A152108 a(n) = ((7+sqrt(5))^n + (7-sqrt(5))^n)/2.

1, 7, 54, 448, 3896, 34832, 316224, 2894528, 26609536, 245174272, 2261620224, 20875015168, 192738922496, 1779844247552, 16437306875904, 151809149370368, 1402086588645376, 12949609668739072, 119602725461950464, 1104655331042787328, 10202654714273202176

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (14, -44).

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((7+r5)^n+(7-r5)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

Magma

I:=[1,7]; [n le 2 select I[n] else 14*Self(n-1)-44*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2018

Mathematica

With[{srt5=Sqrt[5]},Simplify/@Table[((7+srt5)^n+(7-srt5)^n)/2,{n,0,20}]] (* or *) LinearRecurrence[{14,-44},{1,7},20] (* Harvey P. Dale, Jan 16 2012 *)
CoefficientList[Series[(1 - 7 x) / (1 - 14 x + 44 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 04 2018 *)

Formula

From Philippe Deléham, Nov 26 2008: (Start)

a(n) = 14*a(n-1) - 44*a(n-2), n > 1; a(0)=1, a(1)=7.

G.f.: (1-7*x)/(1-14*x+44*x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2*k)*5^(n-k))/7^n. (End)

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008