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 A274757 #21 Apr 13 2025 11:48:43
%S A274757 0,9,15,42,54,99,117,180,204,285,315,414,450,567,609,744,792,945,999,
%T A274757 1170,1230,1419,1485,1692,1764,1989,2067,2310,2394,2655,2745,3024,
%U A274757 3120,3417,3519,3834,3942,4275,4389,4740,4860,5229,5355,5742,5874,6279,6417
%N A274757 Numbers k such that 6*k+1 is a triangular number (A000217).
%C A274757 Numbers of the type floor(3*m*(m+1)/4) for which floor(3*m*(m+1)/4) = 3*floor(m*(m+1)/4). A014601 lists the values of m. - _Bruno Berselli_, Jan 13 2017
%C A274757 Numbers of the form 3*k*(4*k + 1) for k in Z. - _Peter Bala_, Nov 21 2024
%H A274757 Colin Barker, <a href="/A274757/b274757.txt">Table of n, a(n) for n = 1..1000</a>
%H A274757 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A274757 G.f.: 3*x^2*(3 + 2*x + 3*x^2) / ((1 - x)^3*(1 + x)^2).
%F A274757 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
%F A274757 a(n) = 3*(2*n - 1)*(2*n + (-1)^n - 1)/4. Therefore:
%F A274757 a(n) = 3*n*(2*n - 1)/2 for n even,
%F A274757 a(n) = 3*(n-1)*(2*n - 1)/2 for n odd.
%t A274757 Table[3 (2 n - 1) (2 n + (-1)^n - 1)/4, {n, 1, 60}] (* _Bruno Berselli_, Jul 08 2016 *)
%t A274757 LinearRecurrence[{1,2,-2,-1,1},{0,9,15,42,54},50] (* _Harvey P. Dale_, Apr 13 2025 *)
%o A274757 (PARI) isok(n) = ispolygonal(6*n+1, 3)
%o A274757 (PARI) select(n->ispolygonal(6*n+1, 3), vector(7000, n, n-1))
%o A274757 (PARI) concat(0, Vec(3*x^2*(3+2*x+3*x^2)/((1-x)^3*(1+x)^2) + O(x^60)))
%Y A274757 Cf. A000217, A014601.
%Y A274757 Cf. A000096 (k+1), A074377 (2*k+1), A045943 (3*k+1), A274681 (4*k+1), A085787 (5*k+1).
%Y A274757 Cf. similar sequences listed in A274830.
%K A274757 nonn,easy
%O A274757 1,2
%A A274757 _Colin Barker_, Jul 04 2016