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A026741 a(n) = n if n odd, n/2 if n even. 120
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 (psb(AT)maths.warwick.ac.uk), Oct 23 2003

For n>1 a(n) = greatest common divisor of all permutations of {0,1,...,n} treated as base n+1 integers. - David J. Scambler, Nov 08 2006 (see the MathStackExchange 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 Sequence A026741 have to be taken out.

In some cases two three or more sequenced entries of A026741 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 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 = an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0,...) in every column, shifted down twice. A026741 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->inf.} 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

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) = A0227042(n,1). See the formula a(n) = n/gcd(n,2) given below. - Wolfdieter Lang, Jul 04 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

REFERENCES

David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005): 53

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399, 2011.

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2

L. Euler, On the remarkable properties of the pentagonal numbers

Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Kival Ngaokrajang, Illustration of spiral with circle centers 2..5

Kival Ngaokrajang, Illustration of vertex angle of regular n-gon for n = 3..7

StackExchange, Permutations (with no duplicates) of decimal base digits 1,2,…,8,9,0

Eric Weisstein's World of Mathematics, Simplex Simplex Picking

Eric Weisstein's World of Mathematics, Lehmer Number

Index to sequences with linear recurrences with constant coefficients, signature (0,2,0,-1).

FORMULA

G.f.: (x^3+x^2+x)/(1-x^2)^2. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Apr 30 2001

a(n) = +2*a(n-2) - 1*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{2/(i*(i+1))|1<=i<=n-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).

a(n) = Abs[ Numerator[ Det[ DiagonalMatrix[ Table[ 1/i^2 -1, {i, 1, n-1} ] ] + 1 ] ] for n>1. - Alexander Adamchuk, Jun 02 2006

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) = numer(n/(2*n-2)) for n =>2; A022998(n-1) = denom(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

a(n) = A109043(n)/2. Dirichlet g.f. zeta(s-1)*(1-1/2^s). - R. J. Mathar, Apr 18 2011

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) = 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

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, 1, 20} ]]] (* Alexander Adamchuk, Jun 02 2006 *)

nn=40; Riffle[Range[0, nn], Range[1, 2nn+1, 2]] (* Harvey P. Dale, Mar 27 2011 *)

PROG

(PARI) {a(n) = if( n==0, 0, n / gcd(n, 2))} /* Michael Somos, Jun 15 2005 */

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

(Sage) [lcm(n, 2)/2 for n in xrange(0, 77)] # Zerinvary Lajos, Jun 07 2009

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

(Haskell)

import Data.List (transpose)

a026741 n = a026741_list !! n

a026741_list = concat $ transpose [[0..], [1, 3..]]

-- Reinhard Zumkeller, Dec 12 2011

CROSSREFS

Signed version is in A030640. Partial sums give A001318.

Cf. this sequence, A051176, A060819, A060791, A060789 for n / gcd(n, k) with k=2..6.

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

Cf. A013942.

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

Sequence in context: A030640 A176447 A145051 * A105658 A083242 A111618

Adjacent sequences:  A026738 A026739 A026740 * A026742 A026743 A026744

KEYWORD

nonn,easy,nice,frac,mult,changed

AUTHOR

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

EXTENSIONS

More terms from David W. Wilson; better description from Jud McCranie

Edited by Ralf Stephan, Jun 04 2003

STATUS

approved

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Last modified October 31 02:14 EDT 2014. Contains 248845 sequences.