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

1,2

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).

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)

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 *)

Prog

(PARI) a(n)=n*(3*n-5)/2+1+n%2*(2*n-1) \\ Charles R Greathouse IV, Dec 08 2011
(Magma) [(3*n*n-5*n+2)/2+(2*n-1)*(n mod 2): n in [1..50]]; // Vincenzo Librandi, Jan 19 2016

Crossrefs

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

Sequence in context: A200183 A118516 A077103 * A107607 A102975 A120300

Adjacent sequences: A036496 A036497 A036498 * A036500 A036501 A036502

Keyword

nonn,easy

Author

Wouter Meeussen

Extensions

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

Edited by Ray Chandler, Dec 09 2011

Status

approved