A191760 Digital root of the n-th odd square.

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, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1

Offset: 1

Author

Ant King, Jun 17 2011

Keywords

• nonn
• base
• easy

Comments

This sequence is periodic with period <1,9,7,4,9,4,7,9,1> of length nine.

Related to the continued fraction of (153727+sqrt(2207057870693))/1477642 = 1+ 1/(9+1/(7+1/...)). - R. J. Mathar, Jun 27 2011

Examples

			The fifth, odd square number is 81 which has digital root 9. Hence a(5)=9.
		

Links

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

Crossrefs

Cf. A010888, A016754, A056992.

Programs

Mathematica

DigitalRoot[n_Integer?Positive]:=FixedPoint[Plus@@IntegerDigits[#]&,n];DigitalRoot[#] &/@((2#-1)^2 &/@Range[81])
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 9, 7, 4, 9, 4, 7, 9, 1},81] (* Ray Chandler, Aug 25 2015 *)
PadRight[{},120,{1,9,7,4,9,4,7,9,1}] (* Harvey P. Dale, Jun 26 2021 *)

Formula

a(n) = 3*(1+cos(2(n-2)pi/3)+cos(4(n-2)pi/3)) +mod( (1+n)(1+7n-7n^2+7n^3+n^4-n^5+3n^6+3n^7), 9).

a(n) = a(n-9).

a(n) = 51-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) = A010888(A016754(n)).

G.f.: x(1+9x+7x^2+4x^3+9x^4+4x^5+7x^6+9x^7+x^8)/( (1-x)*(1+x+x^2)*(1+x^3+x^6) ) (note that the coefficients of x in the numerator are precisely the terms that constitute the periodic cycle of the sequence).

a(n) = A056992(2n-1). - R. J. Mathar, Jun 27 2011