A160466 -id:A160466 - cp's OEIS frontend

A000466 a(n) = 4*n^2 - 1.

-1, 3, 15, 35, 63, 99, 143, 195, 255, 323, 399, 483, 575, 675, 783, 899, 1023, 1155, 1295, 1443, 1599, 1763, 1935, 2115, 2303, 2499, 2703, 2915, 3135, 3363, 3599, 3843, 4095, 4355, 4623, 4899, 5183, 5475, 5775, 6083, 6399, 6723, 7055, 7395

Offset: 0

Chan Siu Kee (skchan5(AT)hkein.ie.cuhk.hk)

Sum_{n>=1} (-1)^n*a(n)/n! = 1 - 1/e = A068996. - Gerald McGarvey, Nov 06 2007
Sequence arises from reading the line from -1, in the direction -1, 15, ... and the same line from 3, in the direction 3, 35, ..., in the square spiral whose nonnegative vertices are the squares A000290. - Omar E. Pol, May 24 2008
a(n) is the product of the consecutive odd integers 2n-1 and 2n+1 (cf. A005408). - Doug Bell, Mar 08 2009
For n>0: a(n) = A176271(2*n,n); cf. A016754, A053755. - Reinhard Zumkeller, Apr 13 2010
a(n+1) gives the curvature c(n) of the n-th circle touching the two equal semicircles of the symmetric arbelos (1/2, 1/2) and the (n-1)-st circle, with input c(0) = 3 =  A059100(1) (referring to the second circle of the Pappus chain), for n >= 0. - Wolfdieter Lang and Kival Ngaokrajang, Jul 03 2015
After 3, a(n) is pseudoprime to base 2n. For example: (2*2)^(a(2)-1) == 1 (mod a(2)), in fact 4^14 = 15*17895697+1. - Bruno Berselli, Sep 24 2015
Numbers m such that m+1 and (m+1)/4 are squares. - Bruno Berselli, Mar 03 2016
After -1, the least common multiple of 2*m+1 and 2*m-1. - Colin Barker, Feb 11 2017
This sequence contains all products of the twin prime pairs (see A037074). - Charles Kusniec, Oct 03 2019

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
L. B. W. Jolley, Summation of Series, Dover, 2nd ed., 1961.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.
A. Languasco and A. Zaccagnini, Manuale di Crittografia, Ulrico Hoepli Editore (2015), p. 259.

Vincenzo Librandi, Table of n, a(n) for n = 0..900
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.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Kival Ngaokrajang, Illustration of the Pappus chain (downwards direction).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001539, A016286, A016742.
Factor of A160466. Superset of A037074.
Cf. A059100 (curvatures for a Pappus chain).

