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

Author

Ant King, Jun 18 2011

Keywords

• nonn
• base
• easy

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.

Examples

			The fifth even, nonzero square is 100, which has digital root 1. Hence a(5)=1.
		

Links

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

Crossrefs

Cf. A010888, A016742, A191760, A191761.

Programs

Mathematica

DigitalRoot[n_Integer?Positive]:=FixedPoint[Plus@@IntegerDigits[#]&,n];DigitalRoot[(2#)^2] &/@Range[63]

PARI

a(n)=(4*n^2-1)%9+1 \\ Charles R Greathouse IV, Jun 19 2011

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