A093131 - OEIS

%I #94 May 13 2024 11:21:04

%S 0,1,5,20,75,275,1000,3625,13125,47500,171875,621875,2250000,8140625,

%T 29453125,106562500,385546875,1394921875,5046875000,18259765625,

%U 66064453125,239023437500,864794921875,3128857421875,11320312500000,40957275390625,148184814453125

%N Binomial transform of Fibonacci(2n).

%C Second binomial transform of Fibonacci(n). - _Paul Barry_, Apr 22 2005

%H Michael De Vlieger, <a href="/A093131/b093131.txt">Table of n, a(n) for n = 0..1791</a>

%H G. Dresden and Y. Li, <a href="https://arxiv.org/abs/2210.04322">Periodic Weighted Sums of Binomial Coefficients</a>, arXiv:2210.04322 [math.NT], 2022.

%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.

%H M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Griffiths/griffiths20.html">Families of Sequences From a Class of Multinomial Sums</a>, Journal of Integer Sequences, 15 (2012), #12.1.8.

%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 J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/OL13/Pan/pan8.html">Multiple Binomial Transforms and Families of Integer Sequences</a>, J. Int. Seq. 13 (2010), 10.4.2, F^(2) and absolute values of F^(-2).

%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pan/pan12.html">Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences</a>, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.

%H Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).

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

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

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

%F a(n) = (((5 + sqrt(5))/2)^n - ((5 - sqrt(5))/2)^n)/sqrt(5).

%F a(n) = A093130(n)/2^n.

%F a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*Fibonacci(j-k). - _Paul Barry_, Feb 15 2005

%F a(n) = Sum_{k=0..n} C(n, k)*2^k*Fibonacci(n-k) = Sum_{k=0..n} C(n, k)*2^(n-k) * Fibonacci(k). - _Paul Barry_, Apr 22 2005

%F a(n) = A030191(n-1), n > 0. - _R. J. Mathar_, Sep 05 2008

%F E.g.f.: 2*exp(5*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - _Ilya Gutkovskiy_, Aug 11 2017

%F From _Kai Wang_, Dec 22 2019: (Start)

%F a(n) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} 5^i*((i+j)!/(i!*j!)).

%F a(n*k)/a(k) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} (-1)^(j*(k-1))*b(k)^i*((i+j)!/(i!*j!)).

%F a((2*m+1)*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i)*k) + 5^(m*k).

%F a(2*m*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i-1)*k}.

%F a(m+r)*a(n+s) - a(m+s)*a(n+r) = -5^(n+s)*a(m-n)*a(r-s).

%F a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A020876(m+n+r+s) - 5^(n+s)*A020876(m-n)*A020876(r-s))/5.

%F A020876(m+r)*A020876(n+s) - A020876(m+s)*A020876(n+r) = 5^(n+s+1)*a(m-n)*a(r-s).

%F A020876(m+r)*A020876(n+s) - 5*a(m+s)*a(n+r) = 5^(n+s)*A020876(m-n)*A020876(r-s).

%F A020876(m+r)*A020876(n+s) + 5*a(m+s)*a(n+r) = 2*A020876(m+n+r+s) + 5^(n+s+1)*a(m-n)*a(r-s).

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

%F a(n)^2 - a(n+r)*a(n-r) = 5^(n-r)*a(r)^2.

%F a(m)*a(n+1) - a(m+1)*a(n) = 5^n*a(m-n).

%F a(m-n) = (a(m)*A020876(n) - A020876(m)*a(n))/(2*5^n).

%F a(m+n) = (a(m)*A020876(n) + A020876(m)*a(n))/2.

%F A020876(n)^2 - A020876(n+r)*A020876(n-r) = -5^(n-r+1)*a(r)^2.

%F A020876(m)*A020876(n+1) - A020876(m+1)*A020876(n) = -5^(n+1)*a(m-n).

%F A020876(m+n) - 5^(n)*A020876(m-n) = 5*a(m)*a(n).

%F A020876(m-n) = (A020876(m)*A020876(n) - 5*a(m)*a(n))/(2*5^n).

%F A020876(m+n) = (A020876(m)*A020876(n) + 5*a(m)*a(n))/2.  (End)

%F a(2*n) = 5^n*Fibonacci(2*n), a(2*n+1) = 5^n*Lucas(2*n+1). - _G. C. Greubel_, Dec 27 2019

%F a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(2*n, n+k)*(k|5), where (k|5) is the Legendre symbol. - _Greg Dresden_, Oct 14 2022

%p seq(coeff(series(x/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Dec 27 2019

%t CoefficientList[Series[x/(1-5x+5x^2), {x,0,30}], x] (* _Michael De Vlieger_, Dec 22 2019 *)

%t Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n], 5^((n-1)/2)*LucasL[n]], {n,0,30}] (* _G. C. Greubel_, Dec 27 2019 *)

%t LinearRecurrence[{5,-5},{0,1},30] (* _Harvey P. Dale_, Mar 21 2023 *)

%o (PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-5*x+5*x^2))) \\ _G. C. Greubel_, Dec 27 2019

%o (Magma) I:=[0,1]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Dec 27 2019

%o (Sage)

%o def A093131_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x/(1-5*x+5*x^2) ).list()

%o A093131_list(30) # _G. C. Greubel_, Dec 27 2019

%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # _G. C. Greubel_, Dec 27 2019

%Y Cf. A000032, A000045, A020876, A030191.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 23 2004