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A222407 Digital roots of tribonacci numbers A000073.

%I A222407 #34 Aug 21 2024 15:02:28
%S A222407 0,0,1,1,2,4,7,4,6,8,9,5,4,9,9,4,4,8,7,1,7,6,5,9,2,7,9,9,7,7,5,1,4,1,
%T A222407 6,2,9,8,1,9,9,1,1,2,4,7,4,6,8,9,5,4,9,9,4,4,8,7,1,7,6,5,9,2,7,9,9,7,
%U A222407 7,5,1,4,1,6,2,9,8,1,9,9,1,1,2,4,7,4,6,8,9,5,4,9,9,4,4,8,7,1,7,6,5,9,2,7,9,9,7,7,5,1,4,1,6,2,9,8,1,9,9,1,1
%N A222407 Digital roots of tribonacci numbers A000073.
%C A222407 From a(2) onwards, periodic with period length 39.
%C A222407 The period sums to 216 and contains no 3s. When divided into three sets of 13, further patterns are revealed in connection with A100402 (see link below). - _Peter M. Chema_, Dec 21 2016
%H A222407 Peter M. Chema, <a href="/A222407/a222407.pdf">Period 39, divided into three sets of 13, with further digital root patterns</a>
%F A222407 a(n) = A010888(A000073(n)). - _Michel Marcus_, Dec 19 2016
%F A222407 From _Chai Wah Wu_, Jan 30 2018: (Start)
%F A222407 a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-7) - a(n-9) + a(n-10) - a(n-12) + a(n-13) - a(n-15) + a(n-16) - a(n-18) + a(n-19) - a(n-21) + a(n-22) - a(n-24) + a(n-25) - a(n-27) + a(n-28) - a(n-30) + a(n-31) - a(n-33) + a(n-34) - a(n-36) + a(n-37) for n > 38.
%F A222407 G.f.: (-9*x^38 + 8*x^36 - 16*x^35 - x^34 + 15*x^33 - 20*x^32 + 4*x^31 + 12*x^30 - 17*x^29 + 10*x^27 - 17*x^26 - 2*x^25 + 10*x^24 - 15*x^23 + 3*x^22 + 3*x^21 - 11*x^20 + 2*x^19 + 2*x^18 - 5*x^17 - 4*x^16 + x^15 - x^14 - 4*x^13 - 4*x^12 - x^11 + x^10 - 5*x^9 - 5*x^8 + 2*x^7 - 3*x^6 - 3*x^5 - x^4 - x^2)/(x^37 - x^36 + x^34 - x^33 + x^31 - x^30 + x^28 - x^27 + x^25 - x^24 + x^22 - x^21 + x^19 - x^18 + x^16 - x^15 + x^13 - x^12 + x^10 - x^9 + x^7 - x^6 + x^4 - x^3 + x - 1). (End)
%p A222407 f:=proc(n) option remember; if n <= 1 then 0; elif n=2 then 1; else f(n-3)+f(n-2)+f(n-1); fi; end; # A000073
%p A222407 P:=n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # A010888
%p A222407 [seq(P(f(n)),n=0..200)];
%t A222407 FixedPoint[Total@ IntegerDigits@ # &, #] & /@ CoefficientList[ Series[x^2/(1 - x - x^2 - x^3), {x, 0, 81}], x] (* _Michael De Vlieger_, Dec 22 2016 *)
%t A222407 droot[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]; droot/@LinearRecurrence[{1,1,1},{0,0,1},150] (* or *) PadRight[{0,0},150,{9,9,1,1,2,4,7,4,6,8,9,5,4,9,9,4,4,8,7,1,7,6,5,9,2,7,9,9,7,7,5,1,4,1,6,2,9,8,1}] (* _Harvey P. Dale_, Aug 21 2024 *)
%Y A222407 Cf. A000073, A010888, A210456.
%K A222407 nonn
%O A222407 0,5
%A A222407 _N. J. A. Sloane_, Feb 20 2013