This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A123749 #23 Jan 26 2025 09:10:04
%S A123749 1,11,35,965,8755,8783,237449,2138185,6415985,519743405,4677875401,
%T A123749 14033861347,378916960525,3410263045325,3410267502725,30692424759805,
%U A123749 276231889624955,828695755304725,67124359204727825,604119244624305025
%N A123749 Numerators of partial sums of a series for 3/sqrt(5) = (3/5)*sqrt(5).
%C A123749 Denominators are given by A124396.
%C A123749 The sums over central binomial coefficients scaled by powers of 9, r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k have the limit s = lim_{n->infinity} r(n) = 3/sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/9.
%H A123749 G. C. Greubel, <a href="/A123749/b123749.txt">Table of n, a(n) for n = 0..1000</a>
%H A123749 Wolfdieter Lang, <a href="/A123749/a123749.txt">Rationals and more.</a>
%F A123749 a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k in lowest terms.
%F A123749 r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)*(4/9)^k, n>=0, with the double factorials A001147 and A000165.
%e A123749 a(3) = 965 because r(3) = 1 + 2/9 + 2/27 + 20/729 = 965/729 = a(3)/A124396(3).
%p A123749 A123749:=n-> numer(sum(binomial(2*k,k)/9^k, k=0..n)); seq(A123749(n), n=0..20); # _G. C. Greubel_, Aug 10 2019
%t A123749 Table[Numerator[Sum[Binomial[2*k, k]/9^k, {k,0,n}]], {n, 0, 20}] (* _G. C. Greubel_, Aug 10 2019 *)
%o A123749 (PARI) vector(20, n, n--; numerator(sum(k=0,n, binomial(2*k,k)/9^k))) \\ _G. C. Greubel_, Aug 10 2019
%o A123749 (Magma) [Numerator( (&+[Binomial(2*k,k)/9^k: k in [0..n]])): n in [0..20]]; // _G. C. Greubel_, Aug 10 2019
%o A123749 (Sage) [numerator( sum(binomial(2*k,k)/9^k for k in (0..n)) ) for n in (0..20)] # _G. C. Greubel_, Aug 10 2019
%o A123749 (GAP) List([0..20], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/9^k )) ); # _G. C. Greubel_, Aug 10 2019
%Y A123749 Cf. A124396 (denominators).
%Y A123749 Cf. A123747/A123748 partial sums for a series for sqrt(5).
%K A123749 nonn,frac,easy
%O A123749 0,2
%A A123749 _Wolfdieter Lang_, Nov 10 2006