A222407 Digital roots of tribonacci numbers A000073.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Feb 20 2013

Keywords

• nonn

Comments

From a(2) onwards, periodic with period length 39.

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

Links

• Peter M. Chema, Period 39, divided into three sets of 13, with further digital root patterns

Crossrefs

Cf. A000073, A010888, A210456.

Programs

Maple

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:=n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # A010888
[seq(P(f(n)),n=0..200)];

Mathematica

FixedPoint[Total@ IntegerDigits@ # &, #] & /@ CoefficientList[ Series[x^2/(1 - x - x^2 - x^3), {x, 0, 81}], x] (* Michael De Vlieger, Dec 22 2016 *)
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 *)

Formula

a(n) = A010888(A000073(n)). - Michel Marcus, Dec 19 2016

From Chai Wah Wu, Jan 30 2018: (Start)

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.

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)