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 A120714 #26 Feb 16 2025 08:33:01
%S A120714 0,14,42,232,974,4522,20180,91422,411782,1858856,8384078,37827386,
%T A120714 170648724,769875718,3473203086,15669055544,70689396502,318908566562,
%U A120714 1438725432052,6490672907694,29282051536966,132103184740456
%N A120714 Expansion of 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)).
%C A120714 Previous name was: Sequence produced by 7 X 7 Markov chain based on adjacency matrix of 7-vertex graph with 10 edges, derived from the Fano plane.
%C A120714 Take the standard 7-vertex 7-edge Fano plane graph and add three edges that go around the triangle vertices from the middle of the sides ( connecting the middle of the sides without going through the center)
%C A120714 Characteristic polynomial is 6 - 2*x - 24*x^2 - 3*x^3 + 26*x^4 + 15*x^5 - x^7.
%H A120714 G. C. Greubel, <a href="/A120714/b120714.txt">Table of n, a(n) for n = 1..1000</a>
%H A120714 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FanoPlane.html">Fano Plane</a>
%H A120714 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,15,26,-3,-24,-2,6).
%F A120714 a(n) = 15*a(n-2) +26*a(n-3) -3*a(n-4) -24*a(n-5) -2*a(n-6) +6*a(n-7).
%F A120714 O.g.f.: 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)). - _R. J. Mathar_, Dec 05 2007
%p A120714 a[1]:=0: a[2]:=14: a[3]:=42: a[4]:=232: a[5]:=974: a[6]:=4522: a[7]:=20180: a[8]:=91422: for n from 9 to 25 do a[n]:=15*a[n-2]+26*a[n-3]-3*a[n-4]-24*a[n-5]-2*a[n-6]+6*a[n-7] end do: seq(a[n], n=1..25);
%t A120714 LinearRecurrence[{0,15,26,-3,-24,-2,6},{0,14,42,232,974,4522,20180},30] (* _Harvey P. Dale_, Sep 20 2011 *)
%o A120714 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)) )); // _G. C. Greubel_, Jul 22 2023
%o A120714 (SageMath)
%o A120714 def A120714_list(prec):
%o A120714 P.<x> = PowerSeriesRing(ZZ, prec)
%o A120714 return P( 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)) ).list()
%o A120714 a=A120714_list(30); a[1:] # _G. C. Greubel_, Jul 22 2023
%Y A120714 Cf. A111384, A120715.
%K A120714 nonn,easy
%O A120714 1,2
%A A120714 _Roger L. Bagula_, Aug 12 2006
%E A120714 Edited by _N. J. A. Sloane_, Jul 14 2007, Jul 28 2007
%E A120714 New name using g.f. from _Joerg Arndt_, Sep 21 2021