A152749 a(n) = (n+1)*(3*n+1)/4 for n odd, a(n) = n*(3*n+2)/4 for n even.

0, 2, 4, 10, 14, 24, 30, 44, 52, 70, 80, 102, 114, 140, 154, 184, 200, 234, 252, 290, 310, 352, 374, 420, 444, 494, 520, 574, 602, 660, 690, 752, 784, 850, 884, 954, 990, 1064, 1102, 1180, 1220, 1302, 1344, 1430, 1474, 1564, 1610, 1704, 1752, 1850, 1900, 2002

Offset: 0

Author

Vincenzo Librandi, Dec 31 2009

Keywords

• nonn
• easy

Comments

Interleaving of A049450 and A049451 (for n > 0).

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

The sequence terms occur as exponents in the expansion of Sum_{n >= 0} q^(n*(n+1)/2) * Product_{k >= n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - - .... - Peter Bala, Dec 15 2024

Sequence terms occur as exponents in the expansions of Sum_{n >= 0} q^(n*(2*n+1)) * Product_{k >= 2*n+2} 1 - q^k = Sum_{n >= 0} q^(n*(2*n-1)) * Product_{k >= 2*n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - - .... - Peter Bala, Jun 23 2025

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. A049450 (n*(3*n-1)), A049451 (n*(3*n+1)), A153383 (12n+1 is not prime).

Programs

Haskell

a152749 n = a152749_list !! (n-1)
a152749_list = scanl1 (+) a109043_list
-- Reinhard Zumkeller, Mar 31 2012

Magma

[IsOdd(n) select (n+1)*(3*n+1)/4 else n*(3*n+2)/4: n in [0..52]];

Magma

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..30]]; // Bruno Berselli, Nov 13 2012

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 *)

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) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) = A003154(n+1)/8 - (-1)^n*A005408(n)/8.

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)

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