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A123747
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Numerators of partial sums of a series for sqrt(5).
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6
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1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 339895847771, 1700893049407, 42549895540939, 212857129279583
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OFFSET
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0,2
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COMMENTS
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The sum over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k, has the limit lim_{n -> infinity} r(n) = sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/5.
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LINKS
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FORMULA
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a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k, in lowest terms.
r(n) = Sum_{k=0..n} (((2*k-1)!!/((2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.
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EXAMPLE
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a(3)=9 because r(3)= 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).
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MAPLE
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MATHEMATICA
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Table[Numerator[Sum[Binomial[2*k, k]/5^k, {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)
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PROG
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(PARI) vector(25, n, n--; numerator(sum(k=0, n, binomial(2*k, k)/5^k))) \\ G. C. Greubel, Aug 10 2019
(Magma) [Numerator( (&+[Binomial(2*k, k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019
(Sage) [numerator( sum(binomial(2*k, k)/5^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Aug 10 2019
(GAP) List([0..25], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k, k)/5^k )) ); # G. C. Greubel, Aug 10 2019
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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