Search: id:A008619 Results 1-1 of 1 results found. %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 Index entries for sequences related to linear recurrences with constant coefficients %H A008619 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A008619 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 120 %H A008619 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 209 %H A008619 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 351 %H A008619 Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature %H A008619 Index entries for sequences related to Stern's sequences %H A008619 Index entries for related partition-counting sequences %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 Search completed in 0.002 seconds