

A286983


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.


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), 2326.
Index entries for linear recurrences with constant coefficients, signature (3,1,5,1,1).


FORMULA

a(n) = F(n)*(1 + F(n1)) 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(n1) + a(n2)  5*a(n3)  a(n4) + a(n5) 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



