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

%I A057088 #87 Feb 16 2025 08:32:43
%S A057088 1,5,30,175,1025,6000,35125,205625,1203750,7046875,41253125,241500000,
%T A057088 1413765625,8276328125,48450468750,283633984375,1660422265625,
%U A057088 9720281250000,56903517578125,333118994140625,1950112558593750,11416157763671875,66831351611328125,391237546875000000
%N A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.
%C A057088 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 A057088 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 A057088 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 A057088 Indranil Ghosh, <a href="/A057088/b057088.txt">Table of n, a(n) for n = 0..1300</a>
%H A057088 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 A057088 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://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 A057088 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A057088 W. Lang, <a href="https://web.archive.org/web/2024*/https://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 A057088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HoradamSequence.html">Horadam Sequence</a>
%H A057088 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A057088 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,5)
%F A057088 a(n) = 5*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
%F A057088 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 A057088 G.f.: 1/(1 - 5*x - 5*x^2).
%F A057088 a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(k)*3^k. - _Benoit Cloitre_, Oct 25 2003
%F A057088 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 A057088 (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 A057088 a(n) = Sum_{k=0..n} 4^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%F A057088 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 A057088 From _Ehren Metcalfe_, Nov 18 2017: (Start)
%F A057088 With F(n) = A000045(n), L(n) = A000032(n), beta = (1-sqrt(5))/2:
%F A057088 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 A057088 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 A057088 a(n) = round 5^((n+1)/2)*F(2*(n+1))/3.
%F A057088 a(n) = round 5^(n/2)*L(2*(n+1))/3. (End)
%p A057088 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 A057088 LinearRecurrence[{5,5}, {1,5}, 30] (* _G. C. Greubel_, Jan 16 2018 *)
%o A057088 (Sage) [lucas_number1(n,5,-5) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 24 2009
%o A057088 (PARI) x='x+O('x^30); Vec(1/(1 - 5*x - 5*x^2)) \\ _G. C. Greubel_, Jan 16 2018
%o A057088 (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 A057088 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 A057088 nonn,easy
%O A057088 0,2
%A A057088 _Wolfdieter Lang_, Aug 11 2000