0,1

These Esanacci numbers follow the same pattern as Lucas, generalized tribonacci (A001644), generalized tetranacci (A073817) and generalized pentanacci(A074048) numbers. The closed form is a(n)=r1^n+r^2^n+r3^n+r4^n+r5^n+r6^n, with r1, r2, r3, r4, r5, r6 roots of the characteristic polynomial. a(n) is also the trace of A^n, where A is the esamatrix ((1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (1, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 1), (1, 0, 0, 0, 0, 0).

Mario Catalani, "Polymatrix and Generalized Polynacci Numbers", paper in progress.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

T. D. Noe, Table of n, a(n) for n=0..200

M. Catalani, Polymatrix and Generalized Polynacci Numbers

E. Weisstein, Fibonacci n-Step

a(n)=a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6), a(0)=6, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31. G.f.: (6-5x-4x^2-3x^3-2x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)

CoefficientList[Series[(6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6), {x, 0, 40}], x]

Cf. A000078, A001630, A001644, A000032, A073817, A074048.

Sequence in context: A086316 A021167 A085677 this_sequence A101023 A160199 A144540

Adjacent sequences: A074581 A074582 A074583 this_sequence A074585 A074586 A074587

easy,nonn

Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002