A321341 An unbounded sequence which is 1 infinitely often, with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms.

1, 1, 1, 2, 2, 1, 3, 3, 4, 1, 4, 5, 5, 1, 6, 6, 7, 1, 7, 8, 8, 1, 9, 9, 10, 1, 10, 11, 11, 1, 12, 12, 13, 1, 13, 14, 14, 1, 15, 15, 16, 1, 16, 17, 17, 1, 18, 18, 19, 1, 19, 20, 20, 1, 21, 21, 22, 1, 22, 23, 23, 1, 24, 24, 25, 1, 25, 26, 26, 1, 27, 27, 28, 1, 28

Offset: 0

Author

David Nacin, Nov 05 2018

Keywords

• nonn
• easy

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..16383
• 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 (0,0,0,1,0,0,0,1,0,0,0,-1).

Crossrefs

Exhibits a property shared with multiples of A000931.

Programs

Mathematica

{3#+1, 1, 3#+1, 3#+2, 3#+2, 1, 3#+3, 3#+3}& /@ Range[0, 9] // Flatten (* Jean-François Alcover, Nov 24 2018, from Python *)

PARI

a(n)={my(t=n\8*3); [t+1, 1, t+1, t+2, t+2, 1, t+3, t+3][n%8 + 1]} \\ Andrew Howroyd, Nov 19 2018

Python

l=list()
for a in range(10):
   l+=[3*a+1,1,3*a+1,3*a+2,3*a+2,1,3*a+3,3*a+3]

Formula

From Chai Wah Wu, May 08 2025: (Start)

a(n) = a(n-4) + a(n-8) - a(n-12) for n > 11.

G.f.: (-x^9 + x^8 + x^7 + 2*x^6 + x^4 + 2*x^3 + x^2 + x + 1)/(x^12 - x^8 - x^4 + 1). (End)