OFFSET
0,1
COMMENTS
a(n+1)/a(n) converges to 4 + sqrt(17).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (8,1).
FORMULA
a(n) = (4+sqrt(17))^n + (4-sqrt(17))^n.
O.g.f: 2*(-1+4*x)/(-1+8*x+x^2). - R. J. Mathar, Dec 02 2007
a(n) = 2*A088317(n). - R. J. Mathar, Sep 27 2014
E.g.f.: 2*exp(4*x)*cosh(sqrt(17)*x). - Stefano Spezia, Dec 21 2025
EXAMPLE
a(4) = 8*a(3)+a(2) = 8*536+66 = 4354.
MATHEMATICA
LinearRecurrence[{8, 1}, {2, 8}, 30] (* Harvey P. Dale, Sep 21 2014 *)
RecurrenceTable[{a[0] == 2, a[1] == 8, a[n] == 8 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
PROG
(Magma) I:=[2, 8]; [n le 2 select I[n] else 8*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016
(PARI) x='x+O('x^30); Vec(2*(1-4*x)/(1-8*x-x^2)) \\ G. C. Greubel, Nov 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 11 2003
STATUS
approved