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A001477 The nonnegative integers. 604

%I

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,

%T 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,

%U 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77

%N The nonnegative integers.

%C Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - _N. J. A. Sloane_, Mar 13 2010

%C The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - _N. J. A. Sloane_, Jun 16 2010

%C a(n) = A007966(n)*A007967(n). - _Reinhard Zumkeller_, Jun 18 2011

%C Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - _M. F. Hasler_, Nov 20 2013

%C The number of partitions of 2n into exactly 2 parts. - _Colin Barker_, Mar 22 2015

%C The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- _Philippe A.J.G. Chevalier_, Dec 29 2015

%H N. J. A. Sloane, <a href="/A001477/b001477.txt">Table of n, a(n) for n = 0..500000</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

%H Hans Havermann, <a href="/A001477/a001477.txt">Table giving n and American English name for n, for 0 <= n <= 100999, without spaces or hyphens</a>

%H Hans Havermann, <a href="http://chesswanks.com/num/NumberNames.txt">American English number names to one million, without spaces or hyphens</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NaturalNumber.html">Natural Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonnegativeInteger.html">Nonnegative Integer</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = n.

%F a(0) = 0, a(n) = a(n-1) + 1.

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

%F Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01 2001

%F When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2n(n+1) (A046092), antidiagonal sums are n(n+1)(n+2)/2 (A027480). - _Ralf Stephan_, Oct 17 2004

%F Dirichlet generating function: zeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005.

%F E.g.f.: x*e^x. - _Franklin T. Adams-Watters_, Sep 11 2005

%F a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - _Jaume Oliver Lafont_, May 07 2008

%F Alternating partial sums give A001057 = A000217 - 2*(A008794). - _Eric Desbiaux_, Oct 28 2008

%F a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - _Eric Desbiaux_, Nov 15 2009

%F a(n) = Sum_{k>=0} A030308(n,k)*2^k. - _Philippe Deléham_, Oct 20 2011

%F a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - _R. J. Mathar_, Jul 20 2012

%F a(n+1) = det(C(i+1,j), 1 <= i,j <= n), where C(n,k) are binomial coefficients. - _Mircea Merca_, Apr 06 2013

%F a(n-1) = floor(n/e^(1/n)) for n > 0. - _Richard R. Forberg_, Jun 22 2013

%F a(n) = A000027(n) for all n>0.

%F a(n) = floor(cot(1/(n+1))). - _Clark Kimberling_, Oct 08 2014

%F a(0)=0, a(n>0) = 2z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z=A130472(n>0); a 1_to_1 correspondence between integers and naturals. - _Adriano Caroli_, Mar 29 2015

%e Triangular view:

%e 0

%e 1 2

%e 3 4 5

%e 6 7 8 9

%e 10 11 12 13 14

%e 15 16 17 18 19 20

%e 21 22 23 24 25 26 27

%e 28 29 30 31 32 33 34 35

%e 36 37 38 39 40 41 42 43 44

%e 45 46 47 48 49 50 51 52 53 54

%p [ seq(n,n=0..100) ];

%t Table[n, {n, 0, 100}] (* _Stefan Steinerberger_, Apr 08 2006 *)

%t LinearRecurrence[{2, -1}, {0, 1}, 77] (* _Robert G. Wilson v_, May 23 2013 *)

%t CoefficientList[ Series[x/(x - 1)^2, {x, 0, 76}], x] (* _Robert G. Wilson v_, May 23 2013 *)

%o (MAGMA) [ n : n in [0..100]];

%o (PARI) A001477(n)=n /* first term is a(0) */

%o (Haskell)

%o a001477 = id

%o a001477_list = [0..] -- _Reinhard Zumkeller_, May 07 2012

%Y Cf. A000027 (n>=1).

%Y Partial sums of A057427. - _Jeremy Gardiner_, Sep 08 2002

%Y Cf. A038608 (alternating signs), A001787 (binomial transform).

%Y Cf. A055112.

%Y Cf. Boustrophedon transforms: A231179, A000737.

%Y Cf. A245422.

%Y Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.

%K core,nonn,easy,mult,tabl

%O 0,3

%A _N. J. A. Sloane_

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Last modified May 24 13:23 EDT 2018. Contains 304525 sequences. (Running on oeis4.)