A321664 A sequence consisting of three disjoint copies of the Fibonacci sequence, one shifted, with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms.

0, 1, 1, 1, 2, 1, 2, 3, 2, 4, 5, 3, 7, 8, 5, 12, 13, 8, 20, 21, 13, 33, 34, 21, 54, 55, 34, 88, 89, 55, 143, 144, 89, 232, 233, 144, 376, 377, 233, 609, 610, 377, 986, 987, 610, 1596, 1597, 987, 2583, 2584, 1597, 4180, 4181, 2584, 6764, 6765, 4181, 10945

Offset

0,5

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 (0,0,2,0,0,0,0,0,-1).

Formula

G.f.: (1 + x + x^2 + x^3 + x^4)/(1 - x^3 - x^6) - 1/(1 - x^3).

G.f.: (x + x^2 + x^3 - x^5 - x^7)/(1 - 2*x^3 + x^9).

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

a(n) = 2*a(n-3) - a(n-9). - G. C. Greubel, Dec 04 2018

Example

For n=13, as n is 1 (mod 3), we find a(3*4+1) is the 4+2=6th Fibonacci number which is 8.

Maple

seq(coeff(series(((x^4+x^3+x^2+x+1)/(1-x^3-x^6))-(1/(1-x^3)), x, n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Nov 29 2018

Mathematica

CoefficientList[Series[(x+x^2+x^3-x^5-x^7)/(1-2x^3+x^9), {x, 0, 20}], x] (* or *)
LinearRecurrence[{0, 0, 2, 0, 0, 0, 0, 0, -1}, {0, 1, 1, 1, 2, 1, 2, 3, 2}, 50] (* G. C. Greubel, Dec 04 2018 *)

Prog

(Python)
def a(n):
    if n<6:
        return [0, 1, 1, 1, 2, 1][n]
    return a(n-3)+a(n-6)+[1, 0, 0][n%3]
(Racket)
(define (f x) (cond [(< x 6) (list-ref (list 0 1 1 1 2 1) x)]
[else (+ (f (- x 3)) (f (- x 6)) (list-ref (list 1 0 0) (remainder x 3)))]))
(Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9))); // Vincenzo Librandi, Nov 29 2018
(PARI) my(x='x+O('x^70)); Vec((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9)) \\ G. C. Greubel, Dec 04 2018
(Sage) s=((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9)).series(x, 70); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 04 2018

Crossrefs

Exhibits a property shared with multiples of A000931.

Sequence in context: A087154 A029839 A082304 * A250099 A241949 A288126

Adjacent sequences: A321661 A321662 A321663 * A321665 A321666 A321667

Keyword

nonn

Author

David Nacin, Nov 23 2018

Status

approved