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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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61
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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
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OFFSET
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0,2
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COMMENTS
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a(n) is the 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. - R. H. Hardin, Sep 29 2008
Number of length n+1 left factors of Dyck paths having no DUU's (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).
a(n - 1), n > 1, is the number of maximal subsemigroups of the monoid of order-preserving or -reversing partial injective mappings on a set with n elements. - Wilf A. Wilson, Jul 21 2017
Number of monomials of the algebraic normal form of the Boolean function representing the n-th bit of the product 3x in terms of the bits of x. - Sebastiano Vigna, Oct 04 2020
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LINKS
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FORMULA
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a(0)=1, a(1)=2; thereafter a(n) = 2*a(n-2) + 1, n >= 2.
G.f.: (1 + x - x^2)/((1 - x)*(1 - 2*x^2)).
a(n) = -1 + Sum_{alpha = RootOf(-1 + 2*Z^2)} (1/4) * (3 + 4*alpha) * alpha^(-1-n). (That is, the sum is indexed by the roots of the polynomial -1 + 2*Z^2.)
a(n) = 2^(n/2) * (3*sqrt(2)/4 + 1 - (3*sqrt(2)/4 - 1) * (-1)^n) - 1. - Paul Barry, May 23 2004
a(2n+1) = (a(2*n) + a(2*n+2))/2. Combined with a(n) = 2*a(n-2) + 1, n >= 2 and a(0) = 1, this specifies the sequence. - Richard R. Forberg, Nov 30 2013
a(n) = ((5 - (-1)^n)/2)*2^((2*n - 1 + (-1)^n)/4) - 1. - Luce ETIENNE, Sep 20 2014
E.g.f.: (1/4)*exp(-sqrt(2)*x)*(4 - 3*sqrt(2) + (4 + 3*sqrt(2))*exp(2*sqrt(2)*x) - 4*exp(x + sqrt(2)*x)). - Stefano Spezia, Oct 22 2019
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EXAMPLE
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G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 15*x^6 + 23*x^7 + ... - Michael Somos, Jun 24 2018
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MAPLE
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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, Mar 16 2008
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MATHEMATICA
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a[n_]:= If[EvenQ[n], 2^(n/2+1) -1, 3*2^((n-1)/2) -1]; Table[a[n], {n, 0, 41}] (* Robert G. Wilson v, Jun 05 2004 *)
a[0]=1; a[1]=2; a[n_]:= a[n]= 2 a[n-2] +1; Array[a, 42, 0]
a[n_]:= (2 + Mod[n, 2]) 2^Quotient[n, 2] - 1; (* Michael Somos, Jun 24 2018 *)
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PROG
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(Perl)# command line argument tells how high to take n
# Beyond a(38) = 786431 you may need a special code to handle large integers
$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);
}
# Mark A. Mandel (thnidu aT g ma(il) doT c0m), Dec 29 2010
(PARI) {a(n) = (n%2 + 2) * 2^(n\2) - 1}; /* Michael Somos, Jun 24 2018 */
(Haskell)
a052955 n = a052955_list !! n
a052955_list = 1 : 2 : map ((+ 1) . (* 2)) a052955_list
(Magma) [((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1: n in [0..45]]; // G. C. Greubel, Oct 22 2019
(Sage) [((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1 for n in (0..45)] # G. C. Greubel, Oct 22 2019
(GAP) List([0..45], n-> ((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1); # G. C. Greubel, Oct 22 2019
(Python)
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CROSSREFS
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See A016116 for the first differences.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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