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Search: id:A121000
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| A121000 |
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Numerators of partial sums of Catalan numbers scaled by powers of 1/18^2 = 1/324. |
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2
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| 1, 325, 52651, 34117853, 5527092193, 596925956851, 96702005009873, 125325798492795551, 60908338067498638501, 19734301533869558876755, 3196956848486868538038509, 2071628037819490812648983225
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Denominators are given under A121001.
This is the fourth member (p=3) of the first p-family of partial sums of normalized scaled Catalan series CsnI(p):=sum(C(k)/L(2*p)^(2*k),k=0..infinity) with limit L(2*p)*(F(2*p+1) - F(2*p)*phi) = L(2*p)/phi^(2*p), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned first p-family are rI(p;n):=sum(C(k)/L(2*p)^(2*k),k=0..n), n>=0, for p=0,1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.
The limit lim_{n->infinity} r(n) = 18*(13 - 8* phi) = 18/phi^6 = 1.003105620014 (maple10, 15 digits).
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LINKS
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W. Lang: Rationals r(n), limit.
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FORMULA
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a(n)=numerator(r(n)) with r(n) := rI(p=3,n) = sum(C(k)/L(6)^(2*k),k=0..n), with Lucas L(6)=18 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
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EXAMPLE
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Rationals r(n): [1, 325/324, 52651/52488, 34117853/34012224,
5527092193/5509980288, 596925956851/595077871104, ...].
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CROSSREFS
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Sequence in context: A031606 A145414 A166220 this_sequence A048909 A097739 A048918
Adjacent sequences: A120997 A120998 A120999 this_sequence A121001 A121002 A121003
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KEYWORD
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nonn,frac,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 2006
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