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 A114697 #31 May 24 2021 07:35:26
%S A114697 1,3,9,22,55,133,323,780,1885,4551,10989,26530,64051,154633,373319,
%T A114697 901272,2175865,5253003,12681873,30616750,73915375,178447501,
%U A114697 430810379,1040068260,2510946901,6061962063,14634871029,35331704122,85298279275,205928262673
%N A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
%C A114697 Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').
%H A114697 Colin Barker, <a href="/A114697/b114697.txt">Table of n, a(n) for n = 0..1000</a>
%H A114697 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F A114697 a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).
%F A114697 G.f.: (1+x+x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
%F A114697 From _Raphie Frank_, Oct 01 2012: (Start)
%F A114697 a(2*n) = A216134(2*n+1).
%F A114697 a(2*n+1) = A006452(2*n+3)-1.
%F A114697 Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)
%F A114697 a(n) = (2*A078343(n+2) - A010694(n))/4. - _R. J. Mathar_, Oct 02 2012
%F A114697 From _Colin Barker_, May 26 2016: (Start)
%F A114697 a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.
%F A114697 a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)
%F A114697 a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - _G. C. Greubel_, May 24 2021
%t A114697 Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n,0,30}] (* _G. C. Greubel_, May 24 2021 *)
%o A114697 (PARI) Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ _Colin Barker_, Jun 24 2015
%o A114697 (Sage) [(4*lucas_number1(n+2,2,-1) -2*lucas_number1(n+1,2,-1) -3 +(-1)^n)/4 for n in (0..30)] # _G. C. Greubel_, May 24 2021
%Y A114697 Cf. A000129, A002203, A005409, A100828, A111954, A113224.
%Y A114697 Cf. A114647, A114688, A114689, A114695, A116699.
%K A114697 easy,nonn
%O A114697 0,2
%A A114697 _Creighton Dement_, Feb 18 2006