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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
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 (list; graph; refs; listen; history; text; internal format)
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: A000081 A123467 A124497 * A289971 A093637 A068051

Adjacent sequences:  A286980 A286981 A286982 * A286984 A286985 A286986

KEYWORD

easy,nonn

AUTHOR

Milena Harned, May 17 2017

STATUS

approved

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Last modified December 11 19:09 EST 2017. Contains 295919 sequences.