A055850 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=10.

1, 10, 29, 77, 202, 529, 1385, 3626, 9493, 24853, 65066, 170345, 445969, 1167562, 3056717, 8002589, 20951050, 54850561, 143600633, 375951338, 984253381, 2576808805, 6746173034, 17661710297, 46238957857, 121055163274

Offset: 0

Author

Barry E. Williams, Jun 03 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (3,-1).

Crossrefs

Cf. A000032, A000045.

Programs

GAP

List([0..30], n-> Fibonacci(2*n+2) + 7*Fibonacci(2*n) ); # G. C. Greubel, Jan 16 2020

Magma

[Lucas(2*n+1) + 6*Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+7*x)/(1-3*x+x^2) )); // Marius A. Burtea, Jan 16 2020

Maple

with(combinat); seq( fibonacci(2*n+2) + 7*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 16 2020

Mathematica

LinearRecurrence[{3,-1},{1,10},30] (* Harvey P. Dale, Jul 22 2019 *)

PARI

vector(31, n, fibonacci(2*n) + 7*fibonacci(2*(n-1)) ) \\ G. C. Greubel, Jan 16 2020

Sage

[fibonacci(2*n+2) + 7*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 16 2020

Formula

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

G.f.: (1+7*x)/(1-3*x+x^2).

a(n) = Lucas(2n+1) + 6*Fibonacci(2n).

a(n) = Fibonacci(2*n+2) + 7*Fibonacci(2*n). - G. C. Greubel, Jan 16 2020