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A052955
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a(2n) = 2*2^n - 1, a(2n+1) = 3*2^n - 1.
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14
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1, 2, 3, 5, 7, 11, 15, 23, 31, 47, 63, 95, 127, 191, 255, 383, 511, 767, 1023, 1535, 2047, 3071, 4095, 6143, 8191, 12287, 16383, 24575, 32767, 49151, 65535, 98303, 131071, 196607, 262143, 393215, 524287, 786431, 1048575, 1572863, 2097151, 3145727
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) = least k such that A056792(k) = n.
One quarter of the number of positive integer (n+2) X (n+2) arrays with every 2 X 2 subblock summing to 1. [From R. H. Hardin (rhhardin(AT)att.net), Sep 29 2008]
Number of length n+1 left factors of Dyck paths having no DUUs (here U=(1,1) and D=(1,-1)). Example: a(4)=7 because we have UDUDU, UUDDU, UUDUD, UUUDD, UUUDU, UUUUD, and UUUUU (the paths UDUUD, UDUUU, and UUDUU do not qualify).
Number of binary palindromes < 2^n [see A006995]. - Hieronymus Fischer, Feb 03 2012
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1026
Index to sequences with linear recurrences with constant coefficients, signature (1,2,-2).
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FORMULA
| -1+Sum(1/4*(3+4*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z^2))
a(n)=2^(n/2)(3sqrt(2)/4+1-(3sqrt(2)/4-1)(-1)^n)-1. - Paul Barry (pbarry(AT)wit.ie), May 23 2004
G.f.: (1 + x - x^2)/((1 - x)*(1 - 2*x^2)). a(0) = 1, a(1) = 2, a(n+2) = 2*a(n) + 1.
a(n) = 1 + partial sum of A016116(k-1). - Robert G. Wilson v Jun 05 2004
A132340(a(n)) = A027383(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2007
a(n)=A027383(n-1)+1 for n>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
a(n)=A132666(a(n+1)-1). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
a(n)=A132666(a(n-1))+1 for n>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
A132666(a(n))=a(n+1)-1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
a(n) = A027383(n+1)/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
a(n) = 2*a(n-2) + 1 [From Richard Torres (richyrich12887(AT)yahoo.com), Apr 22 2009]
a(n) = 2*a(n-2)+1, a(0)=1, a(1)=2.
a(n) = (5-(-1)^n)/2*2^floor(n/2)-1. - Hieronymus Fischer, Feb 03 2012
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MAPLE
| spec := [S, {S=Prod(Sequence(Prod(Union(Z, Z), Z)), Union(Sequence(Z), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n]/2, n=2..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-2]+2 od: seq(a[n]+1, n=0..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
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MATHEMATICA
| f[n_] := If[EvenQ[n], 2^(n/2 + 1) - 1, 3*2^((n - 1)/2) - 1]; Table[ f[n], {n, 0, 41}] (from Robert G. Wilson v Jun 05 2004)
Clear[a, b, c, m, n]; a[m_] := Table[If[IntegerDigits[n, 2] == Reverse[IntegerDigits[n, 2]], IntegerDigits[n, 2], {0}], {n, 0, 2^m}]; b[m_] := Union[Sort[a[m]]]; c[m_] := Table[FromDigits[b[m][[n]], 2], {n, 1, Length[b[m]]}]; Table[Length[c[m]], {m, 1, 12}] [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 06 2008]
a[0] = 1; a[1] = 2; a[n_] := a[n]= 2 a[n - 2] + 1; Array[a, 42, 0]
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PROG
| (Perl)# command line argument tells how high to take n
# Beyond a(38) = 786431 you may need a special code to handle large integers
# Mark A. Mandel 2010-12-29
$lim = shift;
sub show{};
$n = $incr = $P = 1;
show($n, $incr, $P);
$incr = 1;
for $n (2..$lim) {
$P += $incr;
show($n, $P, $incr, $P);
$incr *=2 if ($n % 2); # double the increment after an odd n
}
sub show {
my($n, $P) = @_;
printf("%4d\t%16g\n", $n, $P);
}
(PARI) a(n)=(2+n%2)<<(n\2)-1 \\ Charles R Greathouse IV, Jun 19 2011
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CROSSREFS
| Cf. A000225 for even terms, A055010 for odd terms. See also A056792.
Essentially 1 more than A027383, 2 more than A060482. [Comment corrected by Klaus Brockhaus, Aug 09 2009]
Union of A000225 & A055010.
For partial sums see A027383.
Cf. A132666.
Sequence in context: A116601 A024792 A055771 * A177485 A165801 A022480
Adjacent sequences: A052952 A052953 A052954 * A052956 A052957 A052958
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KEYWORD
| easy,nonn,changed
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| Formula and more terms from Henry Bottomley (se16(AT)btinternet.com), May 03 2000. Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 29 2001.
Added Perl code to generate sequence -- 2010-12-29, Mark A. Mandel (thnidu aT g ma(il) doT c0m)
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