A075092 Sum of generalized tribonacci numbers (A001644) and reflected generalized tribonacci numbers (A073145).

6, 0, 2, 12, 6, 20, 50, 56, 134, 264, 402, 836, 1542, 2652, 5154, 9392, 16902, 31824, 58082, 106172, 197126, 360932, 662994, 1223784, 2245766, 4130520, 7606770, 13976436, 25711622, 47310252, 86978370, 160002656, 294324230, 541249952

Offset

0,1

Comments

Conjecture: a(n) >= 0.

For n > 2, a(n) is the number of cyclic sequences (q1, q2, ..., qn) consisting of zeros, ones and twos such that each triple contains 0 and 1 at least once, provided the positions of the zeros and ones are fixed on a circle. For example, a(5)=20 because only the sequences (00101), (01001), (01010), (01011), (01012), (01021), (01101), (01201), (02101), (20101) and those obtained from them by exchanging 0 and 1 contain 0 and 1 in each triple (including triples q4, q5, q1 and q5, q1, q2). For n = 1, 2 the statement is still true provided we allow the sequence to wrap around itself on a circle. E.g., a(2) = 2 since only sequences 01 and 10 can be wrapped so one obtains (010) and (101), respectively. - Wojciech Florek, Nov 25 2021

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.

W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31(1) (1993), 2-6.

Index entries for linear recurrences with constant coefficients, signature (0,1,4,1,0,-1).

Formula

a(n) = a(n-2) + 4*a(n-3) + a(n-4) - a(n-6), a(0)=6, a(1)=0, a(2)=2, a(3)=12, a(4)=6, a(5)=20.

G.f.: (6 - 4*x^2 - 12*x^3 - 2*x^4)/(1 - x^2 - 4*x^3 - x^4 + x^6).

Mathematica

CoefficientList[Series[(6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6), {x, 0, 40}], x]

Prog

(PARI) my(x='x+O('x^40)); Vec((6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6)) \\ G. C. Greubel, Apr 13 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6) ));  // G. C. Greubel, Apr 13 2019
(Sage) ((6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 13 2019

Crossrefs

Cf. A001644, A073145, A075091, A294627.

Sequence in context: A371923 A201331 A366349 * A152244 A283634 A179641

Adjacent sequences: A075089 A075090 A075091 * A075093 A075094 A075095

Keyword

easy,nonn

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 31 2002

Status

approved