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

0,2

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).

Formula

G.f.: ( -1-2*x-6*x^2-2*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Feb 28 2011

a(n) = 3*(1+2*n+2*n^2)/4 + (-1)^n*(1+2*n)/4. - R. J. Mathar, Feb 28 2011

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

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 *)

Prog

(Haskell)
a186424 n = a186424_list !! n
a186424_list = filter odd a186423_list
(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

Crossrefs

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

Sequence in context: A154501 A045432 A340541 * A018520 A154933 A197225

Adjacent sequences: A186421 A186422 A186423 * A186425 A186426 A186427

Keyword

nonn,easy

Author

Reinhard Zumkeller, Feb 21 2011

Status

approved