A205650 Period 12: repeat (1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9).

1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9, 1, 6

Offset

1,2

Comments

The members of this sequence are also the digital roots of the indices of those nonzero square numbers that are also triangular.

The coefficients of x^n in the numerator of the generating function form the periodic cycle of the sequence.

Links

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

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

Formula

G.f. x*(1 + 6*x + 8*x^2 + 6*x^3 + x^4 + 9*x^5 + 8*x^6 + 3*x^7 + x^8 + 3*x^9 + 8*x^10 + 9*x^11) / (1 - x^12).

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

a(n) = 63-a(n-1)-a(n-2)-a(n-3)-a(n-4)-a(n-5)-a(n-6)-a(n-7)-a(n-8)-a(n-9)-a(n-10)-a(n-11).

a(n) = 1/4*(21+(-1)^n*(3+14*sin(n*Pi/2))+6*((-1)^n+1)*cos(2*n*Pi/3)-2*sqrt(3)*(-1)^n*(sin(5*n*Pi/6)-sin(n*Pi/6))).

For n>0, a(n) = A010888(A001109(n)) = A010888(sqrt(A001110(n))).

Example

The fourth nonzero square number that is also a triangular number is 204^2. As 204 has digital root 2+4=6, then a(4)=6.

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9}, 86]
PadRight[{}, 90, {1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9}] (* Harvey P. Dale, Apr 10 2013 *)

Prog

(PARI) a(n)=[9, 1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8][n%12+1] \\ Charles R Greathouse IV, Jul 17 2016

Crossrefs

Cf. A010888, A001109, A001110.

Sequence in context: A266092 A143627 A105798 * A153755 A331374 A340725

Adjacent sequences: A205647 A205648 A205649 * A205651 A205652 A205653

Keyword

nonn,easy

Author

Ant King, Jan 30 2012

Status

approved