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A283208 Minimal exponent integer sequence associated with Vietoris sequence. 1
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 (list; graph; refs; listen; history; text; internal format)
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.

I. Cação, M. I. Falcão and H. R. Malonek, Hypercomplex Polynomials, Vietoris' Rational Numbers and a Related Integer Numbers Sequence, Complex Anal. Oper. Theory (2017).

I. Cação, M. I. Falcão, H. R. Malonek, On Vietoris' number sequence and combinatorial identities with quaternions, research paper, 2017.

M. I. Falcão and H. R. Malonek, A note on a one-parameter family of non-symmetric number triangles, Opuscula Mathematica,  32, (2012) 661-673.

Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

S. Ruscheweyh and L. Salinas, Stable functions and Vietoris' theorem, J. Math. Anal. Appl. 291 (2004), 596-604.

FORMULA

Let k = floor(log2(n+1))=A000523(1+n). Then a(n) = n + Sum_{j=1..k} floor((n+1)/(2^j)).

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

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

CROSSREFS

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

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)