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A195315 Centered 32-gonal numbers.

%I A195315 #38 Nov 15 2024 09:05:59
%S A195315 1,33,97,193,321,481,673,897,1153,1441,1761,2113,2497,2913,3361,3841,
%T A195315 4353,4897,5473,6081,6721,7393,8097,8833,9601,10401,11233,12097,12993,
%U A195315 13921,14881,15873,16897,17953,19041,20161,21313,22497,23713,24961,26241,27553,28897,30273
%N A195315 Centered 32-gonal numbers.
%C A195315 Sequence found by reading the line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Semi-axis opposite to A016802 in the same spiral.
%H A195315 Vincenzo Librandi, <a href="/A195315/b195315.txt">Table of n, a(n) for n = 1..10000</a>
%H A195315 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A195315 a(n) = 16*n^2 - 16*n + 1.
%F A195315 G.f.: -x*(1 + 30*x + x^2)/(x-1)^3. - _R. J. Mathar_, Sep 18 2011
%F A195315 Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3)*Pi/4)/(8*sqrt(3)). - _Amiram Eldar_, Feb 11 2022
%F A195315 From _Elmo R. Oliveira_, Nov 14 2024: (Start)
%F A195315 E.g.f.: exp(x)*(16*x^2 + 1) - 1.
%F A195315 a(n) = 2*A069129(n) - 1.
%F A195315 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%t A195315 Table[16*n^2 - 16*n + 1, {n, 1, 41}] (* _Amiram Eldar_, Feb 11 2022 *)
%t A195315 LinearRecurrence[{3,-3,1},{1,33,97},50] (* _Harvey P. Dale_, Feb 11 2024 *)
%o A195315 (Magma) [(16*n^2-16*n+1): n in [1..50]]; // _Vincenzo Librandi_, Sep 19 2011
%o A195315 (PARI) a(n)=16*n^2-16*n+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A195315 Bisection of A195146.
%Y A195315 Cf. A003154, A069129, A069133, A069190, A195314, A195316, A195317, A195318.
%Y A195315 Cf. A016802, A074377.
%K A195315 nonn,easy
%O A195315 1,2
%A A195315 _Omar E. Pol_, Sep 16 2011