A286983 a(n) is the smallest integer that can appear as the n-th term of two distinct nondecreasing sequences of positive integers that satisfy the Fibonacci recurrence relation.

1, 2, 4, 9, 20, 48, 117, 294, 748, 1925, 4984, 12960, 33785, 88218, 230580, 603057, 1577836, 4129232, 10807885, 28291230, 74060636, 193882317, 507572784, 1328814144, 3478834225, 9107631218, 23843966692, 62424118809, 163428146948, 427859929200, 1120151005029, 2932592057430

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

M. Harned, A Fibonacci Related Sequence, Girls' Angle Bulletin, Vol. 10, No. 4 (2017), 23-26.

Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Formula

a(n) = F(n)*(1 + F(n-1)) where F = A000045 (the Fibonacci sequence).

From Colin Barker, May 18 2017: (Start)

G.f.: x*(1 - x - 3*x^2) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).

a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5) for n>5.

(End)

Example

F(4) = 9 since 1, 4, 5, 9 and 3, 3, 6, 9 are the first four terms of distinct nondecreasing sequences of positive integers that satisfy the Fibonacci recurrence relation and there are not two such sequences that have a number less than 9 as their 4th term.

Mathematica

LinearRecurrence[{3, 1, -5, -1, 1}, {1, 2, 4, 9, 20}, 32] (* or *)
Rest@ CoefficientList[Series[x (1 - x - 3 x^2)/((1 + x) (1 - 3 x + x^2) (1 - x - x^2)), {x, 0, 32}], x] (* Michael De Vlieger, May 18 2017 *)

Prog

(PARI) Vec(x*(1 - x - 3*x^2) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)) + O(x^40)) \\ Colin Barker, May 18 2017

Crossrefs

Cf. A000045.

Sequence in context: A123467 A000081 A124497 * A289971 A093637 A068051

Adjacent sequences: A286980 A286981 A286982 * A286984 A286985 A286986

Keyword

easy,nonn

Author

Milena Harned, May 17 2017

Status

approved