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A186424 Odd terms in A186423.

%I A186424 #50 May 11 2025 01:06:50
%S A186424 1,3,11,17,33,43,67,81,113,131,171,193,241,267,323,353,417,451,523,
%T A186424 561,641,683,771,817,913,963,1067,1121,1233,1291,1411,1473,1601,1667,
%U A186424 1803,1873,2017,2091,2243,2321,2481,2563,2731,2817,2993,3083,3267,3361,3553,3651
%N A186424 Odd terms in A186423.
%C A186424 Sum of odd square and half of even square. - _Vladimir Joseph Stephan Orlovsky_, May 20 2011
%C A186424 Numbers m such that 6*m-2 is a square. - _Bruno Berselli_, Apr 29 2016
%H A186424 Vincenzo Librandi, <a href="/A186424/b186424.txt">Table of n, a(n) for n = 0..1000</a>
%H A186424 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A186424 From _R. J. Mathar_, Feb 28 2011: (Start)
%F A186424 G.f.: ( -1-2*x-6*x^2-2*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ).
%F A186424 a(n) = 3*(1+2*n+2*n^2)/4 + (-1)^n*(1+2*n)/4. (End)
%F A186424 a(n+2) = a(n) + A091999(n+2).
%F A186424 Union of A080859 and A126587: a(2*n) = A080859(n) and a(2*n+1) = A126587(n+1).
%F A186424 From _Peter Bala_, Feb 13 2021: (Start)
%F A186424 Appears to be the sequence of exponents in the following series expansion:
%F A186424 Sum_{n >= 0} (-1)^n * x^n/Product_{k = 1..n} 1 - x^(2*k-1) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - .... Cf. A053253.
%F A186424 More generally, for nonnegative integer N, we appear to have the identity
%F A186424 Product_{j = 1..N} 1/(1 + x^(2*j-1))*( P(N,x) + Sum_{n >= 1} (-1)^n * x^((2*N+1)*n-N)/Product_{k = 1..n} 1 - x^(2*k-1) ) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - ..., where P(N,x) is a polynomial in x of degree N^2 - 1, with the first few values given empirically by
%F A186424 P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 - x^2 + x^3, P(3,x) = 1 - x^2 + x^5 - x^7 + x^8 and P(4,x) =  1 - x^2 - x^4 + x^5 + x^8 - x^9 + x^12 - x^14 + x^15. Cf. A203568. (End)
%F A186424 E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/2. - _Stefano Spezia_, May 08 2021
%F A186424 Sum_{n>=0} 1/a(n) = sqrt(2)*Pi*sinh(sqrt(2)*Pi/3)/(1+2*cosh(sqrt(2)*Pi/3)). - _Amiram Eldar_, May 11 2025
%t A186424 Table[If[OddQ[n],n^2+((n+1)^2)/2,(n^2)/2+(n+1)^2],{n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, May 20 2011 *)
%o A186424 (Haskell)
%o A186424 a186424 n = a186424_list !! n
%o A186424 a186424_list = filter odd a186423_list
%o A186424 (Python)
%o A186424 def A186424(n): return (n*(3*n + 2) + 1 if n&1 else n*(3*n + 4) + 2)>>1 # _Chai Wah Wu_, Jan 31 2023
%Y A186424 Cf. A186421, A186423, A203568, A091999, A080859, A126587, A053253.
%K A186424 nonn,easy
%O A186424 0,2
%A A186424 _Reinhard Zumkeller_, Feb 21 2011