|
%I #301 Apr 19 2024 23:15:20
%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 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
%C Partial sums give A000217. - _Omar E. Pol_, Jul 26 2018
%C First differences are A000012 (the "all 1's" sequence). - _M. F. Hasler_, May 30 2020
%C See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - _M. F. Hasler_, Nov 09 2021
%C This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - _Bernard Schott_, Jan 25 2023
%D Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.
%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 David Corneth, <a href="https://www.youtube.com/watch?v=_rinkM0PCOQ">Counting to 13999 visualized | showing changes per digit</a>, YouTube video, 2019.
%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 The IMO Compendium, <a href="https://imomath.com/othercomp/I/Imo1977.pdf">Problem 6</a>, 19th IMO 1977.
%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 László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 12.
%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).
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%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 2*n*(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) = A007966(n)*A007967(n). - _Reinhard Zumkeller_, Jun 18 2011
%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) = 2*z(-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
%o (Python)
%o def a(n): return n
%o print([a(n) for n in range(78)]) # _Michael S. Branicky_, Nov 13 2022
%o (Julia) print([n for n in 0:280]) # _Paul Muljadi_, Apr 15 2024
%Y Cf. A000027 (n>=1).
%Y Cf. A000012 (first differences).
%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.
%Y When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.
%Y Cf. A000290.
%Y Cf. A061579 (transposed matrix / reversed triangle).
%K core,nonn,easy,mult,tabl,changed
%O 0,3
%A _N. J. A. Sloane_
|
