0,3

The floor of the arithmetic mean of the first n+1 positive integers. - Cino Hilliard (hillcino368(AT)gmail.com), Sep 06 2003

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).

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

Number of partitions of n into at most 2 parts. - Jon Perry (perry(AT)globalnet.co.uk), Jun 16 2003

a(n)=#{0<=k<=n: k+n is even} - Paul Barry (pbarry(AT)wit.ie), Sep 13 2003

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

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.

a(n) = A108299(n-1,n)*(-1)^floor(n/2) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005

a(n) = A108561(n+2,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 10 2005

Second outermost diagonal of Losanitsch's triangle (A034851). - Alonso Delarte (alonso.delarte(AT)gmail.com), Mar 12 2006

Arithmetic mean of n-th row of A080511.- Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 20 2003.

a(n) = A125291(A125293(n)) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 26 2006

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

Inverse binomial transform of A045623 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 30 2008]

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

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]

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. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 109, Eq. [6c]; p. 116, P(n,2).

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.

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. Parisse, 'The tower of Hanoi and the Stern-Brocot Array', Thesis, Munich 1997

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.

Index entries for sequences related to linear recurrences with constant coefficients

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 120

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 209

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 351

Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature

Index entries for sequences related to Stern's sequences

Index entries for related partition-counting sequences

Euler transform of [1, 1].

a(n)=1+floor(n/2).

G.f.: 1/((1-x)(1-x^2)). E.g.f.: ((3+2x)exp(x)+exp(-x))/4.

a(n)=a(n-1)+a(n-2)-a(n-3)=-a(-3-n).

a(0)=a(1)=1 and a(n) = floor[ (a(n-1) + a(n-2))/2 + 1].

a(n)=(2n+3+(-1)^n)/4. - Paul Barry (pbarry(AT)wit.ie), May 27 2003

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

E.g.f.: ((1+x)exp(x)+cosh(x))/2; - Paul Barry (pbarry(AT)wit.ie), Sep 13 2003

a(n)=Ceiling (n/2), n>=1. - Mohammad K. Azarian (azarian(AT)evansville.edu), May 22 2007

a(n)=n-a(n-1) (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009]

(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.

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]

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]

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} ]

(PARI) a(n)=n\2+1

Essentially same as A004526.

Harmonic mean of a(n) and A056136 is n.

Cf. A001057, A065033, A001399, A001400, A001401.

a(n)=A010766(n+2, 2).

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]

Sequence in context: A004526 A140106 A123108 this_sequence A110654 A109728 A157271

Adjacent sequences: A008616 A008617 A008618 this_sequence A008620 A008621 A008622

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Additional remarks from Daniele Parisse (daniele.parisse(AT)m.dasa.de).

Edited by N. J. A. Sloane, Sep 06 2009