OFFSET
0,2
COMMENTS
The sequence A084330 is a(0)=0, a(1)=1, a(n)=31a(n-1)-29a(n-2), and the ratio A084330(n+1)/a(n) converges to phi^7 (~29.034441853748633...), where phi is the golden ratio (A001622).
The continued fraction for phi^7 is {29,{29}}, and 29 occurs in the following approximations for n*phi^7: A248786 (29*n+floor(n/29)+0^n-0^(n mod 29)) for A004922 (floor(n*phi^7)), A249079 (29*n+floor(n/29)+0^(1-floor((14+(n mod 29))/29)) for A004942 (round(n*phi^7)), and A248739 (29*n+ceiling(n/29)) for A004962 (ceiling(n*phi^7)).
LINKS
Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Eric W. Weisstein, From MathWorld--A Wolfram Web Resource, Golden Ratio
Eric W. Weisstein, From MathWorld--A Wolfram Web Resource, Golden Ratio Conjugate
Index entries for linear recurrences with constant coefficients, signature (31,-29).
FORMULA
G.f.: (1-29*x)/(29*x^2-31*x+1). - Vincenzo Librandi, Jun 03 2015
EXAMPLE
For n=3, 31*a(2)-29*a(1) = 31*(33)-29*(2) = 1023-58 = 965.
MAPLE
a:= n-> (<<0|1>, <-29|31>>^n. <<1, 2>>)[1, 1]:
seq(a(n), n=0..23); # Alois P. Heinz, Dec 22 2023
MATHEMATICA
LinearRecurrence[{31, -29}, {1, 2}, 50] (* or *) CoefficientList[Series[(1 - 29 x)/(29 x^2 - 31 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 03 2015 *)
PROG
(Python)
print(1, end=', ')
print(2, end=', ')
an = [1, 2]
for n in range(2, 26):
print(31*an[n-1]-29*an[n-2], end=', ')
an.append(31*an[n-1]-29*an[n-2])
(Magma) I:=[1, 2]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 03 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karl V. Keller, Jr., Jun 02 2015
STATUS
approved