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a(n) = ((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)).
1

%I #28 Sep 08 2022 08:45:40

%S 1,14,152,1512,14480,136192,1269568,11781504,109080064,1008734720,

%T 9322763264,86134358016,795679428608,7349600247808,67884508610560,

%U 627000709644288,5791091556155392,53487250561826816,494013479394738176

%N a(n) = ((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)).

%C Fifth binomial transform of A048878.

%C lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(5) = 9.236067977499789696....

%H G. C. Greubel, <a href="/A153884/b153884.txt">Table of n, a(n) for n = 1..1000</a>

%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-44).

%F From _Philippe Deléham_, Jan 03 2009: (Start)

%F a(n) = 14*a(n-1) - 44*a(n-2) for n>1, with a(0)=0, a(1)=1.

%F G.f.: x/(1 - 14*x + 44*x^2). (End)

%F E.g.f.: sinh(sqrt(5)*x)*exp(7*x)/sqrt(5). - _Ilya Gutkovskiy_, Sep 01 2016

%t Join[{a=1,b=14},Table[c=14*b-44*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)

%t LinearRecurrence[{14, -44}, {1, 14}, 25] (* or *) Table[((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)) , {n,0,25}] (* _G. C. Greubel_, Aug 31 2016 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((7+r)^n-(7-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; # _Klaus Brockhaus_, Jan 04 2009

%o (Magma) I:=[1,14]; [n le 2 select I[n] else 14*Self(n-1)-44*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 01 2016

%o (PARI) Vec(x/(1-14*x+44*x^2) + O(x^99)) \\ _Altug Alkan_, Sep 01 2016

%Y Cf. A002163 (decimal expansion of sqrt(5)), A048878.

%K nonn,easy

%O 1,2

%A Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009

%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 04 2009

%E Edited by _Klaus Brockhaus_, Oct 11 2009