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

0,2

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

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A022035 A255119 A005668 * A192807 A076026 A161734

Adjacent sequences: A018901 A018902 A018903 * A018905 A018906 A018907

Keyword

nonn,easy

Author

R. K. Guy

Status

approved