A269226 Period 6: repeat [3, 9, 6, 6, 9, 3].

3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3

Offset: 1

Author

Peter M. Chema, Jul 11 2016

Keywords

• nonn
• base
• easy

Comments

The palindromic sequence arising when the digital root of n alternates diagonally in opposite directions on a square grid. This is the sequence of 3-6-9 appearing every third column on a square grid when A010888 (digital root of n) alternates in both directions diagonally. Other columns are the digital root of 2^n: {1, 2, 4, 8, 7, 5}, or in its opposite direction 5^n: {5,7,8,4,2,1}. All diagonals parallel to the digital roots of n are also {1,2,3,4,5,6,7,8,9} or {9,8,7,6,5,4,3,2,1}.

See the link below for a visual illustration.

This sequence also arises when A180592 (digital root of 2n) is substituted for A010888.

Decimal expansion of 40070/10101. - David A. Corneth, Jul 12 2016

Links

• Peter M. Chema, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Crossrefs

Cf. A010888, A180592.

Programs

PARI

a(n)=[3, 3, 9, 6, 6, 9][n%6+1] \\ Charles R Greathouse IV, Jul 12 2016

Formula

a(n+1) = digital root of 5^n - 2^n.

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) = a(n-6). - Charles R Greathouse IV, Jul 12 2016

a(n) = (12 - 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) - sqrt(3)*sin(n*Pi/3) - 3*sqrt(3)*sin(2*n*Pi/3))/2. - Wesley Ivan Hurt, Oct 05 2018