[4*n^2-1: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

A000466:=n->4*n^2-1; seq(A000466(n), n=0..100); # Wesley Ivan Hurt, Nov 19 2013

4 Range[0,50]^2-1 (* Harvey P. Dale, Jan 23 2011 *)

makelist(4*n^2-1, n, 0, 50); /* Martin Ettl, Nov 12 2012 */

a(n)=4*n^2-1 \\ Charles R Greathouse IV, Oct 27 2011

[4*n^2-1 for n in (0..50)] # Bruno Berselli, Sep 24 2015

O.g.f.: ( 1-6*x-3*x^2 ) / (x-1)^3 . - R. J. Mathar, Mar 24 2011
E.g.f.: (-1 + 4*x + 4*x^2)*exp(x). - Ilya Gutkovskiy, May 26 2016
Sum_{n>=1} 1/a(n) = 1/2 [Jolley eq. 233]. - Benoit Cloitre, Apr 05 2002
Sum_{n>=1} 2/a(n) = 1 = 2/3 + 2/15 + 2/35 + 2/63 + 2/99 + 2/143, ..., with partial sums: 2/3, 4/5, 6/7, 8/9, 10/11, 12/13, 14/15, ... - Gary W. Adamson, Jun 16 2003
1/3 + Sum_{n>=2} 4/a(n) = 1 = 1/3 + 4/15 + 4/35 + 4/63, ..., with partial sums: 1/3, 3/5, 5/7, 7/9, 9/11, ..., (2n+1)/(2n+3). - Gary W. Adamson, Jun 18 2003
Sum_{n>=0} 2/a(2*n+1) = Pi/4 = 2/3 + 2/35 + 2/99, ... = (1 - 1/3) + (1/5 - 2/7) + (1/9 - 1/11) + ... = Sum_{n>=0} (-1)^n/(2*n+1). - Gary W. Adamson, Jun 22 2003
Product(n>=1, (a(n)+1)/a(n)) = Pi/2 (Wallis formula). - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 03 2004
a(n)+2 = A053755(n). - Zak Seidov, Jan 16 2007
a(n)^2 + A008586(n)^2 = A053755(n)^2 (Pythagorean triple). - Zak Seidov, Jan 16 2007
a(n) = a(n-1) + 8*n - 4 for n > 0, a(0)=-1. - Vincenzo Librandi, Dec 17 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 - 1/2 = (A019669-1)/2. [Jolley eq (366)]. - R. J. Mathar, Mar 24 2011
For n>0, a(n) = 2/(Integral_{x=0..Pi/2} (sin(x))^3*(cos(x))^(2*n-2)). - Francesco Daddi, Aug 02 2011
Nonlinear recurrence for c(n) = a(n+1) (see the arbelos comment above) from Descartes' three circle theorem (see the links under A259555): c(n) = 4 + c(n-1) + 4*sqrt(c(n-1) + 1), with input c(0) = 3  =  A059100(1), for n >= 0. The appropriate solution of this recurrence is c(n-1) + 1 = 4*n^2. - Wolfdieter Lang, Jul 03 2015
a(n) = 3*Pochhammer(5/2,n-1)/Pochhammer(1/2,n-1). Hence, the e.g.f. for a(n+1), i.e., dropping the first term, is 3* 1F1(5/2;1/2;x), with 1F1 being the confluent hypergeometric function (also known as Kummer's). - Stanislav Sykora, May 26 2016
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/sqrt(2))/sqrt(2). - Amiram Eldar, Feb 04 2021

A043529 Number of distinct base-2 digits of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Clark Kimberling

Also, if prefixed by 0, the trajectory of 0 under repeated applications of the morphism 0 -> 0,1, 1 -> 1,2, 2 -> 2,2. This is a word that is pure uniform morphic, but neither primitive morphic nor recurrent. - N. J. A. Sloane, Jul 15 2018

Dekking, Michel, Michel Mendes France, and Alf van der Poorten. "Folds." The Mathematical Intelligencer, 4.3 (1982): 130-138 & front cover, and 4:4 (1982): 173-181 (printed in two parts). See Observaion 1.8.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017. See Example 35.
Index entries for sequences related to binary expansion of n

Factor of A160466. Cf. A007456 and A081729. - Johannes W. Meijer, May 24 2009
Cf. A007088, A036987, A043545.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

A043529 := proc(n): if type(ln(n+1)/ln(2), integer) then 1 else 2 fi: end proc: seq(A043529(n), n=0..90); # Johannes W. Meijer, Sep 14 2012

(* Needs version >= 10.2. *)
SubstitutionSystem[{0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 2}}, 0, 7] // Last // Rest (* Jean-François Alcover, Apr 06 2020 *)
Table[Length[Union[IntegerDigits[n,2]]],{n,0,90}] (* Harvey P. Dale, Aug 04 2024 *)

This is 2 unless n = 2^k - 1 for some k in which case it is 1.

a(n) = 2 - A036987(n). - Antti Karttunen, Nov 19 2017

First term added and offset changed by Johannes W. Meijer, May 15 2009

A160464 The Eta triangle.

