%I A026741
%S A026741 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,
%T A026741 29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,25,51,
26,
%U A026741 53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34,69,35,71,36,73,37,75,
38
%N A026741 a(n) = n if n odd, n/2 if n even.
%C A026741 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
%C A026741 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
%C A026741 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
%C A026741 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]
%C A026741 a(n) = A167192(n+2,2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 30 2009]
%C A026741 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:
%C A026741 0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10,
21,...
%C A026741 1, 0, 2, -1, 3, -2, 4, -3, 5, -4, 6, -5, 7, -6, 8, -7, 9, -8, 10, -9,
11,...
%C A026741 -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18,
...
%C A026741 3, -5, 7, -9, 11, -13, 15, -17, 19, -21, 23, -25, 27, -29, 31, -33, 35,
...
%C A026741 -8, 12, -16, 20, -24, 28, -32, 36, -40, 44, -48, 52, -56, 60, -64, 68,
...
%C A026741 20, -28, 36, -44, 52, -60, 68, -76, 84, -92, 100, -108, 116, -124, 132,
... (END)
%D A026741 Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends
in algebraic geometry (Warwick, 1996), 151-233, Cambridge University
Press, 1999.
%H A026741 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">
De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2
%H A026741 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable
properties of the pentagonal numbers</a>
%H A026741 M. Kaneko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer
Sequences, 3 (2000), #00.2.9.
%H A026741 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SimplexSimplexPicking.html">Simplex Simplex Picking</a>
%H A026741 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A026741 G.f.: (x^3+x^2+x)/(1-x^2)^2 - Len Smiley (smiley(AT)math.uaa.alaska.edu),
Apr 30 2001
%F A026741 a(n) = n * 2^((n mod 2) - 1) - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 16 2001
%F A026741 a(n) = 2*n/(3+(-1)^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar
24 2002
%F A026741 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
%F A026741 a(n) = n / gcd(n, 2). a(n)/A04589(n) = n/((n+1)(n+2)).
%F A026741 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
%F A026741 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
%F A026741 Euler transform of finite sequence [1, 2, -1]. - Michael Somos Jun 15
2005
%F A026741 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
%F A026741 a(n)a(n+3) = - 1 + a(n+1)a(n+2). a(-n)=-a(n).
%F A026741 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
%F A026741 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
%F A026741 Equals A126988 * (1, -1, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Apr 17 2007
%F A026741 For n >= 1, a(n) = GCD(n,A000217(n)). - Rick L. Shepherd (rshepherd2(AT)hotmail.com),
Sep 12 2007
%F A026741 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18
2009: (Start)
%F A026741 a(n) = numer(n/(2*n-2)) for n =>2; A022998(n-1) = denom(n/(2*n-2)) for
n =>2.
%F A026741 (End)
%F A026741 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]
%F A026741 a(A131577(n)) = A166444(n). [Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009]
%p A026741 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]
%t A026741 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
%o A026741 (PARI) a(n)=if(n==0, 0, n/gcd(n,2)) /* Michael Somos Jun 15 2005 */
%o A026741 (PARI) a(n) = numerator(n/2) /* Rick Shepherd Sep 12 2007 */ - Rick L.
Shepherd (rshepherd2(AT)hotmail.com), Sep 12 2007
%o A026741 (Other) sage: [lcm(n,2)/2for n in xrange(0, 77)] # [From Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 07 2009]
%Y A026741 Signed version is in A030640. Partial sums give A001318.
%Y A026741 Cf. this sequence, A051176, A060819, A060791, A060789 for n / gcd(n,
k) with k=2..6.
%Y A026741 Cf. A045896, A022998, A060762.
%Y A026741 Cf. A126988.
%Y A026741 Cf. A109007, A130334.
%Y A026741 Cf. A109043 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec
13 2008]
%Y A026741 Contribution from Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009:
(Start)
%Y A026741 Sequence A075888 and the above sequence are fitting together.
%Y A026741 First 2 entries of Sequence A026741 have to be taken out.
%Y A026741 In some cases two three or more sequenced entries of A026741 have to
be added together to get the next entry of A075888.
%Y A026741 Example: Sequences begin with 1,3,2,5,3,7,4,9 (4+9 = 13 next entry in
A075888.
%Y A026741 But it works out well up to prime around 50.000 (havent tested higher
ones).
%Y A026741 As A075888 gives a very regular graph. There seems to be a regularity
in the primes. (End)
%Y A026741 Sequence in context: A076605 A030640 A145051 this_sequence A105658 A083242
A111618
%Y A026741 Adjacent sequences: A026738 A026739 A026740 this_sequence A026742 A026743
A026744
%K A026741 nonn,easy,nice,frac,mult,new
%O A026741 0,4
%A A026741 J. Carl Bellinger (carlb(AT)ctron.com)
%E A026741 More terms from David W. Wilson (davidwwilson(AT)comcast.net); better
description from Jud McCranie (j.mccranie(AT)comcast.net)
%E A026741 Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 04 2003
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