A004187 a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784, 34111385, 233802911, 1602508992, 10983760033, 75283811239, 516002918640, 3536736619241, 24241153416047, 166151337293088, 1138818207635569, 7805576116155895, 53500214605455696, 366695926122033977

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 to A056854 below. - 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

The digital root is A253298, which shares its digital root with A253368. - Peter M. Chema, Jul 04 2016

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 sequences related to Chebyshev polynomials.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (7,-1).

Index entries for Pisot sequences

Formula

G.f.: x/(1-7*x+x^2).

a(n) = F(4*n)/3 = A033888(n)/3, where F=A000045 (the Fibonacci sequence).

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

from Peter 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)

From Peter 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

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

Cf. A000027, A001906, A001353, A004254, A001109, A049685, A033888. a(n)=sqrt((A056854(n)^2 - 4)/45).

Second column of array A028412.

Sequence in context: A036829 A164591 A242630 * A180167 A341425 A186653

Adjacent sequences: A004184 A004185 A004186 * A004188 A004189 A004190

Keyword

nonn,easy

Author

N. J. A. Sloane, R. K. Guy

Extensions

Entry improved by comments from Michael Somos and Wolfdieter Lang, Aug 02 2000

Status

approved