Original entry on oeis.org

-1, -11, 2, -114, 29, -2, -3963, 1156, -122, 4, -104745, 32863, -4206, 222, -4, -3926745, 1287813, -184279, 12198, -366, 4, -198491580, 67029582, -10317484, 781981, -30132, 562, -4

Offset: 2

Johannes W. Meijer, May 24 2009

The ES1 matrix coefficients are defined by ES1[2*m-1,n] = 2^(2*m-1) * int(y^(2*m-1)/(cosh(y))^(2*n),y=0..infinity)/(2*m-1)! for m = 1, 2, 3, .. and n = 1, 2, 3 .. .
This definition leads to ES1[2*m-1,n=1] = 2*eta(2*m-1) and the recurrence relation ES1[2*m-1,n] = ((2*n-2)/(2*n-1))*(ES1[2*m-1,n-1] - ES1[2*m-3,n-1]/(n-1)^2) which we used to extend our definition of the ES1 matrix coefficients to m = 0, -1, -2, .. . We discovered that ES1[ -1,n] = 0.5 for n = 1, 2, .. . As usual eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.
The coefficients in the columns of the ES1 matrix, for m = 1, 2, 3, .. , and n = 2, 3, 4 .. , can be generated with the polynomials GF(z,n) for which we found the following general expression GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n))/p(n).
The CFN1(z,n) polynomials depend on the central factorial numbers A008955.
The ETA(z,n) are the Eta polynomials which lead to the Eta triangle.
The zero patterns of the Eta polynomials resemble a UFO. These patterns resemble those of the Zeta, Beta and Lambda polynomials, see A160474, A160480 and A160487.
The first Maple algorithm generates the coefficients of the Eta triangle. The second Maple algorithm generates the ES1[2*m-1,n] coefficients for m= 0, -1, -2, -3, .. .
The M(n) sequence, see the second Maple algorithm, leads to Gould's sequence A001316 and a sequence that resembles the denominators of the Taylor series for tan(x), A156769(n).
Some of our results are conjectures based on numerical evidence, see especially A160466.

			The first few rows of the triangle ETA(n,m) with n=2,3,.. and m=1,2,... are
  [ -1]
  [ -11, 2]
  [ -114, 29, -2]
  [ -3963, 1156, -122, 4].
The first few ETA(z,n) polynomials are
  ETA(z,n=2) = -1;
  ETA(z,n=3) = -11+2*z^2;
  ETA(z,n=4) = -114 + 29*z^2 - 2*z^4.
The first few CFN1(z,n) polynomials are
  CFN1(z,n=2) = (z^2-1);
  CFN1(z,n=3) = (z^4 - 5*z^2 + 4);
  CFN1(z,n=4) = (z^6 - 14*z^4 + 49*z^2 - 36).
The first few generating functions GF(z;n) are:
  GF(z;n=2) = ((-1)*2*(z^2 - 1)*GF(z;n=1) + (- 1))/3;
  GF(z;n=3) = (4*(z^4 - 5*z^2+4) *GF(z;n=1) + (-11 + 2*z^2))/30;
  GF(z;n=4) = ((-1)*4*(z^6 - 14*z^4 + 49*z^2 - 36)*GF(z;n=1) + (-114 + 29*z^2 - 2*z^4))/315.

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, jpg and pdf, Mar 03 2013.
J. W. Meijer and N.H.G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
Eric. W. Weisstein, Dirichlet Eta Function, Wolfram MathWorld.

The r(n) sequence equals A062383 (n>=1).
The p(n) sequence equals A160473(n) (n>=2).
The GCS(n) sequence equals the Geometric Connell sequence A049039(n).
The M(n-1) sequence equals A001316(n-1)/A156769(n) (n>=1).
The q(n) sequence leads to A081729 and the 'gossip sequence' A007456.
The first right hand column equals A053644 (n>=1).
The first left hand column equals A160465.
The row sums equal A160466.
The CFN1(z, n) and the cfn1(n, k) lead to A008955.
Cf. A094665 and A160468.
Cf. the Zeta, Beta and Lambda triangles A160474, A160480 and A160487.
Cf. A162440 (EG1 matrix).

