A018904 Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(1,6).

1, 6, 37, 229, 1418, 8781, 54377, 336734, 2085253, 12913101, 79965442, 495192589, 3066520913, 18989683446, 117595179557, 728217839669, 4509548979898, 27925753660941, 172932530727097, 1070898946784974, 6631629973859333, 41066915083090461, 254310255712336562

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Comments

a(n) is the number of compositions of n when there are 6 types of ones. - Milan Janjic, Aug 13 2010

Number of words of length n over {0,1,...,7} in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
• Index entries for linear recurrences with constant coefficients, signature (7,-5).
• Index entries for Pisot sequences

Programs

Mathematica

Table[Simplify[(2^(-1 - n) ((7 - #)^n (-5 + #) + (5 + #) (7 + #)^n))/#] &@ Sqrt@ 29, {n, 0, 22}] (* or *)
CoefficientList[Series[-(x - 1)/(5 x^2 - 7 x + 1), {x, 0, 22}], x] (* Michael De Vlieger, Jan 28 2017 *)
LinearRecurrence[{7,-5},{1,6},30] (* Harvey P. Dale, May 01 2022 *)

PARI

S(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=a[n-1]^2\a[n-2]+1); a
S(1, 6, 40) \\ Colin Barker, Feb 16 2016

Formula

a(n) = (a(1)+1)*a(n-1) - (a(1)-1)*a(n-2) = 7*a(n-1) - 5*a(n-2).

G.f.: -(x-1) / (5*x^2-7*x+1). - Colin Barker, Feb 14 2013

a(n) = (2^(-1-n)*((7-sqrt(29))^n*(-5+sqrt(29)) + (5+sqrt(29))*(7+sqrt(29))^n)) / sqrt(29). - Colin Barker, Jan 20 2017