A001477 The nonnegative integers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• core
• nonn
• easy
• mult
• tabl

Comments

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

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

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

The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015

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

Partial sums give A000217. - Omar E. Pol, Jul 26 2018

First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020

See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021

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

Examples

			Triangular view:
   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  26  27
  28  29  30  31  32  33  34  35
  36  37  38  39  40  41  42  43  44
  45  46  47  48  49  50  51  52  53  54
		

References

• Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..500000
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• David Corneth, Counting to 13999 visualized | showing changes per digit, YouTube video, 2019.
• Hans Havermann, Table giving n and American English name for n, for 0 <= n <= 100999, without spaces or hyphens
• Hans Havermann, American English number names to one million, without spaces or hyphens
• The IMO Compendium, Problem 6, 19th IMO 1977.
• Tanya Khovanova, Recursive Sequences
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 12.
• Eric Weisstein's World of Mathematics, Natural Number
• Eric Weisstein's World of Mathematics, Nonnegative Integer
• Index entries for "core" sequences
• Index entries for sequences that are permutations of the natural numbers
• Index entries for linear recurrences with constant coefficients, signature (2,-1).
• Index to sequences related to Olympiads.

Crossrefs

Cf. A000027 (n>=1).

Cf. A000012 (first differences).

Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002

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

Cf. A055112.

Cf. Boustrophedon transforms: A231179, A000737.

Cf. A245422.

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

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.

Cf. A000290.

Cf. A061579 (transposed matrix / reversed triangle).

Programs

Haskell

a001477 = id
a001477_list = [0..]  -- Reinhard Zumkeller, May 07 2012

Julia

print([n for n in 0:280]) # Paul Muljadi, Apr 15 2024

Magma

[ n : n in [0..100]];

Maple

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

Mathematica

Table[n, {n, 0, 100}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{2, -1}, {0, 1}, 77] (* Robert G. Wilson v, May 23 2013 *)
CoefficientList[ Series[x/(x - 1)^2, {x, 0, 76}], x] (* Robert G. Wilson v, May 23 2013 *)
Range[0,100] (* Harvey P. Dale, Dec 29 2024 *)

PARI

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

Python

def a(n): return n
print([a(n) for n in range(78)]) # Michael S. Branicky, Nov 13 2022

Formula

a(n) = n.

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

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

Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001

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

Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005

E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005

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

Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008

a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009

a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011

a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011

a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012

a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013

a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013

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

a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014

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

G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025