nmax:=8; c(2 ):= -1/3: for n from 3 to nmax do c(n) := (2*n-2)*c(n-1)/(2*n-1)-1/((n-1)*(2*n-1)) end do: for n from 2 to nmax do GCS(n-1) := ln(1/(2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2); p(n) := 2^(-GCS(n-1))*(2*n-1)!; ETA(n, 1) := p(n)*c(n); ETA(n, n) := 0 end do: mmax:=nmax: for m from 2 to mmax do for n from m+1 to nmax do q(n) := (1+(-1)^(n-3)*(floor(ln(n-1)/ln(2)) - floor(ln(n-2)/ln(2)))): ETA(n, m) := q(n)*((-1)*ETA(n-1, m-1)+(n-1)^2*ETA(n-1, m)) end do end do: seq(seq(ETA(n,m), m=1..n-1), n=2..nmax);
# End first program.
nmax1:=20; m:=1; ES1row:=1-2*m; with (combinat): cfn1 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j=-k..k) end proc: mmax1:=nmax1: for m1 from 1 to mmax1 do M(m1-1) := 2^(2*m1-2)/((2*m1-1)!); ES1[-2*m1+1,1] := 2*(1-2^(1-(1-2*m1)))*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do ES1[1-2*m1, n] := (-1)^(n-1)*M(n-1)*sum((-1)^(k+1)*cfn1(n-1,k-1)* ES1[2*k-2*n-2*m1+1, 1], k=1..n) od: od: seq(ES1[1-2*m, n], n=1..nmax1-m+1);
# End second program.

We discovered an interesting relation between the Eta triangle coefficients ETA(n,m) = q(n)*((-1)*ETA(n-1,m-1)+(n-1)^2*ETA(n-1,m)), for n = 3, 4, ... and m = 2, 3, ... , with
q(n) = 1 + (-1)^(n-3)*(floor(log(n-1)/log(2)) - floor(log(n-2)/log(2))) for n = 3, 4, ....
See A160465 for ETA(n,m=1) and furthermore ETA(n,n) = 0 for n = 2, 3, ....
The generating functions GF(z;n) of the coefficients in the matrix columns are defined by
GF(z;n) = sum_{m>=1} ES1[2*m-1,n] * z^(2*m-2), with n = 1, 2, 3, .... This leads to
GF(z;n=1) = (2*log(2) - Psi(z) - Psi(-z) + Psi(1/2*z) + Psi(-1/2*z)); Psi(z) is the digamma-function.
GF(z;n) = ((2*n-2)/(2*n-1)-2*z^2/((n-1)*(2*n-1)))*GF(z;n-1)-1/((n-1)*(2*n-1)).
We found for GF(z;n), for n = 2, 3, ..., the following general expression:
GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n) )/p(n) with
r(n) = 2^floor(log(n-1)/log(2)+1) and
p(n) = 2^(-GCS(n))*(2*n-1)! with
GCS(n) = log(1/(2^(-(2*(n-1)-1-floor(log(n-1)/ log(2))))))/log(2).

A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

Original entry on oeis.org

1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120, 272, 153, 342, 190, 420, 231, 506, 276, 600, 325, 702, 378, 812, 435, 930, 496, 1056, 561, 1190, 630, 1332, 703, 1482, 780, 1640, 861, 1806, 946, 1980, 1035, 2162, 1128, 2352, 1225, 2550, 1326, 2756, 1431

Offset: 0

Ralf W. Grosse-Kunstleve, Dec 11 1999

Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.
From Paul Curtz, Apr 17 2014: (Start)
Difference table of A026741/A045896:
   0,      1/6,    1/6,    3/20,     2/15,    5/42, ...
   1/6,      0,  -1/60,   -1/60,    -1/70,   -1/84, ... = 1/6, -A051712/A051713
  -1/6,  -1/60,      0,   1/420,    1/420,   1/504, ...
   3/20,  1/60,  1/420,       0,  -1/2520, -1/2520, ...
  -2/15, -1/70, -1/420, -1/2520,        0, 1/13860, ...
   5/42,  1/84,  1/504,  1/2520, -1/13860,       0, ...
Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ralf W. Grosse-Kunstleve, Origin of EIS sequences A045895 & A045896. [Wayback Machine copy]
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Factor of A160466.

