OFFSET
0,2
COMMENTS
Also, integer values of k*(k+1)/3. - Charles R Greathouse IV, Dec 11 2010
The nonzero coefficients of the expansion of f(a) = Product_{k>=1} (1-a^(2k)), see A194159, occur at the terms of the sequence given above, i.e., f(a) = 1 - a^2 - a^4 + a^10 + a^14 - a^24 - a^30 + a^44 + a^52 - a^70 - a^80 + ... = Sum_{n>=0} (-1)^binomial(n+1,2)*a^A152749(n). - Johannes W. Meijer, Aug 21 2011
Partial sums of A109043. - Reinhard Zumkeller, Mar 31 2012
Nonnegative k such that 12*k+1 is a square. - Vicente Izquierdo Gomez, Jul 22 2013
Equivalently, numbers of the form h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the fifth comment of A062717). - Bruno Berselli, Feb 02 2017
For n > 0, a(n-1) is the sum of the largest parts of the partitions of 2n into two even parts. - Wesley Ivan Hurt, Dec 19 2017
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
From R. J. Mathar, Jan 03-06 2009: (Start)
G.f.: 2*x*(1+x+x^2)/((1+x)^2*(1-x)^3).
a(n) = 2*A001318(n) = ((6*n^2+6*n+1) - (2*n+1)*(-1)^n)/8. (End)
From Amiram Eldar, Mar 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(log(3)-1). (End)
MAPLE
A152749 := proc(n): if type(n, even) then n*(3*n+2)/4 else (n+1)*(3*n+1)/4 fi: end: seq(A152749(n), n=0..51); # Johannes W. Meijer, Aug 21 2011
MATHEMATICA
Table[If[OddQ[n], (n+1)*(3*n+1)/4, n*(3*n+2)/4], {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 2, 4, 10, 14}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
Select[Range[1, 1000], IntegerQ[Sqrt[12#+1]]&] (* Vicente Izquierdo Gomez, Jul 22 2013 *)
PROG
(Magma) [IsOdd(n) select (n+1)*(3*n+1)/4 else n*(3*n+2)/4: n in [0..52]];
(Magma) f:=func<n | n*(3*n+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..30]]; // Bruno Berselli, Nov 13 2012
(Haskell)
a152749 n = a152749_list !! (n-1)
a152749_list = scanl1 (+) a109043_list
-- Reinhard Zumkeller, Mar 31 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Dec 31 2009
EXTENSIONS
Edited, typo corrected and extended by Klaus Brockhaus, Jan 02 2009
Leading term a(0)=0 added by Johannes W. Meijer, Aug 21 2011
STATUS
approved