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

1,2

Comments

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

Links

Table of n, a(n) for n=1..86.

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

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))).

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001108, A000217, A010888, A000290.

Sequence in context: A338574 A011225 A011196 * A105144 A277781 A254767

Adjacent sequences: A205181 A205182 A205183 * A205185 A205186 A205187

Keyword

nonn,easy

Author

Ant King, Jan 23 2012

Status

approved