OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Callan and T. Mansour, Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns, arXiv:1705.00933 [math.CO] (2017), Table 2 No 200 (offset 1 then).
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
Index entries for linear recurrences with constant coefficients, signature (6,-10,5).
FORMULA
a(n) = ((sqrt(5))^(n - 1))*(((sqrt(5) + 1)/2)^n - ((sqrt(5) - 1)/2)^n) + 1. [corrected by Jason Yuen, Sep 06 2024]
a(n) = 1 + 5*(a(n-1) - a(n-2)), a(0) = 1, a(1) = 2.
G.f.: (1 - 2*x)^2/((1 - x)*(1 - 5*x + 5*x^2)). Convolution of A081567 with the sequence 1, -1, -1, -1 (-1 continued). - R. J. Mathar, Sep 06 2008
a(n) = 1 + A030191(n-1). - R. J. Mathar, Jun 20 2011
a(n) = 6*a(n-1) - 10*a(n-2) + 5*a(n-3); a(0) = 1, a(1) = 2, a(2) = 6. - Harvey P. Dale, Aug 20 2011
E.g.f.: exp(x) + (exp((5 + sqrt(5))*x/2) - exp((5 - sqrt(5))*x/2))/sqrt(5). - Franck Maminirina Ramaharo, Nov 09 2018
EXAMPLE
a(2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 2-element chain), namely: the empty map, (1)->(1), (2)->(2), (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2); the mappings are coordinate-wise.
MATHEMATICA
RecurrenceTable[{a[0]==1, a[1]==2, a[n]==1+5(a[n-1]-a[n-2])}, a[n], {n, 30}] (* or *) LinearRecurrence[{6, -10, 5}, {1, 2, 6}, 31] (* Harvey P. Dale, Aug 20 2011 *)
PROG
(Magma) [ n eq 1 select 1 else n eq 2 select 2 else n eq 3 select 6 else 6*Self(n-1)-10*Self(n-2)+ 5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 21 2011
(PARI) Vec((2*x-1)^2/(1-x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Aug 21 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Abdullahi Umar, Aug 25 2008
STATUS
approved