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A193090 Digital roots of the nonzero pentagonal numbers. 1
1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8, 6, 7, 2, 9, 1, 5, 3, 4, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is a periodic sequence with period 9 and cycle 1,5,3,4,8,6,7,2,9 - which are also the coefficients of x in the numerator of the generating function.

Note that the cycle 1,5,3,4,8,6,7,2,9 is a permutation of the first 9 natural numbers A000027. - Omar E. Pol, Aug 15 2011

This sequence is the same as A002450(n+1) mod 9, except with a value of 9 where that would return 0. - Joe Slater, Mar 04 2018

LINKS

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

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

FORMULA

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

As the sum of the terms contained in each cycle is 45, they also satisfy the eighth order inhomogeneous recurrence 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) = cos(8n Pi/9) (1 + 2 cos(2n Pi/9))(1 + 2 cos(2n Pi/3)) + (n + 7n^3 + 5n^4 + n^5 + 5n^6 + 4n^7 + 5n^8) mod 9.

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

a(n) = A010888(A000326(n)). - Jonathan Vos Post, Aug 15 2011

a(n) = 9-((8*(4^n-1)/3) mod 9). - Joe Slater, Mar 04 2018

EXAMPLE

The sixth nonzero pentagonal number is A000326(6) = 51, which has digital root 5 + 1 = 6. Hence a(6) = 6.

MATHEMATICA

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&, n]; DigitalRoot[1/2 # (3#-1)]&/@Range[90]

PadRight[{}, 120, {1, 5, 3, 4, 8, 6, 7, 2, 9}] (* Harvey P. Dale, Sep 12 2017 *)

PROG

(PARI) a(n)=[9, 1, 5, 3, 4, 8, 6, 7, 2][n%9+1] \\ Charles R Greathouse IV, Oct 04 2012

CROSSREFS

Cf. A000326, A002450, A010888.

Sequence in context: A198563 A111889 A004494 * A004162 A319053 A109681

Adjacent sequences:  A193087 A193088 A193089 * A193091 A193092 A193093

KEYWORD

nonn,easy,base

AUTHOR

Ant King, Aug 15 2011

STATUS

approved

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Last modified May 25 19:13 EDT 2019. Contains 323576 sequences. (Running on oeis4.)