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%I #72 Nov 16 2023 12:49:08
%S 1,5,20,75,275,1000,3625,13125,47500,171875,621875,2250000,8140625,
%T 29453125,106562500,385546875,1394921875,5046875000,18259765625,
%U 66064453125,239023437500,864794921875,3128857421875,11320312500000,40957275390625,148184814453125
%N Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.
%C Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+4, s(0) = 1, s(2n+4) = 5. - _Herbert Kociemba_, Jun 14 2004
%C Binomial transform of A002878. - _Philippe Deléham_, Oct 04 2005
%C Diagonal of square array A216219. - _Philippe Deléham_, Mar 15 2013
%C Lim_{n->oo} a(n+1)/a(n) = 2 + phi = A296184, where phi = A001622. - _Wolfdieter Lang_, Nov 16 2023~
%H G. C. Greubel, <a href="/A030191/b030191.txt">Table of n, a(n) for n = 0..1000</a>
%H Santiago Alzate, Oscar Correa, and Rigoberto Flórez, <a href="https://arxiv.org/abs/2009.02639">Fibonacci identities from Jordan Identities</a>, arXiv:2009.02639 [math.NT], 2020.
%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 Wolfdieter 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. (38) and (45), lhs, m=5.
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = (sqrt(5))^n*U(n, sqrt(5)/2).
%F G.f.: 1/(1 - 5*x + 5*x^2).
%F a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).
%F a(2*k) = 5^k*Lucas(2*k+1).
%F a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(2*k). - _Benoit Cloitre_, Jun 21 2003
%F a(n) = 5*a(n-1) - 5*a(n-2). - _Benoit Cloitre_, Oct 23 2003
%F a(n-1) = (((5+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - _Paul Barry_, Mar 23 2004
%F a(n) = Sum_{k-0..n} 5^k*A109466(n,k). - _Philippe Deléham_, Nov 28 2006
%F a(n) = 5*A039717(n), n>0. - _Philippe Deléham_, Mar 12 2013
%F a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - _Philippe Deléham_, Mar 15 2013
%F G.f.: 1/(1-5*x/(1+x/(1-x))). - _Philippe Deléham_, Mar 15 2013
%F a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - _Michael Somos_, Aug 27 2015
%F E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - _Stefano Spezia_, Dec 29 2019
%F a(n) = Sum_{k=0..n} A081567(n-k)*2^k. - _Philippe Deléham_, Mar 10 2023
%e G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...
%p seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Dec 28 2019
%t Table[MatrixPower[{{2,1},{1,3}},n][[1]][[2]],{n,0,44}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* _Michael Somos_, Aug 27 2015 *)
%t Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n,0,30}] (* _G. C. Greubel_, Dec 28 2019 *)
%o (Sage) [lucas_number1(n,5,5) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 22 2009
%o (PARI) {a(n) = imag((quadgen(5) + 2)^(n+1))}; /* _Michael Somos_, Aug 27 2015 *
%o (Magma) I:=[1,5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Dec 28 2019
%o (GAP) a:=[1,5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # _G. C. Greubel_, Dec 28 2019
%Y Cf. A000032, A000045, A001622.
%Y Cf. A002878, A039717, A081567, A109466, A216219.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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