A172968 a(n) = 7*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=2.

1, 2, 13, 89, 610, 4181, 28657, 196418, 1346269, 9227465, 63245986, 433494437, 2971215073, 20365011074, 139583862445, 956722026041, 6557470319842, 44945570212853, 308061521170129, 2111485077978050, 14472334024676221

Offset: 0

Author

Artur Jasinski, Feb 06 2010

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Marta Na Chen, Wenchang Chu, Four classes of symmetric sums over cyclically binomial products, Symmetry 17 (2025) no 2
• Index entries for linear recurrences with constant coefficients, signature (7,-1).

Crossrefs

Cf. A000032, A000045, A081016.

Essentially the same as A033891.

Programs

GAP

List([0..30], n-> Fibonacci(4*n-1)); # G. C. Greubel, Jul 15 2019

Magma

[n le 2 select n else 7*Self(n-1)-Self(n-2): n in [1..30]]; // Bruno Berselli, Mar 29 2016

Maple

with(combinat):F:= n-> fibonacci(n):L:=n-> 2*F(n+1)-F(n):
seq(1/2*(L(4*n)-F(4*n)), n=0..20); # Gary Detlefs, Nov 28 2010

Mathematica

Table[Sqrt[1-2m+5m^2]/.m ->Fibonacci[2n+1]Fibonacci[2n+2], {n, -1, 30}]
CoefficientList[Series[(1-5x)/(1-7x+x^2), {x, 0, 30}], x] (* Michael De Vlieger, Mar 29 2016 *)
Fibonacci[4*Range[0, 30] -1] (* G. C. Greubel, Jul 15 2019 *)

PARI

x='x+O('x^30); Vec((1-5*x)/(1-7*x+x^2)) \\ Altug Alkan, Mar 29 2016

Sage

[fibonacci(4*n-1) for n in (0..30)] # G. C. Greubel, Jul 15 2019

Formula

a(n) = (1/10)*((5+sqrt(5))*((7-3*sqrt(5))/2)^n + ((5-sqrt(5))*((7+3*sqrt(5))/2)^n)).

a(n) = sqrt(1 - 2*F(2n+1)*F(2n+2) + 5*(F(2n+1)*F(2n+2))^2), where F = A000045.

a(n) = sqrt(1 - 2*A081016(n) + 5*A081016(n)^2).

a(n) = A033891(n-1), n>0. - R. J. Mathar, Feb 08 2010

a(n) = (Lucas(4*n) - Fibonacci(4*n))/2, where Lucas = A000032. - Gary Detlefs, Nov 28 2010

G.f.: (1 - 5*x)/(1 - 7*x + x^2). - Bruno Berselli, Mar 29 2016

a(n) = Fibonacci(4*n-1). - G. C. Greubel, Jul 15 2019

a(n) = (a(n-1)^2 + 9)/a(n-2). - Klaus Purath, Aug 30 2020