A194597 Digital roots of the nonzero hexagonal numbers.

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

Offset: 1

Author

Ant King, Aug 30 2011

Keywords

• nonn
• easy
• base

Comments

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

a(n) = A010888(A000384(n)). - Reinhard Zumkeller, Jan 09 2013

Examples

			The sixth nonzero hexagonal number is A000384(6)=66. As 6+6=12 and 1+2=3, this has digital root 3 and so a(6)=3.
		

Crossrefs

Cf. A000384.

Cf. A010878, A010888.

Programs

Haskell

a194597 n = [1,6,6,1,9,3,1,3,9] !! a010878 (n-1)
-- Reinhard Zumkeller, Jan 09 2013

Magma

&cat[ [1,6,6,1,9,3,1,3,9]: k in [1..10] ]; // Vincenzo Librandi, Aug 11 2015

Mathematica

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&,n];DigitalRoot[ # (2#-1)]&/@Range[63]
CoefficientList[Series[(1 + 6 x + 6 x^2 + x^3 + 9 x^4 + 3 x^5 + x^6 + 3 x^7 + 9 x^8)/((1 - x) (1 + x + x^2) (1 + x^3 + x^6)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 11 2015 *)
PadRight[{},120,{1,6,6,1,9,3,1,3,9}] (* Harvey P. Dale, Oct 02 2018 *)

Formula

a(n) = a(n-9), and as the sum of the terms contained in each cycle is 39 they also satisfy the eighth-order inhomogeneous recurrence 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 + cos(2/9*(n-5)*Pi) + cos(4/9*(n-5)*Pi) + cos(2/3*(n-5)*Pi) + cos(8/9*(n-5)*Pi) + cos(4/3*(n-5)*Pi) + cos(14/9*(n-5)*Pi) + cos(16/9*(n-5)*Pi) + cos((2 n Pi)/9) + cos((4 n Pi)/9) + cos((2 n Pi)/3) + cos((8 n Pi)/9) + cos((10 n Pi)/9) + cos((4 n Pi)/3) + cos((14 n Pi)/9) + cos((16 n Pi)/9) + cos(2/9 (2+5 n) Pi) + (8n + 5n^2 + 7n^3 + n^5 + n^7 + 6n^8) mod 9.

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