A087954 (2 + phi)/a(n) is the sum of successive remainders when computing the Euclidean algorithm for (1, A088166(n)/phi) with phi being the golden ratio, where n >= 2.

40, 320, 2160, 15004, 103680, 709804, 4868640, 33385280, 228811000, 1568358004, 10749957120, 73681030804, 505018447960, 3461452808000, 23725145626560, 162614587921804, 1114577054219520, 7639424691459004, 52361396168994000, 358890350005878080

Offset: 2

Author

Thomas Baruchel, Sep 21 2003

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A088166.

Programs

PARI

lucas(n) = fibonacci(n+1) + fibonacci(n-1)
a(n) = if(n%3==0, lucas(4*n)-2, lucas(4*n)-lucas(2*n)) \\ Colin Barker, Mar 10 2016

Formula

For n = 3*k, a(n) = Lucas(4*n) - 2. For all other n, a(n) = Lucas(4*n) - Lucas(2*n).

Empirical g.f.: 4*x^2*(10 + 10*x - 10*x^2 - 139*x^3 + 13*x^4 - 48*x^5 + 144*x^6 - 16*x^7 + 2*x^8 - 7*x^9 + x^10) / ((1 - x)*(1 - 7*x + x^2)*(1 - 3*x + x^2)*(1 + x + x^2)*(1 + 3*x + 8*x^2 + 3*x^3 + x^4)). - Colin Barker, Mar 10 2016