OFFSET
0,3
COMMENTS
Define the sequence T(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}<a_{n+1}/a_n for n >= 0.A004187(with initial 0 omitted) is T(1,7).
This is a divisibility sequence.
For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 7's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). -John M. Campbell,Jul 08 2011
a(n) and b(n):=A056854(n) are the proper and improper nonnegative solutions of the Pell equation b(n)^2 - 5*(3*a(n))^2 = +4. see the cross-reference toA056854below. -Wolfdieter Lang,Jun 26 2013
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,2,3,4,5,6}. -Milan Janjic,Jan 25 2015
Lim_{n->oo} a(n+1)/a(n) = 2 + 3*phi = 1+A090550= 6.854101... -Wolfdieter Lang,Nov 16 2023
LINKS
Vincenzo Librandi,Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru,Polynomial sequences on quadratic curves,Integers, Vol. 15, 2015, #A38.
K. Andersen, L. Carbone, and D. Penta,Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields,Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
D. Birmajer, J. B. Gil, and M. D. Weiner,On the Enumeration of Restricted Words over a Finite Alphabet,J. Int. Seq. 19 (2016) # 16.1.3, example 12
D. W. Boyd,Linear recurrence relations for some generalized Pisot sequences,Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
Zvonko Cerin,Some alternating sums of Lucas numbers,Centr. Eur. J. Math. vol 3 no 1 (2005) 1-13.
R. Flórez, R. A. Higuita, and A. Mukherjee,Alternating Sums in the Hosoya Polynomial Triangle,Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
A. F. Horadam,Special properties of the sequence W_n(a,b; p,q),Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=7, q=-1.
Milan Janjic,On Linear Recurrence Equations Arising from Compositions of Positive Integers,Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova,Recursive Sequences
Wolfdieter Lang,On polynomials related to powers of the generating function of Catalan's numbers,Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=9.
Index entries for linear recurrences with constant coefficients,signature (7,-1).
FORMULA
G.f.: x/(1-7*x+x^2).
a(n) = S(2*n-1, sqrt(9))/sqrt(9) = S(n-1, 7); S(n, x):= U(n, x/2), Chebyshev polynomials of the 2nd kind,A049310.
a(n) = Sum_{i = 0..n-1} C(2*n-1-i, i)*5^(n-i-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
[A049685(n-1), a(n)] = [1,5; 1,6]^n * [1,0]. -Gary W. Adamson,Mar 21 2008
a(n) =A167816(4*n). -Reinhard Zumkeller,Nov 13 2009
a(n) = (((7+sqrt(45))/2)^n-((7-sqrt(45))/2)^n)/sqrt(45). -Noureddine Chair,Aug 31 2011
a(n+1) = Sum_{k = 0..n}A101950(n,k)*6^k. -Philippe Deléham,Feb 10 2012
a(n) = (A081072(n)/3)-1. -Martin Ettl,Nov 11 2012
fromPeter Bala,Dec 23 2012: (Start)
Product {n >= 1} (1 + 1/a(n)) = (1/5)*(5 + 3*sqrt(5)).
Product {n >= 2} (1 - 1/a(n)) = (1/14)*(5 + 3*sqrt(5)). (End)
FromPeter Bala,Apr 02 2015: (Start)
Sum_{n >= 1} a(n)*x^(2*n) = -A(x)*A(-x), where A(x) = Sum_{n >= 1} Fibonacci(2*n)* x^n.
1 + 5*Sum_{n >= 1} a(n)*x^(2*n) = F(x)*F(-x) = G(x)*G(-x), where F(x) = 1 + A(x) and G(x) = 1 + 5*A(x).
1 + Sum_{n >= 1} a(n)*x^(2*n) = H(x)*H(-x) = I(x)*I(-x), where H(x) = 1 + Sum_{n >= 1} Fibonacci(2*n + 3)*x^n and I(x) = 1 + x + x*Sum_{n >= 1} Fibonacci(2*n - 1)*x^n. (End)
E.g.f.: 2*exp(7*x/2)*sinh(3*sqrt(5)*x/2)/(3*sqrt(5)). -Ilya Gutkovskiy,Jul 03 2016
a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*9^k*binomial(n+k, 2*k+1). -Peter Bala,Jul 17 2023
a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*7^(n-2*k)*binomial(n-k, k). -Greg Dresden,Aug 03 2024
EXAMPLE
a(2) = 7*a(1) - a(0) = 7*7 - 1 = 48. -Michael B. Porter,Jul 04 2016
MAPLE
seq(combinat:-fibonacci(4*n)/3, n = 0.. 30); #Robert Israel,Jan 26 2015
MATHEMATICA
LinearRecurrence[{7, -1}, {0, 1}, 30] (*Harvey P. Dale,Jul 13 2011 *)
CoefficientList[Series[x/(1 - 7*x + x^2), {x, 0, 50}], x] (*Vincenzo Librandi,Dec 23 2012 *)
PROG
(MuPAD) numlib::fibonacci(4*n)/3 $ n = 0..25; //Zerinvary Lajos,May 09 2008
(Sage) [lucas_number1(n, 7, 1) for n in range(27)] #Zerinvary Lajos,Jun 25 2008
(Sage) [fibonacci(4*n)/3 for n in range(0, 21)] #Zerinvary Lajos,May 15 2009
(Magma) [Fibonacci(4*n)/3: n in [0..30]]; //Vincenzo Librandi,Jun 07 2011
(PARI) a(n)=fibonacci(4*n)/3 \\Charles R Greathouse IV,Mar 09, 2012
(PARI) concat(0, Vec(x/(1-7*x+x^2) + O(x^99))) \\Altug Alkan,Jul 03 2016
(Maxima)
a[0]:0$ a[1]:1$ a[n]:=7*a[n-1] - a[n-2]$A004187(n):=a[n]$ makelist(A004187(n), n, 0, 30); /*Martin Ettl,Nov 11 2012 */
(Magma) /* By definition: */ [n le 2 select n-1 else 7*Self(n-1)-Self(n-2): n in [1..23]]; //Bruno Berselli,Dec 24 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Entry improved by comments fromMichael SomosandWolfdieter Lang,Aug 02 2000
STATUS
approved