login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008619 Positive integers repeated. 150

%I

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 28 19:08 EST 2014. Contains 250399 sequences.