This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001539 #69 Oct 23 2024 17:21:15
%S A001539 3,35,99,195,323,483,675,899,1155,1443,1763,2115,2499,2915,3363,3843,
%T A001539 4355,4899,5475,6083,6723,7395,8099,8835,9603,10403,11235,12099,12995,
%U A001539 13923,14883,15875,16899,17955,19043,20163,21315,22499,23715,24963,26243,27555,28899
%N A001539 a(n) = (4*n+1)*(4*n+3).
%C A001539 Sequence arises from reading the line from 3, in the direction 3, 35, ... in the square spiral whose vertices are the squares A000290. - _Omar E. Pol_, May 24 2008
%C A001539 log(sqrt(2)+1)/sqrt(2) = 0.62322524... = 2/3 - 2/35 + 2/99 - 2/195 + 2/323, ... = (1 - 1/3) + (1/7 - 1/5) + (1/9 - 1/11) + (1/15 - 1/13) + (1/17 - 1/19) + (1/23 - 1/21) + ... - _Gary W. Adamson_, Mar 01 2009
%C A001539 Numbers k such that k+1 is a square and k+5 is divisible by 8. - _Bruno Berselli_, Sep 27 2017
%C A001539 The concatenation of 8*A000217(n) and 99 is a term of the sequence. Example: for A000217(5) = 15, 8*15 = 120 and 12099 = a(27). In general, a(5*n+2) = 800*A000217(n) + 99. - _Bruno Berselli_, Sep 29 2017
%H A001539 T. D. Noe, <a href="/A001539/b001539.txt">Table of n, a(n) for n = 0..1000</a>
%H A001539 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A001539 a(n) = A016826(n) - 1 = (A001533(n)+5)/4 = (A001538(n)+16)/9.
%F A001539 Sum_{k>=0} 1/a(k) = Pi/8. - _Benoit Cloitre_, Aug 20 2002
%F A001539 G.f.: (3 + 26*x + 3*x^2)/(1 - x)^3. - _Jaume Oliver Lafont_, Mar 07 2009
%F A001539 a(n) = 32*n + a(n-1) for n > 0, a(0)=3. - _Vincenzo Librandi_, Nov 12 2010
%F A001539 a(n) = a(m) + 16*(n-m)*(n+m+1). The previous formula is obtained for m = n-1. - _Bruno Berselli_, Sep 29 2017
%F A001539 From _Amiram Eldar_, Feb 19 2023: (Start)
%F A001539 a(n) = A016813(n)*A004767(n).
%F A001539 Product_{n>=0} (1 - 1/a(n)) = sqrt(2)*cos(Pi/(2*sqrt(2))).
%F A001539 Product_{n>=0} (1 + 1/a(n)) = sqrt(2). (End)
%F A001539 From _Elmo R. Oliveira_, Oct 23 2024: (Start)
%F A001539 E.g.f.: exp(x)*(3 + 16*x*(2 + x)).
%F A001539 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A001539 CoefficientList[Series[(3 + 26 x + 3 x^2)/(1 - x)^3, {x, 0, 41}], x] (* or *) Table[(4 n + 1) (4 n + 3), {n, 0, 41}] (* _Michael De Vlieger_, Sep 29 2017 *)
%o A001539 (Maxima) makelist((4*n+1)*(4*n+3), n, 0, 30); /* _Martin Ettl_, Nov 12 2012 */
%o A001539 (PARI) a(n)=(4*n+1)*(4*n+3) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A001539 Bisection of A000466.
%Y A001539 Cf. A000217, A000290, A001533, A001538, A004767, A016286, A016813, A016826, A157142, A133766, A154633.
%K A001539 nonn,easy
%O A001539 0,1
%A A001539 _N. J. A. Sloane_