A083099 a(n) = 2*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.

0, 1, 2, 10, 32, 124, 440, 1624, 5888, 21520, 78368, 285856, 1041920, 3798976, 13849472, 50492800, 184082432, 671121664, 2446737920, 8920205824, 32520839168, 118562913280, 432250861568, 1575879202816, 5745263575040

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

Keywords

• nonn
• easy

Comments

a(n+1) = a(n) + A083098(n+1). A083098(n+1)/a(n) converges to sqrt(7).

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard, Sep 25 2005

Pisano period lengths: 1, 1, 2, 1, 12, 2, 7, 1, 6, 12, 60, 2,168, 7, 12, 1,288, 6, 18, 12, ... - R. J. Mathar, Aug 10 2012

a(n) is divisible by 2^ceiling(n/2), see formula below. - Ralf Stephan, Dec 24 2013

Connect the center of a regular hexagon with side length 1 with its six vertices. a(n) is the number of paths of length n from the center to any of its vertices. Number of paths of length n from the center to itself is 6*a(n-1). - Jianing Song, Apr 20 2019

References

• John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Project Euler, Problem 752, sequence beta(n).
• Index entries for linear recurrences with constant coefficients, signature (2,6).

Crossrefs

The following sequences (and others) belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A063727, A083098, A083099, A083100, A084057.

Cf. A154245, A291008.

Programs

Magma

[n le 2 select n-1 else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

Maple

A083099 := proc(n)
    option remember;
    if n <= 1 then
        n;
    else
        2*procname(n-1)+6*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

Mathematica

CoefficientList[Series[x/(1-2x-6x^2), {x, 0, 25}], x] (* Adapted for offset 0 by Vincenzo Librandi, Feb 07 2014 *)
Expand[Table[((1 + Sqrt[7])^n - (1 - Sqrt[7])^n)7/(14Sqrt[7]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{2,6}, {0,1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)

PARI

a(n)=([0,1; 6,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

PARI

my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x-6*x^2))) \\ G. C. Greubel, Jan 24 2018

Sage

[lucas_number1(n,2,-6) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

SageMath

A083099=BinaryRecurrenceSequence(2,6,0,1)
[A083099(n) for n in range(41)] # G. C. Greubel, Jun 01 2023

Formula

G.f.: x/(1 - 2*x - 6*x^2).

From Paul Barry, Sep 29 2004: (Start)

E.g.f.: (d/dx)(exp(x)*sinh(sqrt(7)*x)/sqrt(7));

a(n-1) = Sum_{k=0..n} binomial(n, 2k+1)*7^k. (End)

Simplified formula: a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(28). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

a(2n) = 2^n * A154245(n), a(2n+1) = 2^n * (5*A154245(n) - 9*A154245(n-1)). - Ralf Stephan, Dec 24 2013

a(n) = Sum_{k=1,3,5,...<=n} binomial(n,k)*7^((k-1)/2). - Vladimir Shevelev, Feb 06 2014

a(n) = i^(n-1)*6^((n-1)/2)*ChebyshevU(n-1, -i/sqrt(6)). - G. C. Greubel, Jun 01 2023