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A193090
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Digital roots of the nonzero pentagonal numbers.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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) = 9-((8*(4^n-1)/3) mod 9). - Joe Slater, Mar 04 2018
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EXAMPLE
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The sixth nonzero pentagonal number is A000326(6) = 51, which has digital root 5 + 1 = 6. Hence a(6) = 6.
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,base
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AUTHOR
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STATUS
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approved
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