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%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
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