A214887 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=7.

1, 7, 7, 49, 343, 16807, 5764801, 96889010407, 558545864083284007, 54116956037952111668959660849, 30226801971775055948247051683954096612865741943

Offset: 0

Author

Vincenzo Librandi, Aug 01 2012

Keywords

• nonn
• easy

Comments

a(17) has 1350 digits.

From Peter Bala, Nov 01 2013: (Start)

Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 6*sum {n = 1..inf} 1/7^floor(n*phi) (= 36*sum {n = 1..inf} floor(n/phi)/7^n) = 0.87718 67194 00499 51922 ... = 1/(1 + 1/(7 + 1/(7 + 1/(49 + 1/(343 + 1/(16807 + 1/(5764801 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.

Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/7^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..16
• W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
• P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
• D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Crossrefs

Cf. A000045, A000301, A010098, A010099, A010100, A214706, A014565, A215270, A215271, A215272.

Column k=7 of A244003.

Programs

Magma

[7^Fibonacci(n): n in [0..10]];

Maple

a:= n-> 7^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

Mathematica

7^Fibonacci[Range[0,10]]
nxt[{a_,b_}]:={b,a*b}; Transpose[NestList[nxt,{1,7},10]][[1]] (* Harvey P. Dale, Jun 10 2014 *)

Formula

a(n) = 7^Fibonacci(n).