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

0,2

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).

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

Mathematica

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

Prog

(Sage) [lucas_number1(n, 5, -14) for n in range(1, 16)] # Zerinvary Lajos, Apr 24 2009
(PARI) a(n)=n++; (7^n -(-2)^n)/9 \\ Charles R Greathouse IV, Jun 11 2011
(Magma) [(7^(n+1) -(-2)^(n+1))/9: n in [0..30]]; // G. C. Greubel, May 16 2019
(GAP) List([0..30], n-> (7^(n+1) -(-2)^(n+1))/9) # G. C. Greubel, May 16 2019

Crossrefs

Cf. A090409 (binom. transf.)

Sequence in context: A218918 A075135 A202391 * A003482 A221357 A201442

Adjacent sequences: A053570 A053571 A053572 * A053574 A053575 A053576

Keyword

easy,nonn

Author

Barry E. Williams, Jan 18 2000

Status

approved