Cf. A000004, A000384, A000911, A045895, A026741, A051714, A241269.

import Data.Ratio ((%), denominator)
a045896 n = denominator $ n % ((n + 1) * (n + 2))
-- Reinhard Zumkeller, Dec 12 2011

seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
with(combinat): seq(lcm(n+1,binomial(n+2,n)), n=0..50); # Zerinvary Lajos, Apr 20 2008

Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by Amiram Eldar, Sep 14 2022 *)
Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0,60],3,1]] (* Harvey P. Dale, Aug 15 2013 *)

Vec((2*x^3+3*x^2+6*x+1)/(1-x^2)^3+O(x^99)) \\ Charles R Greathouse IV, Mar 23 2016

G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.
a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
a(2*n) = A000384(n+1); a(2*n+1) = A026741(n+1). - Reinhard Zumkeller, Dec 12 2011
Sum_{n>=0} 1/a(n) = 1 + log(2). - Amiram Eldar, Sep 11 2022
From Amiram Eldar, Sep 14 2022: (Start)
a(n) = lcm(2*n+2, n+2)/2.
a(n) = A045895(n+2)/2. (End)
E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - Stefano Spezia, Apr 24 2024

A160467 a(n) = 1 if n is odd; otherwise, a(n) = 2^(k-1) where 2^k is the largest power of 2 that divides n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 32, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16

Offset: 1

Johannes W. Meijer, May 24 2009, Jun 28 2011

Fifth factor of the row sums A160466 of the Eta triangle A160464.
From Peter Luschny, May 31 2009: (Start)
Let odd(n) be the characteristic function of the odd numbers (A000035) and sigma(n) the number of 1's in binary expansion of n (A000120). Then a(n) = 2^(sigma(n-1) - sigma(n) + odd(n)).
Let B_{n} be the Bernoulli number. Then this sequence is also
a(n) = denominator(4*(4^n-1)*B_{2*n}/n). (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000265, A001620, A007814, A026741, A140670, A160464, A220466, A006519, A000010.

nmax:=96: p:= floor(log[2](nmax)): for n from 1 to nmax do a(n):=1 end do: for q from 1 to p do for n from 1 to nmax do if n mod 2^q = 0 then a(n):= 2^(q-1) end if: end do: end do: seq(a(n), n=1..nmax);
From Peter Luschny, May 31 2009: (Start)
a := proc(n) local sigma; sigma := proc(n) local i; add(i,i=convert(n,base,2)) end; 2^(sigma(n-1)-sigma(n)+`if`(type(n,odd),1,0)) end: seq(a(n), n=1..96);
a := proc(n) denom(4*(4^n-1)*bernoulli(2*n)/n) end: seq(a(n), n=1..96); (End)

a[n_] := If[OddQ[n], 1, 2^(IntegerExponent[n, 2] - 1)]; Array[a, 100] (* Amiram Eldar, Jul 02 2020 *)

A160467(n) = 2^max(valuation(n,2)-1,0); \\ Antti Karttunen, Nov 18 2017, after Max Alekseyev's Feb 09 2011 formula.

