A327035 An unbounded sequence consisting solely of Fibonacci numbers with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms.

1, 1, 0, 1, 1, 1, 2, 2, 1, 3, 3, 2, 5, 5, 3, 8, 8, 5, 13, 13, 8, 21, 21, 13, 34, 34, 21, 55, 55, 34, 89, 89, 55, 144, 144, 89, 233, 233, 144, 377, 377, 233, 610, 610, 377, 987, 987, 610, 1597, 1597, 987, 2584, 2584, 1597, 4181, 4181, 2584, 6765, 6765, 4181

Offset

0,7

Comments

This sequence was constructed to show that there are many sequences, besides those merging with multiples of the Padovan sequence A000931, with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms. This refutes a conjecture that was formerly in that entry.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

David Nacin, Van der Laan Sequences and a Conjecture on Padovan Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 1, 0, 0, 1).

Formula

G.f.: (x^5 + x + 1)/(-x^6 - x^3 + 1).

a(3*n) = A000045(n+1), a(3*n+1) = A000045(n+1), a(3*n+2) = A000045(n).

a(n) = a(n-3) + a(n-6).

Example

For n=7, as n is 3(2)+1, a(n) = A000045(2+1) = 2.

Mathematica

LinearRecurrence[{0, 0, 1, 0, 0, 1}, {1, 1, 0, 1, 1, 1}, 50]

Prog

(Python)
a = lambda x:[1, 1, 0, 1, 1, 1][x] if x<6 else a(x-3)+a(x-6)
(Racket)
(define (a x) (cond [(< x 6) (list-ref (list 1 1 0 1 1 1) x)]
[else (+ (a (- x 3)) (a (- x 6)))]))
(Sage)
s=((x^5 + x + 1)/(-x^6 - x^3 + 1)).series(x, 23); s.coefficients(x, sparse=False)

Crossrefs

Cf. A321664, A321341, A328943.

Exhibits a property shared with multiples of A000931.

Sequence in context: A242308 A011373 A321783 * A177352 A210798 A117501

Adjacent sequences: A327032 A327033 A327034 * A327036 A327037 A327038

Keyword

nonn

Author

David Nacin, Nov 28 2019

Status

approved