A110953 Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top.

2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376, 2932030308352, 11728125427712, 46912493322240, 187649990066176, 750599926710272

Offset: 1

Author

Cino Hilliard, Sep 25 2005

Keywords

• easy
• nonn

Comments

The limit of the sequence of fractions used to generate this sequence is 3.

Essentially the same as A003683. - R. J. Mathar, May 25 2009

References

• Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p. 16.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,8).

Programs

PARI

g(n,k,typ) = /* typ = 1 numerator, 2 denominator, k = multiple of denom */ { local(a,b,x,tmp); a=1;b=1; for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1,print1(a","),print1(b",")) ); print(); print(a/b+.) }

Python

from itertools import islice
def A110953_gen(): # generator of terms
    a, b = 1, 1
    while True:
        a, b = a+9*b, a+b
        yield b
A110953_list = list(islice(A110953_gen(),30)) # Chai Wah Wu, Apr 15 2025

Formula

Given a(0)=1, b(0)=1 then for i>=1, a(i)/b(i) = (a(i-1)+ 9*b(i-1)) / (a(i-1) + b(i-1)).

From Chai Wah Wu, Apr 15 2025: (Start)

a(n) = 2*a(n-1) + 8*a(n-2) for n > 2.

G.f.: x*(-8*x - 2)/((2*x + 1)*(4*x - 1)). (End)

Extensions

a(21)-a(25) from Chai Wah Wu, Apr 15 2025