%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 nth 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/bulletinarchive/GABv10n04E.pdf">A Fibonacci Related Sequence, Girls' Angle Bulletin</a>, Vol. 10, No. 4 (2017), 2326.
%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(n1)) 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(n1) + a(n2)  5*a(n3)  a(n4) + a(n5) 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
