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 A379153 #8 Dec 17 2024 08:44:37
%S A379153 1,2,3,6,6,14,15,15,20,19,23,23,27,34,35,44,40,36,40,44,41,48,52,62,
%T A379153 64,66,57,66,72,79,71,75,77,78,79,78,88,86,92,100,103,103,92,116,96,
%U A379153 116,117,113,129,117,123,128,123,126,130,133,129,142,147,134,135,148
%N A379153 The binary weights of the Apéry numbers (A005259).
%H A379153 Amiram Eldar, <a href="/A379153/b379153.txt">Table of n, a(n) for n = 0..10000</a>
%H A379153 Arnold Knopfmacher and Florian Luca, <a href="https://doi.org/10.1016/j.jnt.2011.07.004">Digit sums of binomial sums</a>, Journal of Number Theory, Vol. 132, No. 2 (2012), pp. 324-331.
%H A379153 Florian Luca and Igor E. Shparlinski, <a href="https://doi.org/10.1007/s00026-011-0074-9">On the g-ary expansions of Apéry, Motzkin, Schröder and other combinatorial numbers</a>, Annals of Combinatorics, Vol. 14 (2010), pp. 507-524.
%F A379153 a(n) = A000120(A005259(n)).
%F A379153 a(n) > c * (log(n)/log(log(n)))^(1/4) holds on a set of n of asymptotic density 1, where c > 0 is a constant (Luca and Shparlinski, 2010).
%F A379153 a(n) > c * log(n)/log(log(n)) holds on a set of n of asymptotic density 1, where c > 0 is a constant (Knopfmacher and Luca, 2012).
%F A379153 Conjecture: Limit_{m->oo} (1/m^2) * Sum_{k=1..m} a(k) = log(sqrt(2) + 1)/log(2) = 1.2715533... (Knopfmacher and Luca, 2012).
%t A379153 a[n_] := DigitCount[Sum[(Binomial[n, k] * Binomial[n+k, k])^2, {k, 0, n}], 2, 1]; Array[a, 100, 0]
%o A379153 (PARI) a(n) = hammingweight(sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2));
%Y A379153 Cf. A000120, A005259.
%Y A379153 Similar sequences: A011373, A079584, A082481, A379151, A379152.
%K A379153 nonn,easy,base
%O A379153 0,2
%A A379153 _Amiram Eldar_, Dec 17 2024