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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

%I

%S 1,2,4,9,20,48,117,294,748,1925,4984,12960,33785,88218,230580,603057,

%T 1577836,4129232,10807885,28291230,74060636,193882317,507572784,

%U 1328814144,3478834225,9107631218,23843966692,62424118809,163428146948,427859929200,1120151005029,2932592057430

%N 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.

%H Colin Barker, <a href="/A286983/b286983.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Harned, <a href="http://www.girlsangle.org/page/bulletin-archive/GABv10n04E.pdf">A Fibonacci Related Sequence, Girls' Angle Bulletin</a>, Vol. 10, No. 4 (2017), 23-26.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).

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

%F From _Colin Barker_, May 18 2017: (Start)

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

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

%F (End)

%e 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.

%t LinearRecurrence[{3, 1, -5, -1, 1}, {1, 2, 4, 9, 20}, 32] (* or *)

%t 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 *)

%o (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

%Y Cf. A000045.

%K easy,nonn

%O 1,2

%A _Milena Harned_, May 17 2017

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Last modified February 25 00:39 EST 2018. Contains 299630 sequences. (Running on oeis4.)