0,4

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 (dscambler(AT)bmm.com), Nov 08 2006

a(n) is the second principal diagonal of array with rows 1) A005563, 2) (first bisection of A061037)=A142705, 3) (first trisection of A061039)=A144454, 4) first quadrisection of A061041, 5) first quintisection of A061043, 6) first hexasection A061045, 7)first heptasection of A061047, 8) first octosection of A061049, 9) , 10) , 11) , . From Rydberg-Ritz denominators of spectra of hydrogen atom: 1) 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 2) 0, 3, 2, 15, 6, 35, 12, 63, 20, 99, 39, 143, 3) 0, 1, 8, 5, 8, 35, 16, 7, 80, 11, 40, 143, 4) 0, 3, 1, 15, 3, 35, 3, 63, 5, 99, 15, 143, 5) 0, 3, 8, 3, 24, 7, 48, 63, 16, 99, 6) 0, 1, 2, 5, 2, 35, 4, 7, 20, 7) 0, 3, 8, 15, 24, 5, 48, 9, 8) 0, 3, 1, 15, 3, 35, 3, 63, 5, 99, 15, 143, 9) 0, 1, 8, 5, 8, 35, 16, 7, 80, 11, 40, 143, 10) 0, 3, 2, 3, 6, 7, 12, 63, 4, 99, 6, 11) 0, 3, 8, 15, 24, 35, 48, 63, 80, 9, 120, 13, Thanks to Richard Mathar, April 29, for last four rows. Note first upper principal diagonal 3, 2, 5, 3, 7, 4, 9, =A026741(n+3). [From Paul Curtzz (bpcrtz(AT)free.fr), Sep 13 2009]

a(n) = A167192(n+2,2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 30 2009]

Contribution from Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009: (Start) The array of a(n) and its higher order differences shows essentially 0,0 followed by -3*A001792(.) on the main diagonal:

0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21,...

1, 0, 2, -1, 3, -2, 4, -3, 5, -4, 6, -5, 7, -6, 8, -7, 9, -8, 10, -9, 11,...

-1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18,...

3, -5, 7, -9, 11, -13, 15, -17, 19, -21, 23, -25, 27, -29, 31, -33, 35,...

-8, 12, -16, 20, -24, 28, -32, 36, -40, 44, -48, 52, -56, 60, -64, 68,...

20, -28, 36, -44, 52, -60, 68, -76, 84, -92, 100, -108, 116, -124, 132,... (END)

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

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

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

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

Eric Weisstein's World of Mathematics, Simplex Simplex Picking

Index entries for sequences related to linear recurrences with constant coefficients

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

a(n) = n * 2^((n mod 2) - 1) - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 16 2001

a(n) = 2*n/(3+(-1)^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 24 2002

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

a(n) = n / gcd(n, 2). a(n)/A04589(n) = n/((n+1)(n+2)).

For n>1, a(n) = denominator of sum{2/(i*(i+1))|1<=i<=n-1}, numerator=A026741. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 25 2002

For n > 1, a(n) = GCD of the n-th and (n-1)th triangular numbers (A000217). - Ross La Haye (rlahaye(AT)new.rr.com), 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^(2k)). - Michael Somos Jun 15 2005

a(n)a(n+3) = - 1 + a(n+1)a(n+2). 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 (alex(AT)kolmogorov.com), 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 (rshepherd2(AT)hotmail.com), Jun 05 2006

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

For n >= 1, a(n) = GCD(n,A000217(n)). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 12 2007

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009: (Start)

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

(End)

a(n+1)-a(n) = (-1)^n*A028242(n) (first differences). a(n+2)-2*a(n+1)+a(n) = (-1)^(n+1)*(n+1). (2nd differences). a(n+3)-3*a(n+2)+3*a(n+1)-a(n) = (-1)^n*A144396(n+1). (3rd differences). [Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009]

a(A131577(n)) = A166444(n). [Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009]

a:=n->add(2+add((-1)^j, j=1..n), j=2..n):seq(a(n)/2, n=1..74); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 13 2008]

Numerator[Abs[Table[ Det[ DiagonalMatrix[ Table[ 1/i^2 -1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ]]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 02 2006

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

(PARI) a(n) = numerator(n/2) /* Rick Shepherd Sep 12 2007 */ - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 12 2007

(Other) sage: [lcm(n, 2)/2for n in xrange(0, 77)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2009]

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.

Cf. A126988.

Cf. A109007, A130334.

Cf. A109043 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 13 2008]

Contribution 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 prime around 50.000 (havent tested higher ones).

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

Sequence in context: A076605 A030640 A145051 this_sequence A105658 A083242 A111618

Adjacent sequences: A026738 A026739 A026740 this_sequence A026742 A026743 A026744

nonn,easy,nice,frac,mult,new

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

More terms from David W. Wilson (davidwwilson(AT)comcast.net); better description from Jud McCranie (j.mccranie(AT)comcast.net)

Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 04 2003