Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #49 Mar 12 2024 22:52:55
%S 2,7,30,127,538,2279,9654,40895,173234,733831,3108558,13168063,
%T 55780810,236291303,1000946022,4240075391,17961247586,76085065735,
%U 322301510526,1365291107839,5783465941882,24499154875367,103780085443350,439619496648767,1862258072038418
%N a(n) = 4*a(n-1) + a(n-2), n>=2, a(0) = 2, a(1) = 7.
%C Previous name was: Sequence relates numerators and denominators in the continued fraction convergents to sqrt(5).
%C Floretion Algebra Multiplication Program, FAMP Code: 2lesforcycseq[ ( - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' )*( .5'i + .5i' ) ], 2vesforcycseq = A000004.
%H Vincenzo Librandi, <a href="/A097924/b097924.txt">Table of n, a(n) for n = 0..1000</a>
%H Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, and John Pryce, <a href="http://arxiv.org/abs/1502.03085">On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials</a>, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,1).
%F a(n) = A001077(n+1) - 2*A001076(n).
%F A048875(n) + A001077(n+1)/2 = a(n)/2 + A048876(n).
%F a(n) = ((2*sqrt(5)+3)*(2+sqrt(5))^n + (2*sqrt(5)-3)*(2-sqrt(5))^n)/(2*sqrt(5)).
%F a(n+1) = A001077(n+1) + A015448(n+2) - _Creighton Dement_, Mar 08 2005
%F a(n) = 4*a(n-1) + a(n-2) for n>=2, a(0)=2, a(1)=7. G.f.: (2-x)/(1-4*x-x^2). - _Philippe Deléham_, Nov 20 2008
%F G.f.: G(0)*(2-x)/2, where G(k) = 1 + 1/(1 - x*(8*k + 4 +x)/(x*(8*k + 8 +x) + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Feb 15 2014
%F a(-1 - n) = -(-1)^n * A048875(n). - _Michael Somos_, Feb 23 2014
%e G.f. = 2 + 7*x + 30*x^2 + 127*x^3 + 538*x^4 + 2279*x^5 + 9654*x^6 + 40895*x^7 + ...
%t a[n_] := Expand[((2Sqrt[5] + 3)*(2 + Sqrt[5])^n + (2Sqrt[5] - 3)*(2 - Sqrt[5])^n)/(2Sqrt[5])]; Table[ a[n], {n, 0, 20}] (* _Robert G. Wilson v_, Sep 17 2004 *)
%t a[ n_] := (3 I ChebyshevT[ n + 1, -2 I] + 4 ChebyshevT[ n, -2 I]) I^n / 5; (* _Michael Somos_, Feb 23 2014 *)
%t a[ n_] := If[ n < 0, SeriesCoefficient[ (2 + 7 x) / (1 + 4 x - x^2), {x, 0, -n}], SeriesCoefficient[ (2 - x) / (1 - 4 x - x^2), {x, 0, n}]]; (* _Michael Somos_, Feb 23 2014 *)
%t LinearRecurrence[{4,1}, {2,7}, 50] (* _G. C. Greubel_, Dec 20 2017 *)
%o (PARI) {a(n) = ( 3*I*polchebyshev( n+1, 1, -2*I) + 4*polchebyshev( n, 1, -2*I)) * I^n / 5}; \\ _Michael Somos_, Feb 23 2014
%o (PARI) {a(n) = if( n<0, polcoeff( (2 + 7*x) / (1 + 4*x - x^2) + x * O(x^-n), -n), polcoeff( (2 - x) / (1 - 4*x - x^2) + x * O(x^n), n))}; \\ _Michael Somos_, Feb 23 2014
%o (Magma) I:=[2,7]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%Y Cf. A001076, A001077, A097924.
%K nonn,easy
%O 0,1
%A _Creighton Dement_, Sep 04 2004; corrected Sep 16 2004
%E Edited, corrected and extended by _Robert G. Wilson v_, Sep 17 2004
%E Better name (using formula from _Philippe Deléham_) from _Joerg Arndt_, Feb 16 2014