A026741 a(n) = n if n odd, n/2 if n even.

0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34, 69, 35, 71, 36, 73, 37, 75, 38

Offset: 0

Author

J. Carl Bellinger (carlb(AT)ctron.com)

Keywords

• nonn
• easy
• nice
• frac
• mult

Comments

a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington, Oct 23 2003

For n > 1, a(n) is the greatest common divisor of all permutations of {0, 1, ..., n} treated as base n + 1 integers. - David Scambler, Nov 08 2006 (see the Mathematics Stack Exchange link below).

From Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009: (Start)

Sequence A075888 and the above sequence are fitting together.

First 2 entries of this sequence have to be taken out.

In some cases two three or more sequenced entries of this sequence have to be added together to get the next entry of A075888.

Example: Sequences begin with 1, 3, 2, 5, 3, 7, 4, 9 (4 + 9 = 13, the next entry in A075888).

But it works out well up to primes around 50000 (haven't tested higher ones).

As A075888 gives a very regular graph. There seems to be a regularity in the primes. (End)

Starting with 1 = triangle A115359 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009

From Gary W. Adamson, Dec 11 2009: (Start)

Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column, shifted down twice. This sequence starting with 1 = M * (1, 2, 3, ...)

M =

1;

1, 0;

1, 1, 0;

0, 1, 0, 0;

0, 1, 1, 0, 0;

0, 0, 1, 0, 0, 0;

0, 0, 1, 1, 0, 0, 0;

...

A026741 = M * (1, 2, 3, ...); but A002487 = lim_{n->infinity} M^n, a left-shifted vector considered as a sequence. (End)

A particular case of sequence for which a(n+3) = (a(n+2) * a(n+1)+q)/a(n) for every n > n0. Here n0 = 1 and q = -1. - Richard Choulet, Mar 01 2010

For n >= 2, a(n+1) is the smallest m such that s_n(2*m*(n-1))/(n-1) is even, where s_b(c) is the sum of digits of c in base b. - Vladimir Shevelev, May 02 2011

A001477 and A005408 interleaved. - Omar E. Pol, Aug 22 2011

Numerator of n/((n-1)*(n-2)). - Michael B. Porter, Mar 18 2012

Number of odd terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013

For n >= 3, a(n) is the periodic of integer of spiral length ratio of spiral that have (n-1) circle centers. See illustration in links. - Kival Ngaokrajang, Dec 28 2013

This is the sequence of Lehmer numbers u_n(sqrt(R), Q) with the parameters R = 4 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all natural numbers n and m. Cf. A005013 and A108412. - Peter Bala, Apr 18 2014

The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) = 2 begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [0, 1, 4, 3, 8, 5, 12, ...] is A022998, also a strong divisibility sequence. - Peter Bala, May 19 2014

For n >= 3, (a(n-2)/a(n))*Pi = vertex angle of a regular n-gon. See illustration in links. - Kival Ngaokrajang, Jul 17 2014

For n > 1, the numerator of the harmonic mean of the first n triangular numbers. - Colin Barker, Nov 13 2014

The difference sequence is a permutation of the integers. - Clark Kimberling, Apr 19 2015

From Timothy Hopper, Feb 26 2017: (Start)

Given the function a(n, p) = n/p if n mod p = 0, else n, then a possible formula is: a(n, p) = n*(1 + (p-1)*((n^(p-1)) mod p))/p, p prime, (n^(p-1)) mod p = 1, n not divisible by p. (Fermat's Little Theorem). Examples: p = 2; a(n), p = 3; A051176(n), p = 5; A060791(n), p = 7; A106608(n).

Conjecture: lcm(n, p) = p*a(n, p), gcd(n, p) = n/a(n, p). (End)

Let r(n) = (a(n+1) + 1)/a(n+1) if n mod 2 = 1, a(n+1)/(a(n+1) + 2) otherwise; then lim_{k->oo} 2^(k+2) * Product_{n=0..k} r(n)^(k-n) = Pi. - Dimitris Valianatos, Mar 22 2021

Number of integers k from 1 to n such that gcd(n,k) is odd. - Amiram Eldar, May 18 2025

Examples

			G.f. = x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 3*x^6 + 7*x^7 + 4*x^8 + ...
		

References

• David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 53.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. Penguin (1997), p. 79.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• 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.
• John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.
• Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2.
• Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
• Y. Ito and I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.
• Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.
• Mathematics Stack Exchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.
• Kival Ngaokrajang, Illustration of spiral with circle centers 2..5.
• Kival Ngaokrajang, Illustration of vertex angle of regular n-gon for n = 3..7.
• Eric Weisstein's World of Mathematics, Lehmer Number.
• Eric Weisstein's World of Mathematics, Simplex Simplex Picking.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
• Index to divisibility sequences.

Crossrefs

Signed version is in A030640. Partial sums give A001318.

Cf. A051176, A060819, A060791, A060789 for n / gcd(n, k) with k = 3..6. See also A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Cf. A045896, A022998, A060762, A126988, A109007, A130334, A109043, A115359, A002487, A220466.

Cf. A013942.

Cf. A227042 (first column). Cf. A005013 and A108412.

Cf. A000217, A256095.

