|
| |
|
|
A000051
|
|
2^n + 1.
(Formerly M0717 N0266)
|
|
128
|
|
|
|
2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
Same as Pisot sequence L(2,3).
Length of the continued fraction for sum(k=0,n,1/3^(2^k)). - Benoit Cloitre, Nov 12 2003
See also A004119 for a(n) = 2a(n-1)-1 with first term =1 . - Philippe Deléham, Feb 20 2004
From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg, May 31 2005
Numbers n for which the expression 2^n/(n-1) is an integer. - Paolo P. Lava, May 12 2006
a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007
a(n) = A024036(n)/A000225(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = a(n-1)-th odd numbers (A004273) for n >= 1. [From Jaroslav Krizek, Apr 25 2009]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)charpoly(A,3). [From Milan Janjic, Jan 27 2010]
a(n)*A000225(n) = A000225(2*n); a(n) = A173786(n,0). - Reinhard Zumkeller, Feb 28 2010
First differences of A006127. - Reinhard Zumkeller, Apr 14 2011
The prime numbers in this sequence form A019434, the Fermat primes.
Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4,... - R. J. Mathar, Aug 10 2012
Is the mentioned Pisano period lengths (see above) the same as A007733?
Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012
|
|
|
REFERENCES
|
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
Ivan Panchenko, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Fermat-Lucas Number
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2)
|
|
|
FORMULA
|
a(n) = 2*a(n-1) - 1 = 3*a(n-1) - 2*a(n-2).
G.f.: (2-3*x)/((1-x)*(1-2*x)).
First differences of A052944. - Emeric Deutsch, Mar 04 2004
a(0) = 1, then a(n) = (Sum i=0..n-1 a(i)) - (n-2). - Gerald McGarvey, Jul 10 2004
Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - Ross La Haye, Feb 07 2005
Equals binomial transform of [2, 1, 1, 1,...]. - Gary W. Adamson, Apr 23 2008
a(n) = A000079(n)+1. - Omar E. Pol, May 18 2008
E.g.f.: exp(x) + exp(2*x). - Mohammad K. Azarian, Jan 02 2009
Contribution from Peter Luschny, Apr 20 2009: (Start)
A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)
If p[i]=fibonacci(i-4) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. [From Milan Janjic, May 08 2010]
a(n+2) = a(n) + a(n+1) + A000225(n). [Ivan N. Ianakiev, Jun 24 2012]
G.f.: G(0) where G(k) = 1 + 2^k/(1 - x/(x + 2^k/G(k+1) )) (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 05 2012
E.g.f.: E(0), where E(k)= 1 + 1/(2^k - 2*x*4^k/(2*x*2^k + (k+1)/E(k+1))); (continued fraction).
G.f.: Q(0), where Q(k)= 1 + 1/(2^k - 2*x*4^k/(2*x*2^k + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
|
|
|
MAPLE
|
A000051:=-(-2+3*z)/(2*z-1)/(z-1); [Simon Plouffe in his 1992 dissertation.]
a := n -> add(binomial(n, k)*bernoulli(n-k, 1)*2^(k+1)/(k+1), k=0..n); [Peter Luschny, Apr 20 2009]
|
|
|
MATHEMATICA
|
Table[2^n + 1, {n, 0, 33}]
|
|
|
PROG
|
(PARI) a(n)=if(n<0, 0, 2^n+1)
(Haskell)
a000051 = (+ 1) . a000079
a000051_list = iterate ((subtract 1) . (* 2)) 2
-- Reinhard Zumkeller, May 03 2012
|
|
|
CROSSREFS
|
Apart from the initial 1, identical to A094373..
See A008776 for definitions of Pisot sequences. Cf. A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600 - A074624.
Cf. A052944.
Column 2 of array A103438.
Cf. A000079.
Cf. A005126, A176691, A194455.
Sequence in context: A005257 A091697 A109740 * A094373 A213705 A061902
Adjacent sequences: A000048 A000049 A000050 * A000052 A000053 A000054
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
