The On-Line Encyclopedia of Integer Sequences (OEIS)

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A124397

Numerators of partial sums of a series for sqrt(5)/3.
(history; published version)

#17 by Charles R Greathouse IV at Thu Sep 08 08:45:28 EDT 2022

PROG

(MAGMAMagma) [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#16 by Peter Luschny at Thu Dec 26 05:33:12 EST 2019

STATUS

reviewed

approved

#15 by Michel Marcus at Thu Dec 26 01:13:53 EST 2019

STATUS

proposed

reviewed

#14 by Jon E. Schoenfield at Thu Dec 26 00:16:46 EST 2019

STATUS

editing

proposed

#13 by Jon E. Schoenfield at Thu Dec 26 00:16:43 EST 2019

EXAMPLE

a(3)=17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).

STATUS

proposed

editing

#12 by G. C. Greubel at Wed Dec 25 23:35:43 EST 2019

STATUS

editing

proposed

#11 by G. C. Greubel at Wed Dec 25 23:34:58 EST 2019

COMMENTS

The alternating sums over central binomial coefficients scaled by powers of 5, r(n): = Sum_{k=sum((0..n} (-1)^k)*binomial(2*k,k)/5^k,k=0..n) have the limit s: = lim(r(n),_{n-> infinity} r(n) = sqrt(5)/3. From the expansion of 1/sqrt(1+x) for x=4/5.

LINKS

G. C. Greubel, <a href="/A124397/b124397.txt">Table of n, a(n) for n = 0..1000</a>

FORMULA

a(n) = numerator(r(n)) with the rationals r(n): = Sum_{k=sum((0..n} (-1)^k) * binomial(2*k,k)/5^k,k=0..n) in lowest terms.

r(n) = sum((Sum_{k=0..n} (-1)^k)*((2*k-1)!!/((2*k)!!)*(4/5)^k,k=0..n), n>=0, with the double factorials A001147 and A000165.

MAPLE

seq(numer(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019

MATHEMATICA

Table[Numerator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Dec 25 2019 *)

PROG

(MAGMA) [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019

(Sage) [numerator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

(GAP) List([0..20], n-> NumeratorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019

STATUS

approved

editing

#10 by Wolfdieter Lang at Fri Aug 30 03:49:50 EDT 2019

STATUS

editing

approved

#9 by Wolfdieter Lang at Fri Aug 30 03:49:47 EDT 2019

LINKS

Wolfdieter Lang, <a href="/LANGCHANGE/A124397/a124397.texttxt">Rationals and more</a>.

STATUS

approved

editing

#8 by N. J. A. Sloane at Wed Aug 28 09:25:23 EDT 2019

LINKS

Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/EISpubLANGCHANGE/A124397.text">Rationals and more</a>.

Discussion

Wed Aug 28

09:25

OEIS Server: https://oeis.org/edit/global/2824