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A191762
Digital roots of the nonzero even squares.
1
4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9
OFFSET
1,1
COMMENTS
Period 9: repeat [4, 7, 9, 1, 1, 9, 7, 4, 9]. Bisection of A056992.
The digits in the 9-cycle of this sequence are the same as the digits in the 9-cycle of the digital roots of the odd squares A191760(n). However, these are offset differently (by the first five terms) and hence constitute a different sequence.
FORMULA
a(n) = 3*(1 + cos(2*n*Pi/3) + cos(4*n*Pi/3)) + (4*n^4 + 7*n^6 + 2*n^8) mod 9.
G.f.: x*(4 + 7*x + 9*x^2 + x^3 + x^4 + 9*x^5 + 7*x^6 + 4*x^7 + 9*x^8)/(1-x^9) (note that the coefficients of x in the numerator are precisely the terms that constitute the periodic cycle of the sequence).
a(n) = A010888(A016742(n)). - Michel Marcus, Aug 11 2015
EXAMPLE
The fifth even, nonzero square is 100, which has digital root 1. Hence a(5)=1.
MATHEMATICA
DigitalRoot[n_Integer?Positive]:=FixedPoint[Plus@@IntegerDigits[#]&, n]; DigitalRoot[(2#)^2] &/@Range[63]
PROG
(PARI) a(n)=(4*n^2-1)%9+1 \\ Charles R Greathouse IV, Jun 19 2011
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Ant King, Jun 18 2011
STATUS
approved