A194641 Digital roots of the nonzero heptagonal numbers.

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

Offset

1,2

Comments

This is a periodic sequence with period 9 and cycle 1, 7, 9, 7, 1, 9, 4, 4, 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), and as the sum of the terms contained in each cycle is 51 they also satisfy the eighth-order inhomogeneous recurrence 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) = 3*(1+cos(2*n*Pi/3) +cos(4*n*Pi/3)) +(n^3 +8*n^4 +7*n^5 +5*n^6 +4*n^7 +3*n^8) mod 9.

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

a(n) = A010888(A000566(n)). - Michel Marcus, Aug 10 2015

Example

The sixth nonzero heptagonal number is A000566(6)=81. As 8+1=9, this has digital root 9 and so a(6)=9.

Mathematica

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[DigitalRoot[ n*(5*n - 3)/2], {n, 100}]
PadRight[{}, 120, {1, 7, 9, 7, 1, 9, 4, 4, 9}] (* Harvey P. Dale, Dec 31 2023 *)

Crossrefs

Cf. A000566, A010888.

Sequence in context: A154168 A019644 A100639 * A248674 A108743 A177271

Adjacent sequences: A194638 A194639 A194640 * A194642 A194643 A194644

Keyword

nonn,easy,base

Author

Ant King, Aug 31 2011

Status

approved