A194731 Digital roots of the nonzero octagonal numbers.

1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2, 6, 7, 5, 9, 1, 8, 3, 4, 2

Offset

1,2

Comments

This is a periodic sequence with period 9 and cycle 1, 8, 3, 4, 2, 6, 7, 5, 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) = 45 -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(cos(14*n*Pi/9) +cos(10*n*Pi/9) +cos(2*n*Pi/3) +cos(2*n*Pi/9) +cos(Pi/3)) +(n +8*n^3 +n^4 +8*n^5 +7*n^6 +8*n^7 +4*n^8) mod 9.

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

a(n) = A010888(A000567(n)). - Michel Marcus, Aug 10 2015

Example

The sixth nonzero octagonal number is A000567(6)=96. As 9+6=15 and 1+5=6, this has digital root 6 and so a(6)=6.

Mathematica

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[DigitalRoot[n*(3*n - 2)], {n, 100}]

Crossrefs

Cf. A000567, A010888.

Sequence in context: A117889 A021927 A145594 * A021849 A338462 A201754

Adjacent sequences: A194728 A194729 A194730 * A194732 A194733 A194734

Keyword

nonn,easy,base

Author

Ant King, Sep 02 2011

Status

approved