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A033567 a(n) = (2*n-1)*(4*n-1).

%I A033567 #46 Sep 08 2022 08:44:51
%S A033567 1,3,21,55,105,171,253,351,465,595,741,903,1081,1275,1485,1711,1953,
%T A033567 2211,2485,2775,3081,3403,3741,4095,4465,4851,5253,5671,6105,6555,
%U A033567 7021,7503,8001,8515,9045,9591,10153,10731,11325,11935,12561,13203,13861,14535,15225
%N A033567 a(n) = (2*n-1)*(4*n-1).
%C A033567 a(n+1) = A005563(1), A061037(3), A061039(5), A061041(7), A061043(9), A061045(11), A061047(13), A061049(15). Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, ... spectra of hydrogen. - _Paul Curtz_, Oct 08 2008
%C A033567 Sequence found by reading the segment [1, 3] together with the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - _Omar E. Pol_, Sep 03 2011
%H A033567 G. C. Greubel, <a href="/A033567/b033567.txt">Table of n, a(n) for n = 0..1000</a>
%H A033567 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A033567 a(n) = a(n-1) + 16*n - 14 (with a(0)=1). - _Vincenzo Librandi_, Nov 17 2010
%F A033567 From _G. C. Greubel_, Jul 06 2017: (Start)
%F A033567 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-2).
%F A033567 E.g.f.: (1 + 2*x + 8*x^2)*exp(x).
%F A033567 G.f.: (1 + 15*x^2)/(1 - x)^3. (End)
%F A033567 From _Amiram Eldar_, Jan 03 2022: (Start)
%F A033567 Sum_{n>=0} 1/a(n) = 1 + Pi/4 - log(2)/2.
%F A033567 Sum_{n>=0} (-1)^n/a(n) = 1 + (sqrt(2)-1)*Pi/4 + log(sqrt(2)-1)/sqrt(2). (End)
%t A033567 Table[(2*n - 1)*(4*n - 1), {n, 0, 50}] (* _G. C. Greubel_, Jul 06 2017 *)
%t A033567 LinearRecurrence[{3,-3,1},{1,3,21},50] (* _Harvey P. Dale_, Aug 25 2019 *)
%o A033567 (PARI) vector(60, n, n--; (2*n-1)*(4*n-1)) \\ _Michel Marcus_, Apr 12 2015
%o A033567 (Magma) [(2*n-1)*(4*n-1): n in [0..50]]; // _G. C. Greubel_, Sep 19 2018
%Y A033567 Cf. A045944, A014634.
%K A033567 nonn,easy
%O A033567 0,2
%A A033567 _N. J. A. Sloane_
%E A033567 More terms from _Michel Marcus_, Apr 12 2015