A186424 Odd terms in A186423.

1, 3, 11, 17, 33, 43, 67, 81, 113, 131, 171, 193, 241, 267, 323, 353, 417, 451, 523, 561, 641, 683, 771, 817, 913, 963, 1067, 1121, 1233, 1291, 1411, 1473, 1601, 1667, 1803, 1873, 2017, 2091, 2243, 2321, 2481, 2563, 2731, 2817, 2993, 3083, 3267, 3361, 3553, 3651

Offset: 0

Author

Reinhard Zumkeller, Feb 21 2011

Keywords

• nonn
• easy

Comments

Sum of odd square and half of even square. - Vladimir Joseph Stephan Orlovsky, May 20 2011

Numbers m such that 6*m-2 is a square. - Bruno Berselli, Apr 29 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A186421, A186423, A203568, A091999, A080859, A126587, A053253.

Programs

Haskell

a186424 n = a186424_list !! n
a186424_list = filter odd a186423_list

Mathematica

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 *)

Python

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

Formula

From R. J. Mathar, Feb 28 2011: (Start)

G.f.: ( -1-2*x-6*x^2-2*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ).

a(n) = 3*(1+2*n+2*n^2)/4 + (-1)^n*(1+2*n)/4. (End)

a(n+2) = a(n) + A091999(n+2).

Union of A080859 and A126587: a(2*n) = A080859(n) and a(2*n+1) = A126587(n+1).

From Peter Bala, Feb 13 2021: (Start)

Appears to be the sequence of exponents in the following series expansion:

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.

More generally, for nonnegative integer N, we appear to have the identity

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

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)

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

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