A280308 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=3, a(1)=4, a(2)=5.

3, 4, 5, 12, 21, 38, 71, 130, 239, 440, 809, 1488, 2737, 5034, 9259, 17030, 31323, 57612, 105965, 194900, 358477, 659342, 1212719, 2230538, 4102599, 7545856, 13878993, 25527448, 46952297, 86358738, 158838483, 292149518, 537346739, 988334740, 1817830997, 3343512476, 6149678213, 11311021686, 20804212375

Offset

0,1

Comments

Like other tribonacci sequences, the digital root is period length 39, and is as follows: (3, 4, 5, 3, 3, 2, 8, 4, 5, 8, 8, 3, 1, 3, 7, 2, 3, 3, 8, 5, 7, 2, 5, 5, 3, 4, 3, 1, 8, 3, 3, 5, 2, 1, 8, 2, 2, 3, 7).

Completes the set of tribonacci numbers with 3,4,5 as initial terms, the others being (3,5,4), (4,5,3), (4,3,5), (5,3,4), and (5,4,3). The sum of each of the digital root periods in the above set is 162, except (4,3,5), which sums to 180; the sum of the digital root period of A081172 is also 180.

Each digital root period for tribonacci sequences has triple patterns in cycles of 13, such as period (1,4,7) or digital root of 4^n.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Formula

G.f.: (3+x-2*x^2)/(1-x-x^2-x^3). - Vincenzo Librandi, Jan 01 2017

Mathematica

RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + a[n - 3], a[0] == 3, a[1] == 4, a[2] == 5}, a, {n, 38}] (* Michael De Vlieger, Dec 31 2016 *)
LinearRecurrence[{1, 1, 1}, {3, 4, 5}, 40] (* Vincenzo Librandi, Jan 01 2017 *)

Crossrefs

Cf. A000073, A001590, A081172, A000213.

Sequence in context: A010752 A049929 A262192 * A289121 A060738 A090651

Adjacent sequences: A280305 A280306 A280307 * A280309 A280310 A280311

Keyword

nonn,easy

Author

Peter M. Chema, Dec 31 2016

Status

approved