A104777 - OEIS

%I #35 Dec 19 2020 06:26:40

%S 1,25,49,121,169,289,361,529,625,841,961,1225,1369,1681,1849,2209,

%T 2401,2809,3025,3481,3721,4225,4489,5041,5329,5929,6241,6889,7225,

%U 7921,8281,9025,9409,10201,10609,11449,11881,12769,13225,14161,14641,15625,16129

%N Integer squares congruent to 1 mod 6.

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

%C Odd squares not divisible by 3. - _Reinhard Zumkeller_, Nov 14 2015

%H Reinhard Zumkeller, <a href="/A104777/b104777.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1)

%F A033683(a(n)) = 1.

%F 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

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

%F 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

%F a(n) == 1 (mod 24). - _Joerg Arndt_, Jan 03 2017

%F Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - _Amiram Eldar_, Dec 19 2020

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

%p seq(9*(n-1/2)^2 + 1/4 + (-1)^n * (3*n - 3/2), n = 1 .. 100); # _Robert Israel_, Dec 12 2014

%t 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 *)

%o (PARI) {a(n) = (3*n - 1 - n%2)^2};

%o (Haskell)

%o a104777 = (^ 2) . a007310  -- _Reinhard Zumkeller_, Nov 14 2015

%Y Disjoint union of A016922 and A016970.

%Y Cf. A007310, A001318, A033683, A100044.

%K nonn

%O 1,2

%A _Michael Somos_, Mar 24 2005