A283208 Minimal exponent integer sequence associated with Vietoris sequence.

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, 48, 49, 52, 53, 55, 56, 62, 63, 65, 66, 69, 70, 72, 73, 77, 78, 80, 81, 84, 85, 87, 88, 93, 94, 96, 97, 100, 101, 103, 104, 108, 109, 111, 112, 115, 116, 118, 119

Offset: 0

Author

Isabel Cação and Maria Irene Falcão and Helmuth Malonek, Mar 08 2017

Keywords

• nonn

Comments

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).

Also 2^a(n) is the denominator of 2^(-2n)*A001700(n).

From Rogério Serôdio, Feb 19 2019: (Start)

Sum_{k = -1..2} (-1)^ceiling(k/2 + 1) * a(4*n + k) = 6, for n >= 1.

Sum_{k = 1..4} (-1)^ceiling((k-1 mod 3)/3) * a(2*n + k) = 0, for n >= 0. (End)

Examples

			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.
		

References

• L. Vietoris. Über das Vorzeichen gewisser trigonometrischer Summen. Sitzungsber. Österr. Akad. Wiss., 167 (1958), 125-135.

Links

• R. Askey and J. Steinig, Some positive trigonometric sums, Transactions AMS, 187(1974), 295-307.
• Isabel Cação, Maria Irene Falcão, and Helmuth R. Malonek, Hypercomplex Polynomials, Vietoris' Rational Numbers and a Related Integer Numbers Sequence, Complex Anal. Oper. Theory (2017).
• Isabel Cação, Maria Irene Falcão, and Helmuth R. Malonek, On Vietoris' number sequence and combinatorial identities with quaternions, research paper, 2017.
• Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
• Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
• Regina de Almeida and Paula Catarino, On dual biquaternionic sequence involving Vietoris' numbers, V Int'l Conf. Math. Appl. Sci. Eng. (ICMASE 2024).
• Maria Irene Falcão and Helmuth R. Malonek, A note on a one-parameter family of non-symmetric number triangles, Opuscula Mathematica, 32, (2012) 661-673.
• S. Ruscheweyh and L. Salinas, Stable functions and Vietoris' theorem, J. Math. Anal. Appl. 291 (2004), 596-604.

Crossrefs

Cf. A001405, A000079, A001700, A000523, A001511, A011371, A005187.

Programs

Mathematica

a[n_]:=n+Sum[Floor[(n+1)/(2^j)],{j,1,Log2[n+1]}]
(* or *)
a[n_]:=Log2@Denominator[Binomial[2 n + 1, n+1] 2^(-2 n)]

PARI

for(n=0, 62, print1(n + sum(j=1, logint(n + 1, 2), floor((n+1)/(2^j))), ", ")) \\ Indranil Ghosh, Mar 10 2017

Python

def A283208(n): return n+sum((n+1)//(1<Chai Wah Wu, Jul 16 2022

R

A283208 <- function(n){
soma <- n
for (i in c(1:ceiling(log2(n+1)))){soma <- soma + floor((n+1)/2^i)}
print(soma)} # Rogério Serôdio, Feb 19 2019

Formula

a(n) = n + Sum_{j=1..k} floor((n+1)/(2^j)) where k = floor(log_2(n+1)) = A000523(1+n).

From Rogério Serôdio, Feb 19 2019: (Start)

a(n+4) = a(n) + 6 + b(n), where b(n) = min(k, floor((n+5)/4 mod 2^k) = 1), for n >= 0.

a(n+4) = a(n) + 6 + A001511(ceiling((n+2)/4)), for n >= 0.

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)

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]

a(n) = A005187(n+1) - 1. - Andrey Zabolotskiy, Mar 26 2021