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A185212 a(n) = 12*n^2 - 8*n + 1.

%I A185212 #31 May 28 2022 08:06:47
%S A185212 1,5,33,85,161,261,385,533,705,901,1121,1365,1633,1925,2241,2581,2945,
%T A185212 3333,3745,4181,4641,5125,5633,6165,6721,7301,7905,8533,9185,9861,
%U A185212 10561,11285,12033,12805,13601,14421,15265,16133,17025,17941,18881,19845,20833
%N A185212 a(n) = 12*n^2 - 8*n + 1.
%C A185212 Sequence found by reading the line from 1, in the direction 1, 5, and the same line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - _Omar E. Pol_, May 08 2018
%H A185212 Harvey P. Dale, <a href="/A185212/b185212.txt">Table of n, a(n) for n = 0..1000</a>
%H A185212 Leo Tavares, <a href="/A185212/a185212.jpg">Illustration: Square Block Rays</a>
%H A185212 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A185212 a(n) = 4*A000567(n) + 1.
%F A185212 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=1, a(1)=5, a(2)=33. - _Harvey P. Dale_, Jul 07 2015
%F A185212 G.f.: (-1 - 2*x - 21*x^2)/(-1+x)^3. - _Harvey P. Dale_, Jul 07 2015
%F A185212 E.g.f.: (12*x^2 + 4*x + 1)*exp(x). - _G. C. Greubel_, Jun 25 2017
%F A185212 a(n) = A016754(n-1) + 4*A000384(n). - _Leo Tavares_, May 21 2022
%F A185212 From _Amiram Eldar_, May 28 2022: (Start)
%F A185212 Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/8 - 3*log(3)/8 + 1.
%F A185212 Sum_{n>=0} (-1)^n/a(n) = Pi/8 - sqrt(3)*arccoth(sqrt(3))/2 + 1. (End)
%t A185212 Table[12n^2-8n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,5,33},50] (* _Harvey P. Dale_, Jul 07 2015 *)
%o A185212 (Haskell)
%o A185212 a185212 = (+ 1) . (* 4) . a000567
%o A185212 (PARI) a(n)=12*n^2-8*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A185212 For n > 0: odd terms in A001859.
%Y A185212 Cf. A001082.
%Y A185212 Cf. A016754, A000384.
%K A185212 nonn,easy
%O A185212 0,2
%A A185212 _Reinhard Zumkeller_, Dec 20 2012