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).
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
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Feb 21 2011
STATUS
approved