A114647 - OEIS

%I #13 Sep 08 2022 08:45:23

%S 3,2,7,12,31,70,171,408,987,2378,5743,13860,33463,80782,195027,470832,

%T 1136691,2744210,6625111,15994428,38613967,93222358,225058683,

%U 543339720,1311738123,3166815962,7645370047,18457556052,44560482151,107578520350,259717522851

%N Expansion of (3 -4*x -3*x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

%C Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

%H Colin Barker, <a href="/A114647/b114647.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).

%F G.f.: (3 -4*x -3*x^2)/((1-x)*(1+x)*(1-2*x-x^2)).

%F a(n) = A000129(n+1) + 2*A059841(n). - _R. J. Mathar_, Nov 10 2009

%F From _Colin Barker_, May 26 2016: (Start)

%F a(n) = 1 + (-1)^n + ((1+sqrt(2))^(1+n) - (1-sqrt(2))^(1+n))/(2*sqrt(2)).

%F a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) for n>3. (End)

%F a(n) = A000129(n+1) + 1 + (-1)^n. - _G. C. Greubel_, May 24 2021

%t Table[Fibonacci[n+1, 2] +1+(-1)^n, {n, 0, 30}] (* _G. C. Greubel_, May 24 2021 *)

%o (PARI) Vec((3-4*x-3*x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^50)) \\ _Colin Barker_, May 26 2016

%o (Magma) I:=[3,2,7,12]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // _G. C. Greubel_, May 24 2021

%o (Sage) [lucas_number1(n+1,2,-1) +(1+(-1)^n) for n in (0..30)] # _G. C. Greubel_, May 24 2021

%Y Cf. A000129, A059841, A100828, A114688, A114689, A114695, A114696, A114697.

%K easy,nonn

%O 0,1

%A _Creighton Dement_, Feb 18 2006