A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given. Mohamed Bouhamida, Mar 31 2010įrom the Pinter and Ziegler reference's abstract: authors 'show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. Example: a(5)=5 because we have (HHHH), (HHU)D, (HUH)D, (UHH)D, and (UU)DD (the unique weak ascent is shown between parentheses see A114690). A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps. A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis.

Henry Bottomley, Feb 10 2010į(n+1) = number of Motzkin paths of length n having exactly one weak ascent.

Reinhard Zumkeller, Sep 26 2009ĭifference between number of closed walks of length n+1 from a node on a pentagon and number of walks of length n+1 between two adjacent nodes on a pentagon. Sarah-Marie Belcastro (smbelcas(AT)), Jul 04 2009 Output of Kasteleyn's formula for the number of perfect matchings of an m X n grid specializes to the Fibonacci sequence for m=2. Knuth writes: 'Before Fibonacci wrote his work, the sequence F_ (1 + 4*cos^2 k*Pi/n), where terms = roots to the Fibonacci product polynomials, A152063. Examples: Input: s This problem has been solved See the answer See the answer See the answer done loading. A valid string should contain at least three digits to make one additive sequence.

A000045 - OEIS login The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. A string contains an additive sequence if its digits can make a sequence of numbers in which every number is addition of previous two numbers.

Additive sequence of an array