A006499 Number of restricted circular combinations. (Formerly M2768)

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

0,2

References

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

Links

Table of n, a(n) for n=0..38.

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).

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

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 *)

Crossrefs

Cf. A000032, A056594.

Sequence in context: A103531 A333441 A108860 * A310320 A274676 A310321

Adjacent sequences: A006496 A006497 A006498 * A006500 A006501 A006502

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved