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A213816 Tribonacci sequences A000073 and A001590 interleaved. 3
1, 1, 1, 2, 2, 3, 4, 6, 7, 11, 13, 20, 24, 37, 44, 68, 81, 125, 149, 230, 274, 423, 504, 778, 927, 1431, 1705, 2632, 3136, 4841, 5768, 8904, 10609, 16377, 19513, 30122, 35890, 55403, 66012, 101902, 121415, 187427, 223317, 344732, 410744, 634061, 755476, 1166220 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Bruce (see link) formulated the sequence using the following two equations:

a(2n) = a(2n-1)+a(2n-3),

a(2n+1) = a(2n-1)+a(2n-2),

with n>1 and initial conditions a(1)=a(2)=a(3)= 1.

These equations lead to a pair of tribonacci-type recurrence equations, for n>2:

a(2n+1) = a(2n-1)+a(2n-3)+a(2n-5),

a(2n+2) = a(2n)+a(2n-2)+a(2n-4).

It could be more appropriate to consider the sequence as a kind of two dimensional tribonacci sequence (a(2n-1),(a(2n)), i.e. as (1, 1), (1, 2), (2, 3), (4, 6), (7, 11), (13, 20), (24, 37), (44, 68), (81, 125), (149, 230), (274, 423), (504, 778), (927, 1431), (1705, 2632), (3136, 4841),... since after the first three initial pairs, the next pair can be obtained by adding three previous pairs component-wise. However, the first three initial pairs (1, 1), (1, 2), (2, 3) are redundant in comparison with the original integer sequence that needs only three initial integers 1, 1 and 1.

One method to construct the two dimensional sequence is by using the well-known tribonacci-related morphism f with f(a) = ab, f(b) = ac, f(c) = a on the monoid of strings over the alphabet {a, b, c}. Using component-wise map, the following sequence of pairs is obtained: (c,b), (a, ac), (ab, aba), (abac, abacab), (abacaba, abacabaabac), (abacabaabacab, abacabaabacababacaba), ...; which is initialized by the pair (c,b) and any pair (x,y) is followed by (f(x),f(y)). The length of every string in every component consitutes the two-dimensional sequence.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Ian Bruce, A Modified Tribonacci Sequence, The Fibonacci Quarterly 22, no.3 (1984), 244-246.

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,1).

FORMULA

G.f.: x*(1+x+x^3)/(1-x^2-x^4-x^6). [corrected by G. C. Greubel, Nov 03 2018]

a(1) = a(2) = a(3) = 1; for n>1:

a(2n) = a(2n-1) + a(2n-3),

a(2n+1) = a(2n-1) + a(2n-2).

EXAMPLE

The first 14 pairs of string and its length are

(c, 1);

(b, 1);

(a, 1);

(ac, 2);

(ab, 2);

(aba, 3);

(abac, 4);

(abacab, 6);

(abacaba, 7);

(abacabaabac, 11);

(abacabaabacab, 13);

(abacabaabacababacaba, 20);

(abacabaabacababacabaabac, 24);

(abacabaabacababacabaabacabacabaabacab, 37); ...

MAPLE

with(StringTools):

# The following procedure defines the morphism f

Morphf := proc (x::string) local Start, L, Init, i;

Init := x;

L := length(Init);

Start := 1;

for i from Start to 2*L do

   if Init[i] = "c" then

         Init := Insert(Init, i, "a");     i := i+1; L := L+1;

         Init := Delete(Init, i-1 .. i-1); i := i-1; L := L-1;

   elif Init[i] = "b" then

         Init := Insert(Init, i, "ac");    i := i+2; L := L+2;

         Init := Delete(Init, i-2 .. i-2); i := i-1; L := L-1;

   elif Init[i] = "a" then

         Init := Insert(Init, i, "b");     i := i+1; L := L+1;

   end if;

end do;

eval(Init);

end proc:

#The following procedure is intended to create sequence of

#strings c, b, a, ac, ab, aba, abac, ..., etc, obtained by

#iterating the morphism f n times but it starts from the third

#string "a", i.e. leaving the first two strings "c" and "b"

#behind:

TribWord := proc (x1, x2::string, n) local A, B, C, i;

A := x1; B := x2;

for i to n do

   if type(i, odd) = true then

      A := Morphf(A);

      C := A;

   else

      B := Morphf(B); C := B

   end if;

end do;

eval(C);

end proc;

#The following command will print a(1), a(2), ..., a(30).

for i to 30 do

printf("%d%s", length(TribWord("c", "b", i-2)), `, `);

end do

MATHEMATICA

LinearRecurrence[{0, 1, 0, 1, 0, 1}, {1, 1, 1, 2, 2, 3}, 48] (* Bruno Berselli, Jun 25 2012 *)

PROG

(PARI) x='x+O('x^50); Vec(x*(1+x+x^3)/(1-x^2-x^4-x^6)) \\ G. C. Greubel, Nov 03 2018

(MAGMA) I:=[1, 1, 1, 2, 2, 3]; [n le 6 select I[n] else Self(n-2) + Self(n-4) + Self(n-6): n in [1..50]]; // G. C. Greubel, Nov 03 2018

CROSSREFS

Cf. A000073.

Sequence in context: A320270 A274156 A094860 * A065417 A005860 A266900

Adjacent sequences:  A213813 A213814 A213815 * A213817 A213818 A213819

KEYWORD

nonn,easy

AUTHOR

Loeky Haryanto, Jun 22 2012

STATUS

approved

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Last modified July 9 01:30 EDT 2020. Contains 335537 sequences. (Running on oeis4.)