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A051049
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Number of moves needed to solve an n-ring baguenaudier if the two end rings can be moved simultaneously.
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8
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1, 1, 4, 7, 16, 31, 64, 127, 256, 511, 1024, 2047, 4096, 8191, 16384, 32767, 65536, 131071, 262144, 524287, 1048576, 2097151, 4194304, 8388607, 16777216, 33554431, 67108864, 134217727, 268435456, 536870911, 1073741824
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The row sums of triangle A166692. [Paul Curtz, Oct 20 2009]
The inverse binomial transform equals (-1)^n*A062510(n) with an extra leading term 1. [Paul Curtz, Oct 20 2009]
This is the sequence A(1,1;1,2;1)of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [From Wolfdieter Lang, Oct 18 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
W. Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010]
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FORMULA
| a(n) = (2^(n+1)-(1+(-1)^(n+1)))/2. - Paul Barry, Apr 24 2003
a(n+2) = a(n+1)+2*a(n)+1, a(0)=a(1)=1 - Paul Barry, May 01 2003
G.f.: (1-x+x^2)/((1-x^2)*(1-2*x)); E.g.f.: exp(2*x)-sinh(x). - Paul Barry, Sep 19 2003
a(n) = (sum{k=0..n, 2^k}+(-1)^n)/2=(A000225(n+1)+(-1)^n)/2. - Paul Barry, May 27 2003
(a(n+1)-a(n))/3 = A001045(n) - Paul Barry, May 27 2003
a(n) = sum{k=0..floor(n/2), C(n+1, 2k) } - Paul Barry, May 27 2003
a(n) = sum{k=0..n, C(n, k)+(-1)^(n-k)}-1 - Paul Barry, Jul 21 2003
a(n) = sum{k=0..n, sum{j=0..n-k, if(mod(j-k, 2)=0, binomial(n-k, j), 0}}; - Paul Barry, Jan 25 2005
Row sums of triangle A135221 - Gary W. Adamson, Nov 23 2007
a(n) = A001045(n+1) + A000975(n+1) - A000079(n) [Paul Curtz, Oct 20 2009]
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=1=a(1), a(2)=4. Observed by G. Detlefs. See the W.Lang link. [From Wolfdieter Lang, Oct 18 2010]
a(n) = 3*a(n-1)-2*a(n-2) +3*(-1)^n [From Gary Detlefs, Dec 21 2010]
a(n) = 3* A000975(n-1) + 1, n>0 [From Gary Detlefs, Dec 21 2010]
a(n+2) = A001969(2^n+1) + A000069(2^n); evil + odious [Johannes W. Meijer, Jun 24 2011, Jun 26 2011]
E.g.f.: exp(2x)-sinh(x)=Q(0); Q(k)=1-k!*(x^(k+1))/((2k+1)!*(2^k) -2*(((2k+1)!*(2^k))^2)/( (2k+1)!*(2^(k+1))-(x^k)*((k+1)!)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 16 2011
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MAPLE
| A051049 := proc(n): (2^(n+1)-(1+(-1)^(n+1)))/2 end: seq(A051049(n), n=0..30); [Johannes W. Meijer, Jun 24 2011]
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MATHEMATICA
| b[n_?EvenQ] := 2^(n - 1) - 1; b[n_?OddQ] := 2^(n - 1); Table[b[n], {n, 50}]]
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PROG
| (MAGMA) [(2^(n+1)-(1+(-1)^(n+1)))/2: n in [0..40]]; // Vincenzo Librandi, Aug 14 2011
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CROSSREFS
| Cf. A000975. Row sums of A131086.
Cf. A135221.
Sequence in context: A025619 A093210 A133600 * A108122 A192800 A027609
Adjacent sequences: A051046 A051047 A051048 * A051050 A051051 A051052
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KEYWORD
| nonn
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| Edited and information added by Johannes W. Meijer, Jun 24, 2011.
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