OFFSET
0,2
COMMENTS
Apart from leading term (which should really be 3/2), same as A083329.
Written in binary, a(n) is 1011111...1.
The sequence 2, 5, 11, 23, 47, 95, ... apparently gives values of n such that Nim-factorial(n) = 2. Cf. A059970. However, compare A060152. More work is needed! - John W. Layman, Mar 09 2001
With offset 1, number of (132,3412)-avoiding two-stack sortable permutations.
Number of descents after n+1 iterations of morphism A007413.
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-1). - Milan Janjic, Jan 24 2010
a(n) is the total number of records over all length n binary words. A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all i<j. - Geoffrey Critzer, Jul 18 2020
Called Thabit numbers after the Syrian mathematician Thābit ibn Qurra (826 or 836 - 901). - Amiram Eldar, Jun 08 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric S. Egge and Toufik Mansour, 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, arXiv:math/0205206 [math.CO], 2002.
S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number.
Wikipedia, Thabit number.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
FORMULA
a(n) = A118654(n-1, 4), for n > 0.
a(n) = 2*a(n-1) + 1 = a(n-1) + A007283(n-1) = A007283(n)-1 = A000079(n) + A000225(n + 1) = A000079(n + 1) + A000225(n) = 3*A000079(n) - 1 = 3*A000225(n) + 2.
a(n) = A010036(n)/2^(n-1). - Philippe Deléham, Feb 20 2004
G.f.: x*(2-x)/((1-x)*(1-2*x)). - Philippe Deléham, Oct 04 2011
E.g.f.: (3*exp(2*x) - 2*exp(x) - 1)/2. - Stefano Spezia, Sep 14 2024
EXAMPLE
a(3) = 3*2^2 - 1 = 3*4 - 1 = 11.
MATHEMATICA
Join[{0}, 3*2^Range[0, 34]-1] (* Harvey P. Dale, May 05 2013 *)
PROG
(Magma) [Floor(3*2^(n-1) - 1): n in [0..35]]; // Vincenzo Librandi, May 18 2011
(PARI) a(n)=3*2^n\2 - 1 \\ Charles R Greathouse IV, Apr 08 2016
(Sage) [0]+[3*2^(n-1)-1 for n in (1..35)] # G. C. Greubel, May 06 2019
(GAP) Concatenation([0], List([1..35], n-> 3*2^(n-1)-1)) # G. C. Greubel, May 06 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, May 31 2000
STATUS
approved