A036499 Numbers of the form k*(k+1)/6 for k = 2 or 3 modulo 6.

1, 2, 12, 15, 35, 40, 70, 77, 117, 126, 176, 187, 247, 260, 330, 345, 425, 442, 532, 551, 651, 672, 782, 805, 925, 950, 1080, 1107, 1247, 1276, 1426, 1457, 1617, 1650, 1820, 1855, 2035, 2072, 2262, 2301, 2501, 2542, 2752, 2795, 3015, 3060, 3290, 3337, 3577, 3626

Offset: 1

Author

Wouter Meeussen

Keywords

• nonn
• easy

Comments

Numbers with an odd number of partitions with an extra odd partition; coefficient of z^p in Product_{n >= 1}(1-z^n) has coefficient (-1).

n such that the number of partitions of n into distinct parts with an odd number of parts exceed by 1 the number of partitions of n into distinct parts with an even number of parts. [Euler's 1754/55 pentagonal number theorem, see, e.g., the Freitag-Busam reference (in German). This reference is from Wolfdieter Lang, Jan 18 2016]

In formal power series, A010815=(product(1-x^k),k>0), ranks of coefficients -1. (A001318=ranks of nonzero (1 or -1) in A010815=ranks of odds terms in A000009).

Quasipolynomial of order 2. - Charles R Greathouse IV, Dec 08 2011

Union of A033568 and A033570. - Ray Chandler, Dec 09 2011

References

• Eberhard Freitag and Rolf Busam, Funktionentheorie 1, Springer, Vierte Auflage, 2006, p. 410.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000009, A001318, A010815, A033568, A033570, A036498.

Programs

Magma

[(3*n*n-5*n+2)/2+(2*n-1)*(n mod 2): n in [1..50]]; // Vincenzo Librandi, Jan 19 2016

Maple

seq(seq((6*k+i)*(6*k+i+1)/6,i=2..3),k=0..50); # Robert Israel, Jan 18 2016

Mathematica

Table[ 1/8*(3 + (-1)^k - 6*k)*(1 + (-1)^k - 2*k), {k, 64} ]
LinearRecurrence[{1,2,-2,-1,1},{1,2,12,15,35},50] (* or *)
CoefficientList[Series[(1+x+8x^2+x^3+x^4)/((1-x)^3(1+x)^2),{x,0,100}],x] (* or *)
Table[(2n+1)(3n+{1,2}),{n,0,24}]//Flatten (* Ray Chandler, Dec 09 2011 *)

PARI

a(n)=n*(3*n-5)/2+1+n%2*(2*n-1) \\ Charles R Greathouse IV, Dec 08 2011

Formula

a(n) = (3*n*n-5*n+2)/2 + (2*n-1)*(n mod 2). - Frank Ellermann, Mar 16 2002

G.f.: (1+x+8*x^2+x^3+x^4)/((1-x)^3*(1+x)^2). - Ray Chandler, Dec 09 2011

Bisection: a(2*k+1) = A001318(1+4*k) = (2*k+1)*(3*k+1) = A033570(k), a(2*(k+1)) = A001318(2+4*k) = (2*k+1)*(3*k+2) = A033568(k+1), k >= 0. - Wolfdieter Lang, Jan 18 2016

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. - Wesley Ivan Hurt, Jan 18 2016

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=1} 1/a(n) = Pi/sqrt(3).

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(3) - 4*log(2). (End)

Extensions

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Feb 12 2001

Edited by Ray Chandler, Dec 09 2011