A123747 Numerators of partial sums of a series for sqrt(5).

1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 339895847771, 1700893049407, 42549895540939, 212857129279583

Offset

0,2

Comments

Denominators are given by A123748.

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Wolfdieter Lang, Rationals and more.

Formula

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.

Example

a(3)=9 because r(3)= 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).

Maple

A123747:=n-> numer(sum(binomial(2*k, k)/5^k, k=0..n)); seq(A123747(n), n=0..25); # G. C. Greubel, Aug 10 2019

Mathematica

Table[Numerator[Sum[Binomial[2*k, k]/5^k, {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)

Prog

(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

Crossrefs

Cf. A001077/A001076 continued fraction convergents for sqrt(5).

Cf. A123748, A123749.

Sequence in context: A062727 A223416 A165397 * A144421 A023251 A073501

Adjacent sequences: A123744 A123745 A123746 * A123748 A123749 A123750

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Nov 10 2006

Status

approved