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 A174114 #44 May 12 2025 05:41:11
%S A174114 1,2,8,11,23,28,46,53,77,86,116,127,163,176,218,233,281,298,352,371,
%T A174114 431,452,518,541,613,638,716,743,827,856,946,977,1073,1106,1208,1243,
%U A174114 1351,1388,1502,1541,1661,1702,1828,1871,2003,2048,2186,2233,2377,2426,2576
%N A174114 Even central polygonal numbers (A193868) divided by 2.
%C A174114 Central terms of A170950, seen as a triangle of rows with an odd number of terms.
%C A174114 Equivalently, numbers of the form m*(4*m+3)+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - _Bruno Berselli_, Jan 05 2016
%C A174114 Conjecure: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n} 1 - q^k) = q + q^2 + q^8 + q^11 + q^23 + q^28 + .... Cf. A266883. - _Peter Bala_, May 10 2025
%H A174114 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A174114 a(n+3) - a(n+2) - a(n+1) + a(n) = A010696(n+1).
%F A174114 a(n) = A170950(A002061(n)).
%F A174114 a(n) = A193868(n)/2. - _Omar E. Pol_, Aug 16 2011
%F A174114 G.f.: -x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - _R. J. Mathar_, Aug 18 2011
%F A174114 E.g.f.: ((2 + x + 2*x^2)*cosh(x) + (1 - x + 2*x^2)*sinh(x) - 2)/2. - _Stefano Spezia_, Nov 16 2024
%F A174114 Sum_{n>=1} 1/a(n) = 4*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - _Amiram Eldar_, May 12 2025
%t A174114 Select[Table[(n (n + 1)/2 + 1)/2, {n, 600}], IntegerQ] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2012 *)
%t A174114 (Select[PolygonalNumber@ Range@ 100, OddQ] + 1 )/2 (* Version 10.4, or *)
%t A174114 Rest@ CoefficientList[Series[-x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* _Michael De Vlieger_, Jun 30 2016 *)
%o A174114 (PARI) a(n)=(2*n-1)*(2*n-1-(-1)^n)\4+1 \\ _Charles R Greathouse IV_, Jun 11 2015
%Y A174114 Cf. A002061, A002522, A010696, A170950, A193868, A266883.
%Y A174114 Cf. A033951: numbers of the form m*(4*m+3)+1 for nonnegative m.
%K A174114 nonn,easy
%O A174114 1,2
%A A174114 _Reinhard Zumkeller_, Mar 08 2010
%E A174114 New name from _Omar E. Pol_, Aug 16 2011