A191762 Digital roots of the nonzero even squares.

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.

Links

Table of n, a(n) for n=1..63.

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

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

Cf. A010888, A016742, A191760, A191761.

Sequence in context: A190967 A004787 A210617 * A107905 A258707 A010299

Adjacent sequences: A191759 A191760 A191761 * A191763 A191764 A191765

Keyword

nonn,base,easy

Author

Ant King, Jun 18 2011

Status

approved