

A008619


Positive integers repeated.


218



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, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

The floor of the arithmetic mean of the first n+1 positive integers.  Cino Hilliard, 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, Jan 11 2002
Number of partitions of n into at most 2 parts.  Jon Perry, Jun 16 2003
a(n) = #{k=0..n: k+n is even}.  Paul Barry, 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, 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.
Second outermost diagonal of Losanitsch's triangle (A034851).  Alonso del Arte, Mar 12 2006
Arithmetic mean of nth row of A080511.  Amarnath Murthy, Mar 20 2003
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.  Philippe Deléham, 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 = [(1q^(2m2))/(1+q) + (1q^(2m2))/(1q) +2 (1q^(m1))^2/(1q)^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
Binomial transform of (1)^n*A034008(n) = [1,0,1,2,4,8,16,32,...].  Philippe Deléham, Nov 15 2009
From Jon Perry_, Nov 16 2010: (Start)
Column sums of:
1 1 1 1 1 1...
1 1 1 1...
1 1...
..............

1 1 2 2 3 3... (End)
This sequence is also the halfconvolution of the powers of 1 sequence A000012 with itself. For the definition of halfconvolution see a comment on A201204, where also the rule for the o.g.f. is given.  Wolfdieter Lang, Jan 09 2012
a(n) is also the number of roots of the nth Bernoulli polynomial in the right halfplane for n>0.  Michel Lagneau, Nov 08 2012
a(n) is the number of symmetryallowed, linearlyindependent terms at nth order in the series expansion of the Exe vibronic perturbation matrix, H(Q) (cf. Viel & Eisfeld).  Bradley Klee, Jul 21 2015
a(n) is the number of distinct integers in the nth row of Pascal's triangle.  Melvin Peralta, Feb 03 2016
a(n+1) for n >= 3 is the diameter of the Generalized Petersen Graph G(n, 1).  Nick Mayers, Jun 06 2016
The arithmetic function v_1(n,2) as defined in A289198.  Robert Price, Aug 22 2017
Also, this sequence is the second column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x.  Mohammad K. Azarian, Jul 18 2018
a(n+2) is the least k such that given any k integers, there exist two of them whose sum or difference is divisible by n.  Pablo Hueso Merino, May 09 2020
Column k = 2 of A051159.  John Keith, Jun 28 2021


REFERENCES

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).
D. Parisse, 'The tower of Hanoi and the SternBrocot Array', Thesis, Munich 1997


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 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
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
L. F. Klosinski, G. L. Alexanderson and A. P. Hillman, The William Lowell Putnam Mathematical Competition, Amer. Math. Monthly 91 (1984), 487495. See Problem B2.
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
A. Viel and W. Eisfeld, Effects of higher order JahnTeller coupling on the nuclear dynamics, J. Chem. Phys., 120, 4603 (2004).
Eric Weisstein's World of Mathematics, LegendreGauss Quadrature
Index entries for sequences related to Stern's sequences
Index entries for related partitioncounting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,1).
Index entries for Molien series


FORMULA

Euler transform of [1, 1].
a(n) = 1 + floor(n/2).
G.f.: 1/((1x)(1x^2)).
E.g.f.: ((3+2*x)*exp(x) + exp(x))/4.
a(n) = a(n1) + a(n2)  a(n3) = a(3n).
a(0) = a(1) = 1 and a(n) = floor( (a(n1) + a(n2))/2 + 1 ).
a(n) = (2*n + 3 + (1)^n)/4.  Paul Barry, May 27 2003
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} binomial(j, i)*(2)^i.  Paul Barry, Aug 26 2003
E.g.f.: ((1+x)*exp(x) + cosh(x))/2.  Paul Barry, Sep 13 2003
a(n) = A108299(n1,n)*(1)^floor(n/2) for n > 0.  Reinhard Zumkeller, Jun 01 2005
a(n) = A108561(n+2,n) for n > 0.  Reinhard Zumkeller, Jun 10 2005
a(n) = A125291(A125293(n)) for n>0.  Reinhard Zumkeller, Nov 26 2006
a(n) = ceiling(n/2), n >= 1.  Mohammad K. Azarian, May 22 2007
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
a(n) = A026820(n,2) for n > 1.  Reinhard Zumkeller, Jan 21 2010
a(n) = n  a(n1) + 1 (with a(0)=1).  Vincenzo Librandi, Nov 19 2010
a(n) = A000217(n) / A110654(n).  Reinhard Zumkeller, Aug 24 2011
a(n+1) = A181971(n,n).  Reinhard Zumkeller, Jul 09 2012
1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction))))) = 1/(e1), see A073333.  Philippe Deléham, Mar 09 2013
a(n) = floor(A000217(n)/n), n > 0.  L. Edson Jeffery, Jul 26 2013
a(n) = n*a(n1) mod (n+1) = a(n1) 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
a(n) = (a(0) + a(1) + ... + a(n1))/a(n1), where a(0) = 1.  Melvin Peralta, Jun 16 2015
a(n) = Sum_{k=0..n} (1)^(nk) * (k+1).  Rick L. Shepherd, Sep 18 2020
a(n) = a(n2) + 1 for n >= 2.  Vladimír Modrák, Sep 29 2020
a(n) = A004526(n)+1.  Chai Wah Wu, Jul 07 2022


MAPLE

a:= n> iquo(n+2, 2): seq(a(n), n=0..75);


MATHEMATICA

Flatten[Table[{n, n}, {n, 35}]] (* Harvey P. Dale, Sep 20 2011 *)
With[{c=Range[40]}, Riffle[c, c]] (* Harvey P. Dale, Feb 23 2013 *)
CoefficientList[Series[1/(1  x  x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 75] (* Robert G. Wilson v, Feb 05 2015 *)
Table[QBinomial[n, 2, 1], {n, 2, 75}] (* John Keith, Jun 28 2021 *)


PROG

(PARI) a(n)=n\2+1
(Haskell)
a008619 = (+ 1) . (`div` 2)
a008619_list = concatMap (\x > [x, x]) [1..]
 Reinhard Zumkeller, Apr 02 2012
(Sage)
a = lambda n: 1 if n==0 else a(n1)+1 if 2.divides(n) else a(n1) # Peter Luschny, Feb 05 2015
(Magma) I:=[1, 1, 2]; [n le 3 select I[n] else Self(n1)+Self(n2)Self(n3): n in [1..100]]; // Vincenzo Librandi, Feb 04 2015
(Scala) (2 to 99).map(_ / 2) // Alonso del Arte, May 09 2020
(Python)
def A008619(n): return (n>>1)+1 # Chai Wah Wu, Jul 07 2022


CROSSREFS

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).
Cf. A010551 (partial products).
Cf. A263997 (a block spiral).
Cf. A289187.
Column 2 of A235791.
Sequence in context: A004526 A140106 A123108 * A110654 A350899 A350898
Adjacent sequences: A008616 A008617 A008618 * A008620 A008621 A008622


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Additional remarks from Daniele Parisse (daniele.parisse(AT)m.dasa.de)
Edited by N. J. A. Sloane, Sep 06 2009
Partially edited by Joerg Arndt, Mar 11 2010


STATUS

approved



