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A075427
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a(0) = 1; a(n) = a(n-1)+1 if n is even, otherwise a(n) = 2*a(n-1).
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22
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1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63, 126, 127, 254, 255, 510, 511, 1022, 1023, 2046, 2047, 4094, 4095, 8190, 8191, 16382, 16383, 32766, 32767, 65534, 65535, 131070, 131071, 262142, 262143, 524286, 524287, 1048574, 1048575, 2097150, 2097151, 4194302, 4194303, 8388606
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(0) = 1; for n >= 1, a(2*n) = 2^(n+1)-1, a(2*n-1) = 2^(n+1)-2; a(n) = 2^floor((n+3)/2) - 3/2 + (-1)^n/2. - Benoit Cloitre, Sep 17 2002 [corrected by Robert FERREOL, Jan 26 2011]
a(n) = (-1)^n/2 - 3/2 + 2^(n/2)*(1 + sqrt(2) + (1-sqrt(2))*(-1)^n). - Paul Barry, Apr 22 2004
Interleaved Mersenne numbers: interleaves 2*2^n-1 and 2(2*2^n-1) (A000225(n+1) and 2*A000225(n+1)).
G.f.: (1+2*x)/((1-x^2)*(1-2*x^2));
a(n) = 3*a(n-2) - 2*a(n-4);
a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)). (End)
For n > 0: a(n) = (1 + n mod 2) * a(n-1) + 1 - (n mod 2). - Reinhard Zumkeller, Feb 27 2012
E.g.f.: 2*(cosh(sqrt(2)*x) - sinh(x) + sqrt(2)*sinh(sqrt(2)*x)) - cosh(x). - Stefano Spezia, Jul 11 2023
a(n) = 2^floor((n+3)/2)-1-(n mod 2).
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MAPLE
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A075427 := proc(n) if type(n, 'even') then 2^(n/2+1)-1 ; else 2^(1+(n+1)/2)-2 ; end if; end proc: seq(A075427(n), n=0..40); # R. J. Mathar, Feb 18 2011
isA := proc(n) convert(n, base, 2): 1 - %[1] = nops(%) - add(%) end:
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MATHEMATICA
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a[0]=1; a[n_]:=a[n]=If[EvenQ[n], a[n-1]+1, 2*a[n-1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 20 2011 *)
nxt[{n_, a_}]:={n+1, If[OddQ[n], a+1, 2a]}; Transpose[NestList[nxt, {0, 1}, 40]][[2]] (* or *) LinearRecurrence[{0, 3, 0, -2}, {1, 2, 3, 6}, 50] (* Harvey P. Dale, Mar 12 2016 *)
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PROG
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(Magma) [2^Floor((n+3)/2)-3/2+(-1)^n/2: n in [0..30]]; // Vincenzo Librandi, Aug 17 2011
(Haskell)
a075427 n = a075427_list !! n
a075427_list = 1 : f 1 1 where
f x y = z : f (x + 1) z where z = (1 + x `mod` 2) * y + 1 - x `mod` 2
(Python)
def A075427(n): return (1<<(n>>1)+2)-2 if n&1 else (1<<(n>>1)+1)-1 # Chai Wah Wu, Apr 23 2023
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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