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 A057780 #35 Jan 06 2025 08:59:27
%S A057780 0,3,15,24,48,63,99,120,168,195,255,288,360,399,483,528,624,675,783,
%T A057780 840,960,1023,1155,1224,1368,1443,1599,1680,1848,1935,2115,2208,2400,
%U A057780 2499,2703,2808,3024,3135,3363,3480,3720,3843,4095,4224,4488,4623,4899,5040
%N A057780 Multiples of 3 that are one less than a perfect square.
%C A057780 Also, numbers of the form 9*m^2+6*m, m an integer. - _Jason Kimberley_, Nov 08 2012
%C A057780 k is in this list iff k+1 is in the support of A033684. - _Jason Kimberley_, Nov 13 2012
%C A057780 Exponents in the expansion of Product_{n >= 1} (1 - q^(6*n))^2 * (1 - q ^(9*n)) * (1 - q^(36*n))/((1 - q^(3*n))*(1 - q^(12*n))*(1 - q^(18*n))) = 1 + q^3 + q^15 + q^24 + q^48 + q^63 + q^99 + ... (see Oliver, Theorem 1.1). - _Peter Bala_, Jan 06 2025
%H A057780 Jason Kimberley, <a href="/A057780/b057780.txt">Table of n, a(n) for n = 1..2001</a>
%H A057780 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 A057780 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A057780 a(n) = A001651(n)^2 - 1 = 3 * A001082(n).
%F A057780 G.f.: 3*x^2*(1+4*x+x^2) / ((1-x)^3*(1+x)^2). - _Colin Barker_, Nov 24 2012
%F A057780 From _Colin Barker_, Dec 26 2015: (Start)
%F A057780 a(n) = 3/8*(6*n^2-2*((-1)^n+3)*n+(-1)^n-1).
%F A057780 a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. (End)
%t A057780 Select[3*Range[0,2000],IntegerQ[Sqrt[#+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,3,15,24,48},50] (* _Harvey P. Dale_, Sep 10 2019 *)
%o A057780 (Magma) a:=func<n|9*n^2+6*n>;[0]cat[a(n*m):m in[-1, 1],n in[1..24]]; // _Jason Kimberley_, Nov 09 2012
%o A057780 (PARI) concat(0, Vec(3*x^2*(1+4*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100))) \\ _Colin Barker_, Dec 26 2015
%Y A057780 Numbers of the form 9n^2+kn, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), this sequence (k=6), A218864 (k=8). - _Jason Kimberley_, Nov 08 2012
%K A057780 nonn,easy
%O A057780 1,2
%A A057780 Benjamin Geiger (benjamin_geiger(AT)yahoo.com), Nov 02 2000
%E A057780 Since this is a list, offset corrected to 1 by _Jason Kimberley_, Nov 09 2012