A036498 Numbers of the form m*(6*m-1) and m*(6*m+1), where m is an integer.

0, 5, 7, 22, 26, 51, 57, 92, 100, 145, 155, 210, 222, 287, 301, 376, 392, 477, 495, 590, 610, 715, 737, 852, 876, 1001, 1027, 1162, 1190, 1335, 1365, 1520, 1552, 1717, 1751, 1926, 1962, 2147, 2185, 2380, 2420, 2625, 2667, 2882, 2926, 3151, 3197, 3432, 3480

Offset

1,2

Comments

PartitionQ[ p ] is odd and contains an extra even partition; series term z^p in Product_{n>=1}(1-z^n) has coefficient (+1). - Wouter Meeussen

Numbers k such that the number of partitions of k into distinct parts with an even number of parts exceed by 1 the number of partitions of k into distinct parts with an odd number of parts. [See, e.g., the Freitag-Busam reference given under A036499, p. 410. - Wolfdieter Lang, Jan 18 2016]

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = n(n+1)/6 for n=0 or 5 (mod 6).

a(n) = 1/8*(-1+(-1)^n+2*n)*(-3+(-1)^n+6*n) (see MATHEMATICA code).

G.f.: x^2*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3). - Colin Barker, Apr 02 2012

a(1)=0, a(2)=5, a(3)=7, a(4)=22, a(5)=26, a(n)=a(n-1)+2*a(n-2)- 2*a(n-3)- a(n-4)+a(n-5). - Harvey P. Dale, Aug 13 2012

Bisections: a(2*k+1) = A001318(4*k) = k*(1+6*k) = A049453(k), k >= 0; a(2*k) = A001318(4*k-1) = k*(-1+6*k) = A049452(k), k >= 1. - Wolfdieter Lang, Jan 18 2016

From Amiram Eldar, Feb 13 2024: (Start)

Sum_{n>=2} 1/a(n) = 6 - sqrt(3)*Pi.

Sum_{n>=2} (-1)^n/a(n) = 4*log(2) + 3*log(3) - 6. (End)

Maple

p1 := n->n*(6*n-1): p2 := n->n*(6*n+1): S:={}: for n from 0 to 100 do S := S union {p1(n), p2(n)} od: S

Mathematica

Table[ 1/8*(-1 + (-1)^k + 2*k)*(-3 + (-1)^k + 6*k), {k, 64} ]
CoefficientList[Series[x*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Apr 24 2012 *)
Rest[Flatten[{#(6#-1), #(6#+1)}&/@Range[0, 30]]] (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {0, 5, 7, 22, 26}, 60] (* Harvey P. Dale, Aug 13 2012 *)

Prog

(PARI) \ps 5000; for(n=1, 5000, if(polcoeff(eta(x), n, x)==1, print1(n, ", ")))
(PARI) concat(0, Vec(x^2*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3) + O(x^100))) \\ Altug Alkan, Jan 19 2016
(Magma) [1/8*(-1+(-1)^n+2*n)*(-3+(-1)^n+6*n): n in [1..50]]; // Vincenzo Librandi, Apr 24 2012
(Magma) /* By definition: */ A036498:=func<n | n*(6*n+1)>; [0] cat [A036498(n*m): m in [-1, 1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

Crossrefs

Union of A049452 and A049453.

Cf. A000009, A001318, A010815, A036499.

Sequence in context: A162462 A165144 A084164 * A350193 A248086 A076409

Adjacent sequences: A036495 A036496 A036497 * A036499 A036500 A036501

Keyword

nonn,easy

Author

Wouter Meeussen

Extensions

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

Additional comments and more terms from James A. Sellers, Feb 14 2001

Status

approved