A205184 Period 12: repeat (1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9).

1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9, 1, 8

Offset: 1

Author

Ant King, Jan 23 2012

Keywords

• nonn
• easy

Comments

The terms of this sequence are the digital roots of the indices of those nonzero triangular numbers that are also perfect squares (A001108).

Examples

			As the fourth nonzero triangular number that is also a perfect square is A000217(288), and 288 has digital root A010888(288)=9, then a(4)=9.
		

Links

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

Crossrefs

Cf. A001108, A000217, A010888, A000290.

Programs

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9}, 86]
LinearRecurrence[{0, 1, 0, 0, 0, -1, 0, 1},{1, 8, 4, 9, 7, 8, 7, 9},86] (* Ray Chandler, Aug 03 2015 *)

PARI

a(n)=[9, 1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1][n%12+1] \\ Charles R Greathouse IV, Jul 17 2016

Formula

G.f.: x*(1+8*x+3*x^2+x^3+3*x^4-x^5+x^6+9*x^7) / ((1-x)*(1+x)*(1+x^2)*(1-sqrt(3)*x+x^2)*(1+sqrt(3)*x+x^2)).

a(n) = a(n-12).

a(n) = 25-a(n-1)-a(n-6)-a(n-7).

a(n) = a(n-2)-a(n-6)+a(n-8).

a(n) = 1/4*(25+(-1)^n*(9+4*sqrt(3)*(cos(n*Pi/6)-cos(5*n*Pi/6))+2*cos(n*Pi/2))).