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

Author

Milena Harned, May 17 2017

Keywords

• easy
• nonn

Examples

			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.
		

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).

Crossrefs

Cf. A000045.

Programs

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 *)

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

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)