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A005843
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The even numbers: a(n) = 2n.
(Formerly M0985)
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347
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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; internal format)
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OFFSET
| 0,2
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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 R. Janjic (agnus(AT)blic.net), Sep 19 2007
A134452(a(n)) = 0; A134451(a(n)) = 2 for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (rayjchandler(AT)sbcglobal.net), Aug 21 2003
A059841(a(n))=1, A000035(a(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 29 2008]
Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Oct 28 2008: (Start)
(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)
A116471=2*A008794
(End)
A056753(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 23 2009]
Twice the nonnegative numbers. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 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). [From Paul Muljadi (paulmuljadi(AT)yahoo.com), 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. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 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. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), May 28 2010]
Union of A179082 and A179083. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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). [From Eric 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]
The identity (18n^2+1)^2-(81n^2+9)*(2n)^2=1 can be written as A157889(n)^2-A157888(n)*a(n+1)^2=1. Also, the identity (6n^2-1)^2-(9n^2-3)*(2n)^2=1 can be written as A140811(n+1)^2-A157872(n)*a(n+1)^2=1. - Vincenzo Librandi, Feb 05 2012
The identity (18*n^2-1)^2-(81*n^2-9)*(2*n)^2 = 1 can be written as A157910(n)^2-A157909(n)*a(n)^2=1 for n>0. Also, the identity (16*n^2+1)^2-(64*n^2+8)*(2*n)^2 = 1 can be written as A108211(n)^2-A158488(n)*a(n)^2 = 1 for n>0. - Vincenzo Librandi, Feb 08 2012
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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).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Hosoya Index
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Eric Weisstein's World of Mathematics, Star Graph
Wikipedia, Alkane
Index entries for "core" sequences
Index to sequences with linear recurrences with constant coefficients, signature (2,-1).
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FORMULA
| G.f.: 2*x/(1-x)^2.
Inverse binomial transform of A036289, n*2^n. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jan 13 2006
a(0)=0, a(1)=2, a(n) = 2a(n-1)-a(n-2). - Jaume Oliver i Lafont (joliverlafont(AT)gmail.com), May 07 2008
a(n)=Sum{k=1,n}floor(6n/4^k+1/2) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 04 2009]
a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 05 2009]
a(n) = Sum_k>=0 {A030308(n,k)*A000079(k+1)}. - From DELEHAM Philippe, 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
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MAPLE
| A005843 := n->2*n;
A005843:=2/(z-1)**2; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| lst={}; Do[AppendTo[lst, 2*n], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 29 2008]
Range[0, 120, 2] (* From Harvey P. Dale, Aug 16 2011 *)
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PROG
| (MAGMA) [ 2*n : n in [0..100]];
(R) seq(0, 200, 2)
(PARI) A005843(n) = 2*n
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CROSSREFS
| a(n)=2*A001477(n). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 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
Sequence in context: A087113 A004275 A119432 * A076032 A004277 A122080
Adjacent sequences: A005840 A005841 A005842 * A005844 A005845 A005846
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KEYWORD
| nonn,easy,core,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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