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A004187 a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1. 49
0, 1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784, 34111385, 233802911, 1602508992, 10983760033, 75283811239, 516002918640, 3536736619241, 24241153416047, 166151337293088, 1138818207635569, 7805576116155895, 53500214605455696, 366695926122033977 (list; graph; refs; listen; history; text; internal format)
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 . A004178 (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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

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, 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.

M. 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

W. 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).

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<=k<=n} A101950(n,k)*6^k. - Philippe Deléham, Feb 10 2012

a(n) = (A081072(n)/3)-1. - Martin Ettl, Nov 11 2012

Product {n >= 1} (1 + 1/a(n)) = 1/5*(5 + 3*sqrt(5)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/14*(5 + 3*sqrt(5)). - Peter Bala, Dec 23 2012

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

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 xrange(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 A186653 A231378

Adjacent sequences:  A004184 A004185 A004186 * A004188 A004189 A004190

KEYWORD

nonn,easy,changed

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified June 26 23:21 EDT 2017. Contains 288777 sequences.