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%I #44 Dec 12 2023 08:24:32
%S 0,1,3,6,1,6,3,1,9,9,1,3,6,1,6,3,1,9,9,1,3,6,1,6,3,1,9,9,1,3,6,1,6,3,
%T 1,9,9,1,3,6,1,6,3,1,9,9,1,3,6,1,6,3,1,9,9,1,3,6,1,6,3,1,9,9,1,3,6,1,
%U 6,3,1,9,9,1,3,6,1,6,3,1,9,9,1,3,6,1,6
%N Digital roots of triangular numbers.
%C Decimal expansion of 45387733/3333333330. - _Enrique Pérez Herrero_, Nov 14 2021
%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) = A010888(A000217(n)).
%F Periodic sequence for n>0: a(n+9) = a(n);
%F a(A016777(n)) = 1; a(A007494(n)) <> 1;
%F a(A090570(n)) = A010888(A090570(n)).
%F a(n) = 1 + ((n^2 + n - 2)/2) mod 9. - _Ant King_, Apr 25 2009
%F G.f.: x(1 + 3x + 6x^2 + x^3 + 6x^4 + 3x^5 + x^6 + 9x^7 + 9x^8)/((1-x)(1 + x + x^2)(1 + x^3 + x^6)). - _Ant King_, Nov 16 2010
%t digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&,n]; Table[If[n==0,0,digitalRoot[n(n+1)/2]], {n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, May 02 2011 *)
%o (PARI) a(n)=if(n, n=n*(n+1)/2%9; if(n, n, 9), 0) \\ _Charles R Greathouse IV_, Dec 19 2016
%o (Python)
%o def A145389(n): return (9, 1, 3, 6, 1, 6, 3, 1, 9)[n%9] if n else 0 # _Chai Wah Wu_, Feb 09 2023
%Y Cf. A000217, A004157, A010888.
%K nonn,base,easy
%O 0,3
%A _Reinhard Zumkeller_, Oct 10 2008
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