%I A008619
%S A008619 1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,
%T A008619 15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,
%U A008619 27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,37,38
%N A008619 Positive integers repeated.
%C A008619 The floor of the arithmetic mean of the first n+1 positive integers.
- Cino Hilliard (hillcino368(AT)gmail.com), Sep 06 2003
%C A008619 Number of partitions of n into powers of 2 where no power is used more
than three times, or 4th binary partition function (see A072170).
%C A008619 Number of partitions of n in which the greatest part is at most 2. -
Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 11 2002
%C A008619 Number of partitions of n into at most 2 parts. - Jon Perry (perry(AT)globalnet.co.uk),
Jun 16 2003
%C A008619 a(n)=#{0<=k<=n: k+n is even} - Paul Barry (pbarry(AT)wit.ie), Sep 13
2003
%C A008619 Number of symmetric Dyck paths of semilength n+2 and having two peaks.
E.g. a(6)=4 because we have UUUUUUU*DU*DDDDDDD, UUUUUU*DDUU*DDDDDD,
UUUUU*DDDUUU*DDDDD and UUUU*DDDDUUUU*DDDD, where U=(1,1), D=(1,-1)
and * indicates a peak. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Jan 12 2004
%C A008619 Smallest positive integer whose harmonic mean with another positive integer
is n (for n > 0). For example, a(6)=4 is already given (as 4 is the
smallest positive integer such that the harmonic mean of 4 (with
12) is 6) - but the harmonic mean of 2 (with -6) is also 6 and 2
< 4, so the two positive integer restrictions need to be imposed
to rule out both 2 and -6.
%C A008619 a(n) = A108299(n-1,n)*(-1)^floor(n/2) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 01 2005
%C A008619 a(n) = A108561(n+2,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 10 2005
%C A008619 Second outermost diagonal of Losanitsch's triangle (A034851). - Alonso
Delarte (alonso.delarte(AT)gmail.com), Mar 12 2006
%C A008619 Arithmetic mean of n-th row of A080511.- Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
Mar 20 2003.
%C A008619 a(n) = A125291(A125293(n)) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 26 2006
%C A008619 a(n) is the number of ways to pay n euros (or dollars) with coins of
one and two euros (respectively dollars). - Richard Choulet and Robert
G. Wilson v, Dec 31 2007
%C A008619 Inverse binomial transform of A045623 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Dec 30 2008]
%C A008619 Coefficient of q^n in the expansion of (m choose 2)_q as m goes to infinity.
- Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%C A008619 This Itakura comment follows from a partial fraction decomposition (m
choose 2)_q = [(1-q^(2m-2))/(1+q) + (1-q^(2m-2))/(1-q) +2 (1-q^(m-1))^2/
(1-q)^2]/4. Interpreted as generating functions in q, they have convolution
structures; the first term in the numerator creates +1,-1,+1,-1 etc,
the 2nd term creates +1,+1,+1,+1 etc, the 3rd term 2,4,6,8 etc. as
m->infinity. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep
25 2008]
%C A008619 Binomial transform of (-1)^n*A034008(n) = [1,0,1,-2,4,-8,16,-32,...].
[From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 15 2009]
%D A008619 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993,
p. 100.
%D A008619 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 109, Eq. [6c]; p.
116, P(n,2).
%D A008619 Gerzson Keri and Patric R. J. Ostergard, The Number of Inequivalent (2R+3,
7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9
(2006), Article 06.4.7.
%D A008619 Klosinski, L.F., G. L. Alexanderson and A. P.Hillman, The William Lowell
Putnam Mathematical Competition, Amer. Math. Monthly 91 (1984), 487-495.
See Problem B2.
%D A008619 D. Parisse, 'The tower of Hanoi and the Stern-Brocot Array', Thesis,
Munich 1997
%D A008619 B. Reznick, Some binary partition functions, in "Analytic number theory"
(Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477,
Progr. Math., 85, Birkhaeuser Boston, Boston, MA, 1990.
%H A008619 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A008619 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A008619 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=120">
Encyclopedia of Combinatorial Structures 120</a>
%H A008619 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=209">
Encyclopedia of Combinatorial Structures 209</a>
%H A008619 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=351">
Encyclopedia of Combinatorial Structures 351</a>
%H A008619 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>
%H A008619 <a href="Sindx_St.html#Stern">Index entries for sequences related to
Stern's sequences</a>
%H A008619 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%F A008619 Euler transform of [1, 1].
%F A008619 a(n)=1+floor(n/2).
%F A008619 G.f.: 1/((1-x)(1-x^2)). E.g.f.: ((3+2x)exp(x)+exp(-x))/4.
%F A008619 a(n)=a(n-1)+a(n-2)-a(n-3)=-a(-3-n).
%F A008619 a(0)=a(1)=1 and a(n) = floor[ (a(n-1) + a(n-2))/2 + 1].
%F A008619 a(n)=(2n+3+(-1)^n)/4. - Paul Barry (pbarry(AT)wit.ie), May 27 2003
%F A008619 a(n)=sum{k=0..n, sum{j=0..k, sum{i=0..j, C(j, i)(-2)^i }}} - Paul Barry
(pbarry(AT)wit.ie), Aug 26 2003
%F A008619 E.g.f.: ((1+x)exp(x)+cosh(x))/2; - Paul Barry (pbarry(AT)wit.ie), Sep
13 2003
%F A008619 a(n)=Ceiling (n/2), n>=1. - Mohammad K. Azarian (azarian(AT)evansville.edu),
May 22 2007
%F A008619 a(n)=n-a(n-1) (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 21 2009]
%e A008619 (2 choose 2)_q = 1, (3 choose 2)_q = q^2 + q + 1, (4 choose 2)_q = q^4
+ q^3 + 2*q^2 + q + 1, (5 choose 2)_q = q^6 + q^5 + 2*q^4 + 2*q^3
+ 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to
1, q^2 to 2 and so on.
%e A008619 For n=2, a(2)=2-1=1; n=3, a(3)=3-1=2; n=4, a(4)=4-2=2; n=5, a(5)=5-2=3
[From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009]
%p A008619 B:=[S,{S = Set(Sequence(Z,1 <= card),card <=2)},unlabelled]: seq(combstruct[count](B,
size=n), n=0..74);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Mar 21 2009]
%t A008619 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)), {x, 0, 65} ], x ] or
Table[ Floor[(n + 1)/2], {n, 1, 100} ] or a[1] = a[2] = 1; a[n_]
:= a[n] = Floor[(a[n - 1] + a[n - 2])/2 + 1]; Table[ a[n], {n, 1,
76} ]
%o A008619 (PARI) a(n)=n\2+1
%Y A008619 Essentially same as A004526.
%Y A008619 Harmonic mean of a(n) and A056136 is n.
%Y A008619 Cf. A001057, A065033, A001399, A001400, A001401.
%Y A008619 a(n)=A010766(n+2, 2).
%Y A008619 INVERT transformation yields A006054 without leading zeros. INVERTi transformation
yields negative of A124745 with the first 5 terms there dropped.
[From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 11 2008]
%Y A008619 Sequence in context: A004526 A140106 A123108 this_sequence A110654 A109728
A157271
%Y A008619 Adjacent sequences: A008616 A008617 A008618 this_sequence A008620 A008621
A008622
%K A008619 nonn,easy,nice,new
%O A008619 0,3
%A A008619 N. J. A. Sloane (njas(AT)research.att.com).
%E A008619 Additional remarks from Daniele Parisse (daniele.parisse(AT)m.dasa.de).
%E A008619 Edited by N. J. A. Sloane, Sep 06 2009
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