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

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).

Problem B2 in Klosinski, L.F.,G. L. Alexanderson and A. P.Hillman, The William Lowell Putnam Mathematical Competition, Amer. Math. Monthly 91 (1984), 487-495.

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.

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

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.

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

Adjacent sequences: A008616 A008617 A008618 this_sequence A008620 A008621 A008622

Sequence in context: A130472 A004526 A123108 this_sequence A110654 A109728 A025162

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]

nonn,easy,nice

njas

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