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A055010
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a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.
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39
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0, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887
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OFFSET
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0,2
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COMMENTS
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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.
a(n) = A164874(n,1), n>0; subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009
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+1) = A196168(A000079(n)). - Reinhard Zumkeller, Oct 28 2011
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
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LINKS
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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).
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FORMULA
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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
a(n) = A099258(A033484(n)-1) = floor(A033484(n)/2). - Reinhard Zumkeller, Oct 09 2004
a(n) = (-1 + 3*2^(n-1))*(1-(C(2*n,n) mod 2)), with n>=0. - Paolo P. Lava, Nov 20 2008
G.f.: x*(2-x)/((1-x)*(1-2*x)). - Philippe Deléham, Oct 04 2011
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EXAMPLE
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a(3) = 3*2^2 - 1 = 3*4 - 1 = 11.
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MATHEMATICA
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Join[{0}, 3*2^Range[0, 34]-1] (* Harvey P. Dale, May 05 2013 *)
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PROG
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(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
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CROSSREFS
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Cf. A007505 for primes in this sequence. Apart from initial term, same as A052940 and A083329.
Cf. A266550 (independence number of the n-Mycielski graph).
Sequence in context: A086219 A153893 A083329 * A266550 A081973 A334276
Adjacent sequences: A055007 A055008 A055009 * A055011 A055012 A055013
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley, May 31 2000
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STATUS
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approved
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