021 a(n) = A244082(n) + 2, n >= 1. - _Omar E. Pol_, Apr 21 2021
%F A010021 Sum_{n>=0} 1/a(n) = 3/4 + Pi/16*coth(Pi/4) = 1.04940725316131.. - _R. J. Mathar_, May 07 2024
%F A010021 a(n) = 2*A108211(n). - _R. J. Mathar_, May 07 2024
%F A010021 a(n) = A195315(n)+A195315(n+1). - _R. J. Mathar_, May 07 2024
%t A010021 Join[{1}, 32 Range[40]^2 + 2] (* _Bruno Berselli_, Feb 07 2012 *)
%t A010021 CoefficientList[Series[(1 + x) (1 + 30 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 25 2014 *)
%Y A010021 Cf. A206399, A158563, A158575, A244082.
%Y A010021 Cf. A274979 (generalized 18-gonal numbers).
%K A010021 nonn,easy
%O A010021 0,2
%A A010021 _N. J. A. Sloane_