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A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence. 25

%I

%S 1,5,30,175,1025,6000,35125,205625,1203750,7046875,41253125,241500000,

%T 1413765625,8276328125,48450468750,283633984375,1660422265625,

%U 9720281250000,56903517578125,333118994140625,1950112558593750,11416157763671875,66831351611328125,391237546875000000

%N Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.

%C a(n) gives the length of the word obtained after n steps with the substitution rule 0->11111, 1->111110, starting from 0. The number of 1's and 0's of this word is 5*a(n-1) and 5*a(n-2), resp.

%C a(n) / a(n-1) converges to (5 + (3 * sqrt(5))) / 2 as n approaches infinity. (5 + (3 * sqrt(5))) / 2 can also be written as phi^2 + (2 * phi), phi^3 + phi, phi + sqrt(5) + 2, (3 * phi) + 1, (3 * phi^2) - 2, phi^4 - 1 and (5 + (3 * (L(n) / F(n)))) / 2, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity. - _Ross La Haye_, Aug 18 2003, on another version

%C Pisano period lengths: 1, 3, 3, 6, 1, 3, 24, 12, 9, 3, 10, 6, 56, 24, 3, 24,288, 9, 18, 6, ... - _R. J. Mathar_, Aug 10 2012

%H Indranil Ghosh, <a href="/A057088/b057088.txt">Table of n, a(n) for n = 0..1300</a>

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=5, q=5.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H W. Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=5.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HoradamSequence.html">Horadam Sequence</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,5)

%F a(n) = 5*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.

%F a(n) = S(n, i*sqrt(5))*(-i*sqrt(5))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.

%F G.f.: 1/(1 - 5*x - 5*x^2).

%F a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(k)*3^k. - _Benoit Cloitre_, Oct 25 2003

%F a(n) = ((5 + 3*sqrt(5))/2)^n(1/2 + sqrt(5)/6) + (1/2 - sqrt(5)/6)((5 - 3*sqrt(5))/2)^n. - _Paul Barry_, Sep 22 2004

%F (a(n)) appears to be given by the floretion - 0.75'i - 0.5'j + 'k - 0.75i' + 0.5j' + 0.5k' + 1.75'ii' - 1.25'jj' + 1.75'kk' - 'ij' - 0.5'ji' - 0.75'jk' - 0.75'kj' - 1.25e ("jes"). - _Creighton Dement_, Nov 28 2004

%F a(n) = Sum_{k=0..n} 4^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006

%F G.f.: G(0)/(2-5*x), where G(k)= 1 + 1/(1 - x*(9*k-5)/(x*(9*k+4) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 17 2013

%F From _Ehren Metcalfe_, Nov 18 2017: (Start)

%F With F(n) = A000045(n), L(n) = A000032(n), beta = (1-sqrt(5))/2:

%F a(2*n-1) = 5^n*F(4*n)/3 = (5^(n-1/2)*L(4*n) - 2*5^(n-1/2)*beta^(4*n))/3.

%F a(2*n) = 5^n*L(4*n+2)/3 = (5^(n+1/2)*F(4*n+2) + 2*5^n*beta^(4*n+2))/3.

%F a(n) = round 5^((n+1)/2)*F(2*(n+1))/3.

%F a(n) = round 5^(n/2)*L(2*(n+1))/3. (End)

%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+5*a[n-2]od: seq(a[n], n=1..33); # _Zerinvary Lajos_, Dec 14 2008

%t LinearRecurrence[{5,5}, {1,5}, 30] (* _G. C. Greubel_, Jan 16 2018 *)

%o (Sage) [lucas_number1(n,5,-5) for n in xrange(1, 22)] # _Zerinvary Lajos_, Apr 24 2009

%o (PARI) x='x+O('x^30); Vec(1/(1 - 5*x - 5*x^2)) \\ _G. C. Greubel_, Jan 16 2018

%o (MAGMA) I:=[1, 5]; [n le 2 select I[n] else 5*Self(n-1) + 5*Self(n-2): n in [0..30]]; // _G. C. Greubel_, Jan 16 2018

%Y Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 11 2000

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Last modified February 22 23:03 EST 2019. Contains 320411 sequences. (Running on oeis4.)