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
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))).
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
KEYWORD
nonn,easy
AUTHOR
Ant King, Jan 30 2012
STATUS
approved