A104777 Integer squares congruent to 1 mod 6.

1, 25, 49, 121, 169, 289, 361, 529, 625, 841, 961, 1225, 1369, 1681, 1849, 2209, 2401, 2809, 3025, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 8281, 9025, 9409, 10201, 10609, 11449, 11881, 12769, 13225, 14161, 14641, 15625, 16129

Offset

1,2

Comments

Exponents of powers of q in expansion of eta(q^24).

Odd squares not divisible by 3. - Reinhard Zumkeller, Nov 14 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Formula

A033683(a(n)) = 1.

G.f.: ( -1-24*x-22*x^2-24*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Feb 20 2011

a(n) = A007310(n)^2 = 1 + 24*A001318(n-1).

a(n) = 9*n^2 - 9*n + 5/2 + (-1)^n * (3*n - 3/2). a(n+4) = 2*a(n+2) - a(n) + 72. - Robert Israel, Dec 12 2014

a(n) == 1 (mod 24). - Joerg Arndt, Jan 03 2017

Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - Amiram Eldar, Dec 19 2020

Example

eta(q^24) = q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + ...

Maple

seq(9*(n-1/2)^2 + 1/4 + (-1)^n * (3*n - 3/2), n = 1 .. 100); # Robert Israel, Dec 12 2014

Mathematica

Select[Range[130]^2, Mod[#, 6]==1&] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {1, 25, 49, 121, 169}, 50] (* Harvey P. Dale, Mar 09 2017 *)

Prog

(PARI) {a(n) = (3*n - 1 - n%2)^2};
(Haskell)
a104777 = (^ 2) . a007310  -- Reinhard Zumkeller, Nov 14 2015

Crossrefs

Disjoint union of A016922 and A016970.

Cf. A007310, A001318, A033683, A100044.

Sequence in context: A348754 A106564 A308177 * A289829 A358060 A131706

Adjacent sequences: A104774 A104775 A104776 * A104778 A104779 A104780

Keyword

nonn

Author

Michael Somos, Mar 24 2005

Status

approved