A194825 Digital roots of the nonzero 9-gonal (nonagonal) numbers.

1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 9, 6, 1, 3

Offset

1,2

Comments

This is a periodic sequence with period 9 and cycle 1,9,6,1,3,3,1,6,9, which are also the coefficients of x in the numerator of the generating function.

Links

Table of n, a(n) for n=1..86.

Formula

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

a(n) = 39-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) = 2*(1+cos(2*n*Pi/9)+cos(2*n*Pi/3)+cos(4*n*Pi/9)+cos(8*n*Pi/9)+cos(4*(n-2)*Pi/9)+cos(2*(n-2)*Pi/3)+cos(8*(n-2)*Pi/9))+cos(2*(n-2)*Pi/9)+cos(4*(4*n+1)*Pi/9) + mod(8*n+5*n^2+8*n^3+5*n^4+8*n^5+2*n^6+5*n^7+5*n^8,9).

G.f.: x*(1+9*x+6*x^2+x^3+3*x^4+3*x^5+x^6+6*x^7+9*x^8)/((1-x)*(1+x+x^2)*(1+x^3+x^6)).

a(n) = A010888(A001106(n)). - Michel Marcus, Aug 10 2015

Example

The sixth nonzero 9-gonal (nonagonal) number is A001106(6)=111. As 1+1+1=3, this has digital root 3 and so a(6)=3.

Mathematica

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[DigitalRoot[n*(7*n-5)/2], {n, 100}]
dr[n_]:=FixedPoint[Total[IntegerDigits[#]]&, n]; dr/@PolygonalNumber[ 9, Range[ 90]] (* Requires Mathematica version 10 or later *)  (* or *) PadRight[{}, 90, {1, 9, 6, 1, 3, 3, 1, 6, 9}] (* Harvey P. Dale, Jun 26 2021 *)

Prog

(PARI) a(n)=n*(7*n-5)/2%9 \\ Charles R Greathouse IV, Sep 15 2013

Crossrefs

Cf. A010888, A001106.

Sequence in context: A075567 A019883 A085574 * A092732 A342680 A176532

Adjacent sequences: A194822 A194823 A194824 * A194826 A194827 A194828

Keyword

nonn,easy,base

Author

Ant King, Sep 03 2011

Status

approved