A006499 Number of restricted circular combinations.

1, 3, 9, 12, 16, 28, 49, 77, 121, 198, 324, 522, 841, 1363, 2209, 3572, 5776, 9348, 15129, 24477, 39601, 64078, 103684, 167762, 271441, 439203, 710649, 1149852, 1860496, 3010348, 4870849, 7881197, 12752041, 20633238, 33385284, 54018522, 87403801, 141422323, 228826129

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n >= 2, a(n) is also the number of ways to use white dominoes and black and white squares to tile this strip of length n which has a 4-cell zig-zag in the center with the rule that black squares must appear exactly twice and can only appear in the four center zig-zag cells. Here is the strip of length 7:

_

___|_|___

|||_|||_|_|,

|_|

and here is one of the a(7) = 77 ways to tile it according to our rules (the two black squares in the center are identified with X):

_

___|X|___

|||___||__|. - Greg Dresden and Emma Li, Sep 06 2024

|X|

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978), 113-118.
• T. Guardia and D. Jiménez, Fiboquadratic Sequences and Extensions of the Cassini Identity Raised From the Study of Rithmomachia, arXiv preprint arXiv:1509.03177 [math.HO], 2015-2016.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Crossrefs

Cf. A000032, A056594.

Programs

Maple

A006499:=-(1+2*z+6*z**2+2*z**3)/((z**2+z-1)*(1+z**2)); # [conjectured (correctly) by Simon Plouffe in his 1992 dissertation]

Mathematica

CoefficientList[ Series[(1 + 2x + 6x^2 + 2x^3)/((1 + x^2)(1 - x - x^2)), {x, 0, 35}], x] (* Robert G. Wilson v, Feb 25 2005 *)

Formula

a(n) = A000032(n+2) - 2*A056594(n) - A056594(n-1).

G.f.: (1+2x+6x^2+2x^3)/((1+x^2)*(1-x-x^2)). - Ralf Stephan, Apr 23 2004

From Ralf Stephan, Jun 09 2005: (Start)

a(n) = Lucas(n+2) - i^n - (-i)^n - (1/2)*i^(n-1) - (1/2)*(-i)^(n-1) where i = sqrt(-1).

a(n) = (1/2)*(Lucas(n+2) - 3*(-1)^floor(n/2) + (-1)^floor((n-1)/2)). (End)

From Greg Dresden, Jan 15 2024: (Start)

a(n) = Lucas(floor(n/2+1))*Lucas(ceiling(n/2+1));

a(2*n) = Lucas(n+1)^2;

a(2*n+1) = Lucas(n+1)*Lucas(n+2). (End)

E.g.f.: exp(x/2)*(3*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)) - 2*cos(x) - sin(x). - Stefano Spezia, Mar 12 2024

Extensions

a(36)-a(38) from Stefano Spezia, Mar 12 2024