def A160467(n): return max(1,(n&-n)>>1) # Chai Wah Wu, Jul 08 2022

a(n) = A026741(n)/A000265(n). - Paul Curtz, Apr 18 2010
a(n) = 2^max(A007814(n) - 1, 0). - Max Alekseyev, Feb 09 2011
a((2*n-1)*2^p) = A011782(p), p >= 0 and n >= 1. - Johannes W. Meijer, Jan 25 2013
a(n) = (1 + A140670(n))/2. - Antti Karttunen, Nov 18 2017
From Amiram Eldar, Dec 31 2022: (Start)
Dirichlet g.f.: zeta(s)*(2^s-2+1/2^s)/(2^s-2).
Sum_{k=1..n} a(k) ~ (1/(4*log(2)))*n*log(n) + (5/8 + (gamma-1)/(4*log(2)))*n, where gamma is Euler's constant (A001620). (End)
a(n) = A006519(n)/gcd(n,2). - Ridouane Oudra, Feb 08 2025
a(n) = A000010(A006519(n)). - Ridouane Oudra, Jul 27 2025

Keyword mult added by Max Alekseyev, Feb 09 2011

Name changed by Antti Karttunen, Nov 18 2017

A119951 Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108.

Original entry on oeis.org

1, 3, 29, 65, 281, 595, 9949, 20613, 84883, 173965, 1421113, 2894229, 11762641, 23859587, 773201629, 1564082093, 6321150767, 12761711209, 102977321267, 207595672639, 836499257311, 1684433835077, 27122471168057, 54567418372945, 219485160092143, 441266239318305, 3547513302275441

Offset: 1

Wolfdieter Lang, Jul 20 2006

For the corresponding denominator sequence see A120069.
The asymptotics for C(n)/2^(2*(k-1)) is 4/(sqrt(Pi)*k^(3/2)) (see the E. Weisstein link, also for references). The sum over the asymptotic values from k=1..infinity is (4/sqrt(Pi))*Zeta(3/2) = 5.895499840 (maple10, 10 digits).
The partial sums r(n) = Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
The above partial sums are equal to 4 - binomial(2n+2,n+1)/2^(2n-1). - Pieter Mostert, Oct 12 2012
The series s = Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers), converges by J. L. Raabe's criterion. See the Meschkowski reference for Raabe's criterion and the example given there. The series he gives as an example can be rewritten as (1 + 4*s)/2. From the expansion of sqrt(1+x) for |x|<=1 one finds for x=-1 the value s=4 (see the W. Lang link).
This sequence was essential for unraveling the structure of the row sums A160466 of the Eta triangle A160464. - Johannes W. Meijer, May 24 2009

			Rationals r(n): [1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, ...]

H. Meschkowski, Unendliche Reihen, 2., verb. u. erw. Aufl., Mannheim, Bibliogr. Inst., 1982, p. 32.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, Rationals r(n) and more.
Eric Weisstein's World of Mathematics, Catalan numbers, see eq.(10).

A160464 is the Eta triangle.

Factor of A160466.

Numerator[Table[(1/4^n)*Sum[Binomial[2*(i + 1), i + 1]*Binomial[2*(n - i), n - i], {i, 0, n - 1}], {n, 1, 50}]] (* G. C. Greubel, Jan 31 2017 *)

for(n=1,25, print1(numerator(sum(i=0,n-1, binomial(2*(i+1),i+1)* binomial(2*(n-i), n-i))/4^n), ", ")) \\ G. C. Greubel, Jan 31 2017

a(n) = numerator of Sum_{k=1..n} C(k)/2^(2*(k-1)).
a(n-1) = numerator of (1/4^n)*Sum_{i=0..n-1} (binomial(2*(i+1), i+1)*binomial(2*(n-i), n-i)), for n>=1. - Johannes W. Meijer, May 24 2009
a(n) = (2^n-(2*n+2)!/(2^(n+1)*(n+1)!^2))*gcd((n+1)!,2^(n+1)). - Gary Detlefs, Nov 06 2020

cp's OEIS Frontend

A000466 a(n) = 4*n^2 - 1.

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A043529 Number of distinct base-2 digits of n.

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A160464 The Eta triangle.

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A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

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A160467 a(n) = 1 if n is odd; otherwise, a(n) = 2^(k-1) where 2^k is the largest power of 2 that divides n.

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A119951 Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108.

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Mathematica