This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A104777 #40 Jan 21 2025 09:03:01
%S A104777 1,25,49,121,169,289,361,529,625,841,961,1225,1369,1681,1849,2209,
%T A104777 2401,2809,3025,3481,3721,4225,4489,5041,5329,5929,6241,6889,7225,
%U A104777 7921,8281,9025,9409,10201,10609,11449,11881,12769,13225,14161,14641,15625,16129
%N A104777 Integer squares congruent to 1 mod 6.
%C A104777 Exponents of powers of q in expansion of eta(q^24).
%C A104777 Odd squares not divisible by 3. - _Reinhard Zumkeller_, Nov 14 2015
%C A104777 From _Peter Bala_, Jan 03 2025: (Start)
%C A104777 Exponents of q in the expansion of q*Product_{n >= 1} (1 - q^(24*n))^5/(1 - q^(48*n))^2 = q - 5*q^(5^2) + 7*q^(7^2) - 11*q^(11^2) + 13*q^(13^2) - 17*q^(17^2) + 19*q^(19)^2 - + ... (a consequence of the quintuple product identity).
%C A104777 Also, exponents in the expansion of q*Product_{n >= 1} (1 - q^(48*n))^13 / ( (1 - q^(24*n))*(1 - q^(96*n)) )^5 = q + 5*q^(5^2) + 7*q^(7^2) + 11*q^(11^2) - 13*q^(13^2) - 17*q^(17^2) - 19*q^(19^2) - 23*q^(23^2) + + + + - - - - ... (see Oliver, Theorem 1.1). (End)
%H A104777 Reinhard Zumkeller, <a href="/A104777/b104777.txt">Table of n, a(n) for n = 1..10000</a>
%H A104777 Robert J. Lemke Oliver, <a href="https://doi.org/10.1016/j.aim.2013.03.019">Eta quotients and theta functions</a>, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
%H A104777 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1)
%F A104777 A033683(a(n)) = 1.
%F A104777 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 A104777 a(n) = A007310(n)^2 = 1 + 24*A001318(n-1).
%F A104777 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 A104777 a(n) == 1 (mod 24). - _Joerg Arndt_, Jan 03 2017
%F A104777 Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - _Amiram Eldar_, Dec 19 2020
%e A104777 eta(q^24) = q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + ...
%p A104777 seq(9*(n-1/2)^2 + 1/4 + (-1)^n * (3*n - 3/2), n = 1 .. 100); # _Robert Israel_, Dec 12 2014
%t A104777 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 A104777 (PARI) {a(n) = (3*n - 1 - n%2)^2};
%o A104777 (Haskell)
%o A104777 a104777 = (^ 2) . a007310 -- _Reinhard Zumkeller_, Nov 14 2015
%Y A104777 Disjoint union of A016922 and A016970.
%Y A104777 Cf. A007310, A001318, A033683, A100044.
%K A104777 nonn,easy
%O A104777 1,2
%A A104777 _Michael Somos_, Mar 24 2005