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A121449

Expansion of (1 - 3*x + 2*x^2)/(1 - 4*x + 3*x^2 + x^3).
(history; published version)

#41 by Joerg Arndt at Fri Feb 03 01:36:17 EST 2023

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reviewed

approved

#40 by Michel Marcus at Fri Feb 03 01:35:45 EST 2023

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proposed

reviewed

#39 by Jon E. Schoenfield at Thu Feb 02 23:34:06 EST 2023

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editing

proposed

#38 by Jon E. Schoenfield at Thu Feb 02 23:33:42 EST 2023

NAME

Expansion of (1 - 3*x + 2*x^2)/(1 - 4*x + 3*x^2 + x^3).

COMMENTS

In the cited Witula-Slota-Warzynski paper three so-called quasi-Fibonacci numbers A(n;d), B(n;d) and C(n;d), where n = 0,1,..., d \in C are discussed. These numbers are created by each of the following relations:

(1+d*c(j))^n = A(n;d) + B(n;d)*c(j) + C(n;d)*c(2*j), for every j=1,2,4, where c(j):=2*cos(2Pi2*Pi*j/7).

A(n;-1) =sum Sum_{k=0,..,n} binomial(n,k)*(A(k;1)*A(n-k;1)-A(k;1)*B(n-k;1)-B(k;1)*C(n-k;1)-A(n-k;1)*C(k;1)+2*B(n-k;1)*C(k;1)-C(k;1)*C(n-k;1)),

B(n;-1) =sum Sum_{k=0,..,n} binomial(n,k)*(-A(k;1)*B(n-k;1)+A(k;1)*C(n-k;1)+B(k;1)*B(n-k;1)-A(n-k;1)*C(k;1)+B(n-k;1)*C(k;1)-C(k;1)*C(n-k;1)), and

C(n;-1) =sum Sum_{k=0,..,n} binomial(n,k)*(-A(k;1)*B(n-k;1)+A(n-k;1)*B(k;1)+B(k;1)*B(n-k;1)-B(k;1)*C(n-k;1)-A(n-k;1)*C(k;1)) (see identities (3.50-52) and (3.61-63) in the Witula-Slota-Warzynski paper).

FORMULA

7*a(n) = (2-c(4))*(1-c(1))^n + (2-c(1))*(1-c(2))^n + (2-c(2))*(1-c(4))^n = (s(2))^2*(1-c(1))^n + (s(4))^2*(1-c(2))^n + (s(1))^2*(1-c(4))^n, where c(j):=2*cos(2Pi2*Pi*j/7) and s(j):=2*sin(2Pi2*Pi*j/7) -- it is the special case, for d=-1, of the Binet's formula for the respective quasi-Fibonacci number A(n;d) discussed in the Witula-Slota-Warzynski paper. - Roman Witula, Aug 07 2012

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approved

editing

#37 by Charles R Greathouse IV at Thu Sep 08 08:45:27 EDT 2022

PROG

(MAGMAMagma) I:=[1, 1, 3]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#36 by Peter Luschny at Fri Jul 02 16:46:57 EDT 2021

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reviewed

approved

#35 by Michel Marcus at Fri Jul 02 16:44:17 EDT 2021

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proposed

reviewed

#34 by Michael De Vlieger at Fri Jul 02 16:09:15 EDT 2021

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editing

proposed

#33 by Michael De Vlieger at Fri Jul 02 16:09:13 EDT 2021

LINKS

Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.

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approved

editing

#32 by Bruno Berselli at Thu Apr 19 03:26:49 EDT 2018

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reviewed

approved