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 A121442 #36 Sep 08 2022 08:45:27
%S A121442 1,1,9,17,97,241,1097,3169,12801,40225,152265,501489,1831649,6192785,
%T A121442 22176137,76079553,269472001,932011841,3281180297,11399814865,
%U A121442 39998425697,139315579185,487901595593,1701743382561,5953542163713,20781331011169,72661467102025
%N A121442 Expansion of (1-x^2)/(1-x-9*x^2+x^3).
%C A121442 From _Roman Witula_, Aug 08 2012: (Start)
%C A121442 We have a(n)=A(n;2), where A(n;2), B(n;2) and C(n;2) are the special cases of so-called quasi-Fibonacci numbers A(n;d), B(n;d), and C(n;d) for the value of argument d=2 - for details see Witula's comments to A121449 or the paper of Witula-Slota-Warzynski's. The sequences A(n;2), B(n;2) and C(n;2) are defined by the following system of recurrence equations:
%C A121442 A(0;2)=1, B(0;2)=C(0;2)=0,
%C A121442 A(n+1;2)=A(n;2)+4*B(n;2)-2*C(n;2), B(n+1;2)=2*A(n;2)+B(n;2), and C(n+1;2)=2*B(n;2)-C(n;2).
%C A121442 We note that A(n;1)=A077998(n), B(n;1)=A006054(n+1), and C(n;1)=A006054(n). We know (see formulas (3.61-63) in Witula et al.'s paper) that the sequences: (-2)^(-n)*(A(n;1)*(A(n;2)-C(n;2))-B(n;1)*(B(n;2)+C(n;2))+C(n;1)*B(n;2)), (-2)^(-n)*(-A(n;1)*C(n;2)+B(n;1)*(A(n;2)-C(n;2))-C(n;1)*(B(n;2)-C(n;2))), and (-2)^(-n)*(A(n;1)*(B(n;2)-C(n;2))-B(n;1)*B(n;2)+C(n;1)*(A(n;2)-B(n;2)+C(n;2))) are the binomial transforms of the sequences (-2)^(-n)*A(n;1), (-2)^(-n)*B(n;1), and (-2)^(-n)*C(n;1) respectively. Moreover the elements of the sequences A(n;1/2)=2^(-n)*A052975, B(n;1/2)=2^(-n)*A094789, and C(n;1/2) could be described by certain convolutions type identities for the elements of A(n;2), B(n;2), and C(n;2) (see identities (3.58-60) in Witula et al.'s paper). (End)
%H A121442 G. C. Greubel, <a href="/A121442/b121442.txt">Table of n, a(n) for n = 0..1000</a>
%H A121442 Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H A121442 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 9, -1).
%F A121442 a(0)=a(1)=1, a(2)=9, a(n+1) = a(n)+9*a(n-1)-a(n-2) for n>=2.
%F A121442 7*a(n) = (2-c(4))*(1-2*c(1))^n + (2-c(1))*(1-2*c(2))^n + (2-c(2))*(1-2*c(4))^n = (s(2))^2*(1-2*c(1))^n + (s(4))^2*(1-2*c(2))^n + (s(1))^2*(1-2*c(4))^n, where c(j):=2*Cos(2Pi*j/7) and s(j):=2*Sin(2Pi*j/7) - it is the special case, for d=2, of the Binet's formula for the respective quasi-Fibonacci number A(n;d) discussed in Witula-Slota-Warzynski's paper (see also A121449). - _Roman Witula_, Aug 08 2012
%t A121442 LinearRecurrence[{1,9,-1},{1,1,9},50] (* Roman Witula, Aug 08 2012 *)
%t A121442 CoefficientList[Series[(1 - x^2)/(1 - x - 9 x^2 + x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 18 2015 *)
%o A121442 (PARI) Vec((1-x^2)/(1-x-9*x^2+x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A121442 (Magma) I:=[1,1,9]; [n le 3 select I[n] else Self(n-1)+9*Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Sep 18 2015
%Y A121442 Cf. A077998, A006054, A121449, A085810.
%K A121442 nonn,easy
%O A121442 0,3
%A A121442 _Philippe Deléham_, Sep 06 2006
%E A121442 Corrected by _T. D. Noe_, Oct 25 2006
%E A121442 More terms from _Vincenzo Librandi_, Sep 18 2015