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A283208 Minimal exponent integer sequence associated with Vietoris sequence.

%I A283208 #97 Aug 09 2024 11:49:53
%S A283208 0,2,3,6,7,9,10,14,15,17,18,21,22,24,25,30,31,33,34,37,38,40,41,45,46,
%T A283208 48,49,52,53,55,56,62,63,65,66,69,70,72,73,77,78,80,81,84,85,87,88,93,
%U A283208 94,96,97,100,101,103,104,108,109,111,112,115,116,118,119
%N A283208 Minimal exponent integer sequence associated with Vietoris sequence.
%C A283208 a(n) is the least nonnegative integer such that 2^a(n) (2n+1)/(n+1) c(2n) is a nonnegative integer, where c(2n) = c(2n-1) = Pochhammer(1/2,n)/(n!) used in [Vietoris, Askey & Steinig, Ruscheweyh & Salinas] and named Vietoris sequence in [Cação et al.]. Also c(n) = A001405(n)/A000079(n).
%C A283208 Also 2^a(n) is the denominator of 2^(-2n)*A001700(n).
%C A283208 From _Rogério Serôdio_, Feb 19 2019: (Start)
%C A283208 Sum_{k = -1..2} (-1)^ceiling(k/2 + 1) * a(4*n + k) = 6, for n >= 1.
%C A283208 Sum_{k = 1..4} (-1)^ceiling((k-1 mod 3)/3) * a(2*n + k) = 0, for n >= 0. (End)
%D A283208 L. Vietoris. Über das Vorzeichen gewisser trigonometrischer Summen. Sitzungsber. Österr. Akad. Wiss., 167 (1958), 125-135.
%H A283208 R. Askey and J. Steinig, <a href="https://doi.org/10.1090/S0002-9947-1974-0338481-3 ">Some positive trigonometric sums</a>, Transactions AMS, 187(1974), 295-307.
%H A283208 Isabel Cação, Maria Irene Falcão, and Helmuth R. Malonek, <a href="https://dx.doi.org/10.1007/s11785-017-0649-5">Hypercomplex Polynomials, Vietoris' Rational Numbers and a Related Integer Numbers Sequence</a>, Complex Anal. Oper. Theory (2017).
%H A283208 Isabel Cação, Maria Irene Falcão, and Helmuth R. Malonek, <a href="https://repositorium.sdum.uminho.pt/bitstream/1822/47203/1/CMMSE2017-Repositorium.pdf">On Vietoris' number sequence and combinatorial identities with quaternions</a>, research paper, 2017.
%H A283208 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H A283208 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Falcao/falcao5.html">Intrinsic Properties of a Non-Symmetric Number Triangle</a>, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
%H A283208 Regina de Almeida and Paula Catarino, <a href="https://icmase.com/uploadfiles/files/3aafa289-e39d-4b38-bce9-339c1655b8fa.pdf">On dual biquaternionic sequence involving Vietoris' numbers</a>, V Int'l Conf. Math. Appl. Sci. Eng. (ICMASE 2024).
%H A283208 Maria Irene Falcão and Helmuth R. Malonek, <a href="http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-article-AGHS-0007-0003">A note on a one-parameter family of non-symmetric number triangles</a>, Opuscula Mathematica,  32, (2012) 661-673.
%H A283208 S. Ruscheweyh and L. Salinas, <a href="http://dx.doi.org/10.1016/j.jmaa.2003.11.035">Stable functions and Vietoris' theorem</a>, J. Math. Anal. Appl. 291 (2004), 596-604.
%F A283208 a(n) = n + Sum_{j=1..k} floor((n+1)/(2^j)) where k = floor(log_2(n+1)) = A000523(1+n).
%F A283208 From _Rogério Serôdio_, Feb 19 2019: (Start)
%F A283208 a(n+4) = a(n) + 6 + b(n), where b(n) = min(k, floor((n+5)/4 mod 2^k) = 1), for n >= 0.
%F A283208 a(n+4) = a(n) + 6 + A001511(ceiling((n+2)/4)), for n >= 0.
%F A283208 G.f.: ((2*x + x^2 + 3*x^3 + x^4)/(1 - x) + Sum_{k >= 0} (Sum_{i = 0..3}  x^(8*2^k-1+i))/(1 - x^(8*2^k)))/(1 - x^4). (End)
%F A283208 a(n) = n + log_2((n+1)!-((n+1)! AND (n+1)!-1)) (empirical). - _Gary Detlefs_, Apr 29 2019 [True: equivalent to a(n) = n + A011371(n+1), which is equivalent to the top formula here. - _Andrey Zabolotskiy_, Mar 26 2021]
%F A283208 a(n) = A005187(n+1) - 1. - _Andrey Zabolotskiy_, Mar 26 2021
%e A283208 For n=3, a(3)=6 and 2^a(n)(2n+1)/(n+1) c(2n) = (2^6)*7/4*c(6) = 64*35/64 = 35.
%t A283208 a[n_]:=n+Sum[Floor[(n+1)/(2^j)],{j,1,Log2[n+1]}]
%t A283208 (* or *)
%t A283208 a[n_]:=Log2@Denominator[Binomial[2 n + 1, n+1] 2^(-2 n)]
%o A283208 (PARI) for(n=0, 62, print1(n + sum(j=1, logint(n + 1, 2), floor((n+1)/(2^j))), ", ")) \\ _Indranil Ghosh_, Mar 10 2017
%o A283208 (R)
%o A283208 A283208 <- function(n){
%o A283208 soma <- n
%o A283208 for (i in c(1:ceiling(log2(n+1)))){soma <- soma + floor((n+1)/2^i)}
%o A283208 print(soma)} # _Rogério Serôdio_, Feb 19 2019
%o A283208 (Python)
%o A283208 def A283208(n): return n+sum((n+1)//(1<<j) for j in range(1,(n+1).bit_length()+2)) # _Chai Wah Wu_, Jul 16 2022
%Y A283208 Cf. A001405, A000079, A001700, A000523, A001511, A011371, A005187.
%K A283208 nonn
%O A283208 0,2
%A A283208 Isabel Cação and _Maria Irene Falcão_ and Helmuth Malonek, Mar 08 2017