Programs

Haskell

import Data.List (transpose)
a026741 n = a026741_list !! n
a026741_list = concat $ transpose [[0..], [1,3..]]
-- Reinhard Zumkeller, Dec 12 2011

Magma

[2*n/(3+(-1)^n): n in [0..70]]; // Vincenzo Librandi, Aug 14 2011

Maple

A026741 := proc(n) if type(n,'odd') then n; else n/2; end if; end proc: seq(A026741(n), n=0..76); # R. J. Mathar, Jan 22 2011

Mathematica

Numerator[Abs[Table[Det[DiagonalMatrix[Table[1/i^2 - 1, {i, 1, n - 1}]] + 1], {n, 20}]]] (* Alexander Adamchuk, Jun 02 2006 *)
halfMax = 40; Riffle[Range[0, halfMax], Range[1, 2halfMax + 1, 2]] (* Harvey P. Dale, Mar 27 2011 *)
a[ n_] := Numerator[n / 2]; (* Michael Somos, Jan 20 2017 *)
Array[If[EvenQ[#],#/2,#]&,80,0] (* Harvey P. Dale, Jul 08 2023 *)

PARI

a(n) = numerator(n/2) \\ Rick L. Shepherd, Sep 12 2007

Python

def A026741(n): return n if n % 2 else n//2 # Chai Wah Wu, Apr 02 2021

Sage

[lcm(n, 2) / 2 for n in range(77)] # Zerinvary Lajos, Jun 07 2009

Formula

G.f.: x*(1 + x + x^2)/(1-x^2)^2. - Len Smiley, Apr 30 2001

a(n) = 2*a(n-2) - a*(n-4) for n >= 4.

a(n) = n * 2^((n mod 2) - 1). - Reinhard Zumkeller, Oct 16 2001

a(n) = 2*n/(3 + (-1)^n). - Benoit Cloitre, Mar 24 2002

Multiplicative with a(2^e) = 2^(e-1) and a(p^e) = p^e, p > 2. - Vladeta Jovovic, Apr 05 2002

a(n) = n / gcd(n, 2). a(n)/A045896(n) = n/((n+1)*(n+2)).

For n > 0, a(n) = denominator of Sum_{i=1..n-1} 2/(i*(i+1)), numerator=A022998. - Reinhard Zumkeller, Apr 21 2012, Jul 25 2002 [thanks to Phil Carmody who noticed an error]

For n > 1, a(n) = GCD of the n-th and (n-1)-th triangular numbers (A000217). - Ross La Haye, Sep 13 2003

Euler transform of finite sequence [1, 2, -1]. - Michael Somos, Jun 15 2005

G.f.: x * (1 - x^3) / ((1 - x) * (1 - x^2)^2) = Sum_{k>0} k * (x^k - x^(2*k)). - Michael Somos, Jun 15 2005

a(n+3) + a(n+2) = 3 + a(n+1) + a(n). a(n+3) * a(n) = - 1 + a(n+2) * a(n+1). a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 15 2005

For n > 1, a(n) is the numerator of the average of 1, 2, ..., n - 1; i.e., numerator of A000217(n-1)/(n-1), with corresponding denominators [1, 2, 1, 2, ...] (A000034). - Rick L. Shepherd, Jun 05 2006

Equals A126988 * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007

For n >= 1, a(n) = gcd(n,A000217(n)). - Rick L. Shepherd, Sep 12 2007

a(n) = numerator(n/(2*n-2)) for n >= 2; A022998(n-1) = denominator(n/(2*n-2)) for n >= 2. - Johannes W. Meijer, Jun 18 2009

a(n) = A167192(n+2, 2). - Reinhard Zumkeller, Oct 30 2009

a(n) = A106619(n) * A109012(n). - Paul Curtz, Apr 04 2011

From R. J. Mathar, Apr 18 2011: (Start)

a(n) = A109043(n)/2.

Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s). (End)

a(n) = A001318(n) - A001318(n-1) for n > 0. - Jonathan Sondow, Jan 28 2013

a((2*n+1)*2^p - 1) = 2^p - 1 + n*A151821(p+1), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 03 2013

a(n+1) = denominator(H(n, 1)), n >= 0, with H(n, 1) = 2*n/(n+1) the harmonic mean of n and 1. a(n+1) = A227042(n, 1). See the formula a(n) = n/gcd(n, 2) given above. - Wolfdieter Lang, Jul 04 2013

a(n) = numerator(n/2). - Wesley Ivan Hurt, Oct 02 2013

a(n) = numerator(1 - 2/(n+2)), n >= 0; a(n) = denominator(1 - 2/n), n >= 1. - Kival Ngaokrajang, Jul 17 2014

a(n) = Sum_{i = floor(n/2)..floor((n+1)/2)} i. - Wesley Ivan Hurt, Apr 27 2016

Euler transform of length 3 sequence [1, 2, -1]. - Michael Somos, Jan 20 2017

G.f.: x / (1 - x / (1 - 2*x / (1 + 7*x / (2 - 9*x / (7 - 4*x / (3 - 7*x / (2 + 3*x))))))). - Michael Somos, Jan 20 2017

From Peter Bala, Mar 24 2019: (Start)

a(n) = Sum_{d|n, n/d odd} phi(d), where phi(n) is the Euler totient function A000010.

O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 - x^(2*n)). (End)

a(n) = A256095(2*n,n). - Alois P. Heinz, Jan 21 2020

E.g.f.: x*(2*cosh(x) + sinh(x))/2. - Stefano Spezia, Apr 28 2023

From Ctibor O. Zizka, Oct 05 2023: (Start)

For k >= 0, a(k) = gcd(k + 1, k*(k + 1)/2).

If (k mod 4) = 0 or 2 then a(k) = (k + 1).

If (k mod 4) = 1 or 3 then a(k) = (k + 1)/2. (End)

Sum_{n=1..oo} 1/a(n)^2 = 7*Pi^2/24. - Stefano Spezia, Dec 02 2023

a(n)*a(n+1) = A000217(n). - Rémy Sigrist, Mar 19 2025

Extensions

Better description from Jud McCranie

Edited by Ralf Stephan, Jun 04 2003