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A120777
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One half of denominators of partial sums of a series for sqrt(2).
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9
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1, 4, 8, 64, 128, 512, 1024, 16384, 32768, 131072, 262144, 2097152, 4194304, 16777216, 33554432, 1073741824, 2147483648, 8589934592, 17179869184, 137438953472, 274877906944, 1099511627776, 2199023255552
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also denominators of partial sums sum(C(k)/(-4)^k,k=0..n)= A120788(n)/A120777(n).
One half of denominators of partial sums which involve Catalan numbers A000108(k) divided by 4^k with alternating signs.
The listed numbers coincide with the denominators of sum(C(k)/4^k,k=0..n). See numerators A120778. In general these denominators may be different. See e.g. A120783 versus A120793 and A120787 versus A120796.
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FORMULA
| a(n)=denominator(r(n)), with the rationals r(n) defined under A120088.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)
a(n) = denom(binomial(2*n+2,n+1)/2^(2*n+1))
If b(n) = log(a(n))/log(2) then c(n) = b(n+1)-b(n) = A001511(n+1) i.e. the ruler function.
(End)
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MAPLE
| Conjecture: The following Maple program appears to generate this sequence! Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(denom(coeff(gser, z, n))/2, n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 21 2008
restart; nmax:=23; A001511 (1):=1: for n from 2 to nmax do A001511(n):= A001511(n-1)+(-1)^n* A001511(floor(n/2)) od: b(0):=0: for n from 1 to nmax do b(n):=b(n-1)+ A001511(n+1) od: for n from 0 to nmax-1 do a(n):=2^b(n) od: seq(a(n), n=0..nmax-1); [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009]
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CROSSREFS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)
Appears in A162446.
(End)
Sequence in context: A123288 A192200 A063083 * A091095 A075787 A086891
Adjacent sequences: A120774 A120775 A120776 * A120778 A120779 A120780
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KEYWORD
| nonn,easy,frac
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 20 2006
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