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Search: id:A120794
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| A120794 |
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Numerators of partial sums of Catalan numbers scaled by powers of -1/16. |
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+0
4
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| 1, 15, 121, 3867, 30943, 495067, 3960569, 253475987, 2027808611, 32444935345, 259559486959, 8305903553295, 66447228478363, 1063155655468083, 8505245244078969, 1088671391232413187
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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From the expansion of sqrt(1+1/4) = 1+(1/8)*sum(C(k)/(-16)^k,k=0..infinity) one has, with the partial sums r(n) are defined below, r:=limit(r(n),n to infinity)= 4*(sqrt(5)-2) = 4*(2*phi-3)) = 0.944271909...
Denominators coincide with the listed numbers of A120785 but may differ for higher n values.
This is the first member (p=1) of the fourth famliy of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.
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LINKS
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W. Lang: Rationals r(n) and limit.
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FORMULA
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a(n)=numerator(r(n)), with the rationals r(n):=sum(((-1)^k)*C(k)/16^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
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EXAMPLE
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Rationals r(n): [1, 15/16, 121/128, 3867/4096, 30943/32768,
495067/524288, 3960569/4194304,...].
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CROSSREFS
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The second member (p=2) of this p-family is A121012/A121013.
Sequence in context: A022580 A081079 A138424 this_sequence A038743 A027839 A034271
Adjacent sequences: A120791 A120792 A120793 this_sequence A120795 A120796 A120797
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KEYWORD
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nonn,easy,frac
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 20 2006
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