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

0,2

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)

References

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

Links

Table of n, a(n) for n=0..62.

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.

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.

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

Example

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.

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)]

Prog

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

Crossrefs

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

Sequence in context: A190847 A201734 A287775 * A259726 A259587 A304107

Adjacent sequences: A283205 A283206 A283207 * A283209 A283210 A283211

Keyword

nonn

Author

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

Status

approved