A249576 -id:A249576 - cp's OEIS frontend

A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

2, -1, 1, -4, 3, -2, 10, -7, 5, -24, 17, -12, 58, -41, 29, -140, 99, -70, 338, -239, 169, -816, 577, -408, 1970, -1393, 985, -4756, 3363, -2378, 11482, -8119, 5741, -27720, 19601, -13860, 66922, -47321, 33461, -161564, 114243, -80782, 390050, -275807, 195025, -941664, 665857, -470832

Offset: 0

Russell Walsmith, Nov 01 2014

The sequence comprises, in reverse order, numbers to the right of a(0) in A249576.

			M^-1 = [[1,-4,4][-1,3,-2][1,-2,1]]. sqrt(M[1,3]) = 2; M[3,3] = M[1,1] = -1; M[3,1] = 1. Hence a(0) = 2; a(1) = -1; a(2) = 1.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,-2,0,0,1).

Cf. A249576, A000129, A001333, A163271, A077985.

LinearRecurrence[{0,0,-2,0,0,1},{2,-1,1,-4,3,-2},50] (* Harvey P. Dale, Aug 02 2024 *)

Vec(-(x^4+x^2-x+2)/(x^6-2*x^3-1) + O(x^100)) \\ Colin Barker, Nov 02 2014

a(n) = -2*a(n-3)+a(n-6); G.f.: -(x^4+x^2-x+2) / (x^6-2*x^3-1). - Colin Barker, Nov 02 2014

A092550 Expansion of -x(1+x+x^2+x^4)/(-1+2x^3+x^6).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 7, 5, 12, 17, 12, 29, 41, 29, 70, 99, 70, 169, 239, 169, 408, 577, 408, 985, 1393, 985, 2378, 3363, 2378, 5741, 8119, 5741, 13860, 19601, 13860, 33461, 47321, 33461, 80782, 114243, 80782, 195025, 275807, 195025, 470832, 665857

Offset: 1

Roger L. Bagula, Apr 08 2004

nonn

If prefaced with a 1: denominators in convergents to barover:[1, 0, 1] as follows:
1,....0,....1,....1,....0,....1,....1,....0,....1,....
1/1,..0/1,..1/2,..1/3...1/2...2/5...3/7...2/5...5/12,...;
Gary W. Adamson, Mar 25 2014
For k(n), a term in A249576, k(n+6) mod (k(n+5)) = a(n). - Russell Walsmith, Nov 27 2014

Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.
D. Panario, M. Sahin, Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1).

Cf. A000045.

m=3 fib[n_Integer?Positive] :=fib[n] =If[Mod[n, m]==0, fib[n-2], fib[n-1]+fib[n-2]] fib[0]=fib[1] = fib[2] = 1 digits=50 a=Table[fib[n], {n, 1, digits}]
LinearRecurrence[{0,0,2,0,0,1},{1,1,1,2,3,2},50] (* Harvey P. Dale, Jan 13 2015 *)

a(n) = a(n-2) if 3|n, otherwise a(n)= a(n-1)+a(n-2).
From R. J. Mathar, Dec 08 2010: (Start)
a(n)= +2*a(n-3) +a(n-6).
G.f.: -x*(1+x+x^2+x^4)/(-1+2*x^3+x^6).
a(3n+1) = A000129(n+1). a(3n)=A000129(n). a(3n+2)= A078057(n). (End)

Edited, and new name, Joerg Arndt, Sep 17 2013

cp's OEIS Frontend

A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

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A092550 Expansion of -x(1+x+x^2+x^4)/(-1+2x^3+x^6).

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cp's OEIS Frontend

A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A092550 Expansion of -x*(1+x+x^2+x^4)/(-1+2*x^3+x^6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A092550 Expansion of -x(1+x+x^2+x^4)/(-1+2x^3+x^6).