OFFSET
0,2
COMMENTS
a(n)=( A000045(k+2) )^3 if n=3k, a(n)=( A000045(k+2) )^3 * A000045(k+3) if n=3k+1, a(n)= A000045(k+2) * ( A000045(k+3) )^2 if n=3k+2. Number of all subsets of the set {1,2,...,n} which do not contain two elements whose difference is 3. a(n) is number of compositions of n+3 into elements of the set {1,2,4,5,6}, but with condition that 2 succeed only 2 or 4. Number of all permutations of {1,2,...,n+3} satisfying p(i)-i in {-3,0,3}. - Vladimir Baltic, Feb 17 2003
REFERENCES
M. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461 doi:10.1016/j.amc.2009.12.069, Table 1 k=3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 16.
G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978), 113-118.
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
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,1,1,-1,-1).
FORMULA
Recurrence: a(n) = a(n-1)+a(n-2)-a(n-3)+a(n-4)+a(n-5)+a(n-6)-a(n-7)-a(n-8) G.f.: -(x^7+2*x^6+x^5-x^4-3*x^3-x^2-x-1)/(x^8+x^7-x^6-x^5-x^4+x^3-x^2-x+1). - Vladimir Baltic, Feb 17 2003
a(n) = F(floor(n/3) + 3)^(n mod 3)*F(floor(n/3) + 2)^(3 - (n mod 3)) where F(n) is the n-th Fibonacci number. - David Nacin, Feb 29 2012
EXAMPLE
For example, a_4=12 and 12 subsets are: emptyset, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {2,3}, {2,4}, {3,4}, {1,2,3}, {2,3,4}. Corresponding compositions of 7=4+3 are: 1+1+1+1+1+1+1+1, 4+1+1+1, 1+4+1+1, 1+1+4+1, 1+1+1+4, 5+1+1, 4+2+1, 1+5+1, 1+4+2, 1+1+5, 6+1 and 1+6.
MAPLE
A006500:=-(2*z**6+z**7-z**4+z**5-3*z**3-z**2-z-1)/(z**6-z**3-1)/(z**2+z-1); # Conjectured by Simon Plouffe in his 1992 dissertation.
MATHEMATICA
Table[Fibonacci[Floor[n/3] + 3]^Mod[n, 3] * Fibonacci[Floor[n/3] + 2]^(3 - Mod[n, 3]), {n, 0, 40}] (* David Nacin, Feb 29 2012 *)
Table[Product[Fibonacci[Floor[(n + i)/3] + 2], {i, 0, 2}], {n, 0, 30}] (* David Nacin, Mar 07 2012 *)
LinearRecurrence[{1, 1, -1, 1, 1, 1, -1, -1}, {1, 2, 4, 8, 12, 18, 27, 45}, 40] (* David Nacin, Mar 07 2012 *)
PROG
(Python)
def a(n, adict={0:1, 1:2, 2:4, 3:8, 4:12, 5:18, 6:27, 7:45}):
if n in adict:
return adict[n]
adict[n]=a(n-1)+a(n-2)-a(n-3)+a(n-4)+a(n-5)+a(n-6)-a(n-7)-a(n-8)
return adict[n] # David Nacin, Mar 07 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved