A164593 - OEIS

%I #26 Sep 08 2022 08:45:47

%S 5,22,108,520,2512,12128,58560,282752,1365248,6592000,31828992,

%T 153683968,742051840,3582943232,17299980288,83531694080,403326697472,

%U 1947433566208,9403041054720,45401898483712,219219758153728

%N a(n) = ((5 + sqrt(18))*(2 + sqrt(8))^n + (5 - sqrt(18))*(2 - sqrt(8))^n)/2.

%C Binomial transform of A096980 without initial 1. Second binomial transform of A164737. Inverse binomial transform of A101386.

%H G. C. Greubel, <a href="/A164593/b164593.txt">Table of n, a(n) for n = 0..1000</a> (* terms 0..100 from Vincenzo Librandi)

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://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 href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).

%F a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 5, a(1) = 22.

%F G.f.: (5 + 2*x)/(1-4*x-4*x^2).

%F E.g.f.: exp(2*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 12 2017

%F a(n) = 2^n*(Pell(n+2) + 3*Pell(n+1)), where Pell(n) = A000129(n). - _G. C. Greubel_, Apr 16 2020

%p seq(coeff(series( (5+2*x)/(1-4*x-4*x^2) , x, n+1), x, n), n = 0..25); # _G. C. Greubel_, Apr 16 2020

%t LinearRecurrence[{4, 4}, {5, 22}, 25] (* _G. C. Greubel_, Aug 12 2017 *)

%t Table[2^n*(Fibonacci[n+2, 2] + 3*Fibonacci[n+1, 2]), {n,0,25}] (* _G. C. Greubel_, Apr 16 2020 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(2+2*r)^n+(5-3*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 24 2009

%o (PARI) my(x='x+O('x^25)); Vec((5+2*x)/(1-4*x-4*x^2)) \\ _G. C. Greubel_, Aug 12 2017

%o (Sage) [2^n*(lucas_number1(n+2, 2, -1) + 3*lucas_number1(n+1, 2, -1)) for n in range(25)] # _G. C. Greubel_, Apr 16 2020

%Y Cf. A000129, A096980, A101386, A164737.

%K nonn,easy

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009