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A005843 The even numbers: a(n) = 2n.
(Formerly M0985)
467
0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

-2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008

If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 2-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007

A134452(a(n)) = 0; A134451(a(n)) = 2 for n>0. - Reinhard Zumkeller, Oct 27 2007

Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - Ray Chandler, Aug 21 2003

A059841(a(n))=1, A000035(a(n))=0. - Reinhard Zumkeller, Sep 29 2008

(APSO) Alternating partial sums of (a-b+c-d+e-f+g...)=(a+b+c+d+e+f+g...)-2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - Eric Desbiaux, Oct 28 2008

A056753(a(n)) = 1. - Reinhard Zumkeller_, Aug 23 2009

Twice the nonnegative numbers. - Juri-Stepan Gerasimov, Dec 12 2009

The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - Paul Muljadi, Feb 18 2010

For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - Jaroslav Krizek, Feb 15 2010

For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010

Union of A179082 and A179083. - Reinhard Zumkeller, Jun 28 2010

The Hosoya index H(n) of the n-star graph S_n is given by H(2n-1) = 0 and H(2n) = a(n). - Eric W. Weisstein, Jul 11 2011

a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - Jason Kimberley, Oct 30 2011

For n > 0: A048272(a(n)) <= 0. - Reinhard Zumkeller, Jan 21 2012

Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - Ivan N. Ianakiev, Sep 18 2013

For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - Derek Orr, Jul 02 2014

a(n) when n>2 is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014

It appears that for n>2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - Bob Selcoe, Mar 08 2015

Number of partitions of 4n into exactly 2 parts. - Colin Barker, Mar 23 2015

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test

Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786, 2012. - From N. J. A. Sloane, Sep 17 2012

Milan Janjic, Two Enumerative Functions

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Even Number, Hamiltonian Cycle, Hosoya Index, Riemann Zeta Function Zeros, Star Graph

Wikipedia, Alkane

Index entries for "core" sequences

Index to sequences with linear recurrences with constant coefficients, signature (2,-1).

FORMULA

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

E.g.f.: 2*x*exp(x). - Geoffrey Critzer, Aug 25 2012

G.f. with interpolated zeros: 2x*^2/((1-x)^2 * (1+x)^2; e.g.f. with interpolated zeros: x*(exp(x)-exp(-x)/2. - Geoffrey Critzer, Aug 25 2012.

Inverse binomial transform of A036289, n*2^n. - Joshua Zucker, Jan 13 2006

a(0)=0, a(1)=2, a(n) = 2a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008

a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - Vladimir Shevelev, Jun 04 2009

a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - Jaroslav Krizek, Sep 05 2009

a(n) = Sum_k>=0 {A030308(n,k)*A000079(k+1)}. - Philippe Deléham, Oct 17 2011

Digit sequence 22 read in base n-1. - Jason Kimberley, Oct 30 2011

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Dec 23 2011

a(n) = 2*n = product(2*sin(Pi*k/(2*n)), k=1..2*n-1), n>=0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013

MAPLE

A005843 := n->2*n;

A005843:=2/(z-1)**2; # Simon Plouffe in his 1992 dissertation

MATHEMATICA

Range[0, 120, 2] (* Harvey P. Dale, Aug 16 2011 *)

PROG

(MAGMA) [ 2*n : n in [0..100]];

(R) seq(0, 200, 2)

(PARI) A005843(n) = 2*n

(Haskell)

a005843 = (* 2)

a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

CROSSREFS

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009

Cf. A000027, A005408, A001358, A005843, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011

Cf. A231200 (boustrophedon transform).

Sequence in context: A087113 A004275 A119432 * A076032 A004277 A122080

Adjacent sequences:  A005840 A005841 A005842 * A005844 A005845 A005846

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 27 22:20 EDT 2015. Contains 257884 sequences.