A053573 a(n) = 5*a(n-1) + 14*a(n-2), a(0)=1, a(1)=5.

1, 5, 39, 265, 1871, 13065, 91519, 640505, 4483791, 31386025, 219703199, 1537920345, 10765446511, 75358117385, 527506838079, 3692547833785, 25847834902031, 180934844183145, 1266543909544159, 8865807366284825, 62060651565042351

Offset: 0

Author

Barry E. Williams, Jan 18 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,14).

Crossrefs

Cf. A090409 (binomial transform).

Programs

GAP

List([0..30], n-> (7^(n+1) -(-2)^(n+1))/9); # G. C. Greubel, May 16 2019

Magma

[(7^(n+1) -(-2)^(n+1))/9: n in [0..30]]; // G. C. Greubel, May 16 2019

Mathematica

LinearRecurrence[{5,14},{1,5},30] (* Harvey P. Dale, May 29 2017 *)

PARI

a(n)=n++;(7^n -(-2)^n)/9 \\ Charles R Greathouse IV, Jun 11 2011

Sage

[lucas_number1(n,5,-14) for n in range(1, 16)] # Zerinvary Lajos, Apr 24 2009

Formula

a(n) = (7^(n+1) - (-2)^(n+1))/9.

a(n) = 5*a(n-1) + 14*a(n-2), with a(0)=1, a(1)=5.

G.f.: 1/(1-5*x-14*x^2). - Zerinvary Lajos, Apr 24 2009

E.g.f.: (7*exp(7*x) - 2*exp(-2*x))/9. - G. C. Greubel, May 16 2019