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A008619 Positive integers repeated. 150


%S 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 15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,

%U 27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,37,38

%N Positive integers repeated.

%C The floor of the arithmetic mean of the first n+1 positive integers. - _Cino Hilliard_, Sep 06 2003

%C 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 Number of partitions of n in which the greatest part is at most 2. - Robert G. Wilson v, Jan 11 2002

%C Number of partitions of n into at most 2 parts. - Jon Perry, Jun 16 2003

%C a(n)=#{0<=k<=n: k+n is even} - Paul Barry , Sep 13 2003

%C 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, Jan 12 2004

%C 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 a(n) = A108299(n-1,n)*(-1)^floor(n/2) for n>0. - _Reinhard Zumkeller_, Jun 01 2005

%C a(n) = A108561(n+2,n) for n>0. - _Reinhard Zumkeller_, Jun 10 2005

%C Second outermost diagonal of Losanitsch's triangle (A034851). - _Alonso del Arte_, Mar 12 2006

%C Arithmetic mean of n-th row of A080511.- _Amarnath Murthy_, Mar 20 2003.

%C a(n) = A125291(A125293(n)) for n>0. - _Reinhard Zumkeller_, Nov 26 2006

%C 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 Inverse binomial transform of A045623 . [_Philippe Deléham_, Dec 30 2008]

%C 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 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. [R. J. Mathar, Sep 25 2008]

%C Binomial transform of (-1)^n*A034008(n) = [1,0,1,-2,4,-8,16,-32,...]. [_Philippe Deléham_, Nov 15 2009]

%C a(n) = A026820(n,2) for n>1. [_Reinhard Zumkeller_, Jan 21 2010]

%C Column sums of:

%C 1 1 1 1 1 1...

%C 1 1 1 1...

%C 1 1...

%C ..............

%C --------------

%C 1 1 2 2 3 3... [_Jon Perry_, Nov 16 2010]

%C a(n) = A000217(n) / A110654(n). [_Reinhard Zumkeller_, Aug 24 2011]

%C This sequence is also the half-convolution of the powers of 1 sequence A000012 with itself. For the definition of half-convolution see a comment on A201204, where also the rule for the o.g.f. is given. [_Wolfdieter Lang_, Jan 09 2012]

%C a(n+1) = A181971(n,n). - _Reinhard Zumkeller_, Jul 09 2012

%C a(n) is also the number of roots of the n-th Bernoulli polynomial in the right half-plane for n>0. [Michel Lagneau, Nov 08 2012]

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

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

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

%H Charles R Greathouse IV, <a href="/A008619/b008619.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 Gerzson Keri and Patric R. J. Ostergard, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Keri/keri6.html">The Number of Inequivalent (2R+3,7)R Optimal Covering Codes</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=120">Encyclopedia of Combinatorial Structures 120</a>

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=209">Encyclopedia of Combinatorial Structures 209</a>

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=351">Encyclopedia of Combinatorial Structures 351</a>

%H L. F. Klosinski, G. L. Alexanderson and A. P. Hillman, <a href="http://www.jstor.org/stable/2322570">The William Lowell Putnam Mathematical Competition</a>, Amer. Math. Monthly 91 (1984), 487-495. See Problem B2.

%H B. Reznick, <a href="http://dx.doi.org/10.1007/978-1-4612-3464-7_29">Some binary partition functions</a>, 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>

%H <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%H <a href="/index/Rec#order_03">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F Euler transform of [1, 1].

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

%F G.f.: 1/((1-x)(1-x^2)).

%F E.g.f.: ((3+2*x)*exp(x)+exp(-x))/4.

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

%F a(0)=a(1)=1 and a(n) = floor( (a(n-1) + a(n-2))/2 + 1 ).

%F a(n)=(2*n+3+(-1)^n)/4. - Paul Barry, May 27 2003

%F a(n)=sum{k=0..n, sum{j=0..k, sum{i=0..j, C(j, i)(-2)^i }}} - Paul Barry, Aug 26 2003

%F E.g.f.: ((1+x)*exp(x)+cosh(x))/2; - Paul Barry, Sep 13 2003

%F a(n)=Ceiling (n/2), n>=1. - _Mohammad K. Azarian_, May 22 2007

%F INVERT transformation yields A006054 without leading zeros. INVERTi transformation yields negative of A124745 with the first 5 terms there dropped. [R. J. Mathar, Sep 11 2008]

%F a(n)=n-a(n-1)+1 (with a(0)=1). [Vincenzo Librandi, Nov 19 2010]

%F 1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction))))) = 1/(e-1), see A073333. - _Philippe Deléham_, Mar 09 2013

%F a(n) = floor(A000217(n)/n), n > 0. - _L. Edson Jeffery_, Jul 26 2013

%F a(n) = n*a(n-1) mod (n+1) = -a(n-1) mod (n+1), the least positive residue modulo n+1 for each expression for n > 0, with a(0) = 1 (basically restatements of Vincenzo Librandi's formula). - _Rick L. Shepherd_, Apr 02 2014

%p with(numtheory): for n from 1 to 80 do:it:=0:

%p y:=[fsolve(bernoulli(n,x) , x, complex)] : for m from 1 to nops(y) do : if Re(y[m]) >0 then it:=it+1:else fi:od: printf(`%d, `,it):od: [Michel Lagneau, Nov 08 2012].

%t Flatten[Table[{n,n},{n,35}]] (* _Harvey P. Dale_, Sep 20 2011 *)

%t With[{c=Range[40]},Riffle[c,c]] (* _Harvey P. Dale_, Feb 23 2013 *)

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

%o (Haskell)

%o a008619 = (+ 1) . (`div` 2)

%o a008619_list = concatMap (\x -> [x,x]) [1..]

%o -- _Reinhard Zumkeller_, Apr 02 2012

%Y Essentially same as A004526.

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

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

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

%Y Cf. A010551 (partial products).

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_.

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

%E Edited by _N. J. A. Sloane_, Sep 06 2009

%E Partially edited by _Joerg Arndt_, Mar 11 2010

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Last modified December 19 08:59 EST 2014. Contains 252186 sequences.