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%I #50 Jan 31 2021 20:43:32
%S 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,
%T 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,
%U 6,7,2,9,1,5,3,4,8,6,7,2,9,1,5,3,4,8
%N Digital roots of the nonzero pentagonal numbers.
%C 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.
%C 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
%C 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
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,1).
%F a(n) = a(n-9).
%F 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).
%F 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.
%F 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)).
%F a(n) = A010888(A000326(n)). - _Jonathan Vos Post_, Aug 15 2011
%F a(n) = 9-((8*(4^n-1)/3) mod 9). - _Joe Slater_, Mar 04 2018
%e The sixth nonzero pentagonal number is A000326(6) = 51, which has digital root 5 + 1 = 6. Hence a(6) = 6.
%t DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&,n]; DigitalRoot[1/2 # (3#-1)]&/@Range[90]
%t PadRight[{},120,{1,5,3,4,8,6,7,2,9}] (* _Harvey P. Dale_, Sep 12 2017 *)
%o (PARI) a(n)=[9, 1, 5, 3, 4, 8, 6, 7, 2][n%9+1] \\ _Charles R Greathouse IV_, Oct 04 2012
%Y Cf. A000326, A002450, A010888.
%K nonn,easy,base
%O 1,2
%A _Ant King_, Aug 15 2011
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