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 A226956 #44 Sep 08 2022 08:46:05
%S A226956 1,1,2,2,3,5,9,15,24,38,61,99,161,261,422,682,1103,1785,2889,4675,
%T A226956 7564,12238,19801,32039,51841,83881,135722,219602,355323,574925,
%U A226956 930249,1505175,2435424,3940598,6376021,10316619,16692641,27009261,43701902,70711162,114413063,185124225,299537289
%N A226956 a(0)=a(1)=1, a(n+2) = a(n+1) + a(n) - A128834(n).
%C A226956 a(n) and differences:
%C A226956 1, 1, 2, 2, 3, 5, 9, 15, 24, 38, ... a(n)
%C A226956 0, 1, 0, 1, 2, 4, 6, 9, 14, 23, 38, ... b(n)
%C A226956 1, -1, 1, 1, 2, 2, 3, 5, 9, 15, 24, 38, ... a(n-2)
%C A226956 -2, 2, 0, 1, 0, 1, 2, 4, 6, 9, 14, 23, 38, ... b(n-2)
%C A226956 4, -2, 1,-1, 1, 1, 2, 2, 3, 5, 9, ... a(n-4)
%C A226956 -6, 3,-2, 2, 0, 1, 0, 1, 2, 4, 6, 9, ... b(n-4)
%C A226956 9, -5, 4,-2, 1,-1, 1, 1, 2, 2, 3, 5, 9, ... a(n-6)
%C A226956 -14, 9,-6, 3,-2, 2, 0, 1 0, 1, 2, ... b(n-6)
%C A226956 23,-15, 9,-5, 4,-2, 1, -1, 1, 1, 2, 2, ... a(n-8)
%C A226956 a(n)-b(n+1) = period 6: repeat 0, 1, 1, 0, -1, -1 = A128834(n).
%C A226956 Diagonals with the same number give 1, 2, 9, 38, ... = A001077(n).
%C A226956 Second column: the (n+2)-th term is identical to a(n+1) signed.
%C A226956 a(n+1) is identical to its twice shifted inverse binomial transform signed.
%C A226956 a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.
%H A226956 G. C. Greubel, <a href="/A226956/b226956.txt">Table of n, a(n) for n = 0..1000</a>
%H A226956 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1)
%F A226956 a(n+6) - a(n-6) = 20*A000045(n).
%F A226956 a(n) = 2*a(n-1) - a(n-2) + a(n-4).
%F A226956 a(n) = 3*a(n-3) + 5*a(n-6) + a(n-9) (plus many similar by telescoping the fundamental recurrence).
%F A226956 a(n+3) - a(n-3) = 2*A000032(n).
%F A226956 G.f.: (x-1)*(1+x^2) / ( (x^2+x-1)*(x^2-x+1) ). - _R. J. Mathar_, Jun 26 2013
%F A226956 2*a(n) = A000032(n) + A010892(n-1). - _R. J. Mathar_, Jun 26 2013
%F A226956 a(n+5) = a(n+4) + a(n+2) + A108014(n).
%F A226956 a(2n+1) + A226447(2n+2) = 2*A182895(n).
%F A226956 a(n) - a(n-2) = 0,2,1,1,1,3,6,... = abs(A111734(n-2)).
%e A226956 a(0) = a(1) = 1.
%e A226956 a(2) = a(3) = 2.
%e A226956 a(4) = 2*a(3) - a(2) + a(0) = 4-2+1 = 3.
%e A226956 a(5) = 6-2+1 = 5.
%t A226956 a[n_] := (LucasL[n] + {0, 1, 1, 0, -1, -1}[[Mod[n, 6] + 1]])/2; Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Jun 28 2013, after _R. J. Mathar_ *)
%t A226956 LinearRecurrence[{2,-1,0,1}, {1,1,2,2}, 30] (* _G. C. Greubel_, Jan 15 2018 *)
%o A226956 (PARI) x='x+O('x^30); Vec((x-1)*(1+x^2)/((x^2+x-1)*(x^2-x+1))) \\ _G. C. Greubel_, Jan 15 2018
%o A226956 (Magma) I:=[1, 1, 2, 2]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%Y A226956 Cf. Diagonals in A024490.
%K A226956 nonn,easy
%O A226956 0,3
%A A226956 _Paul Curtz_, Jun 24 2013