A040001 -id:A040001 - cp's OEIS frontend

A005408 The odd numbers: a(n) = 2*n + 1.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 0

N. J. A. Sloane

Leibniz's series: Pi/4 = Sum_{n>=0} (-1)^n/(2n+1) (cf. A072172).
Beginning of the ordering of the natural numbers used in Sharkovski's theorem - see the Cielsielski-Pogoda paper.
The Sharkovski ordering begins with the odd numbers >= 3, then twice these numbers, then 4 times them, then 8 times them, etc., ending with the powers of 2 in decreasing order, ending with 2^0 = 1.
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(6).
Also continued fraction for coth(1) (A073747 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
a(1) = 1; a(n) is the smallest number such that a(n) + a(i) is composite for all i = 1 to n-1. - Amarnath Murthy, Jul 14 2003
Smallest number greater than n, not a multiple of n, but containing it in binary representation. - Reinhard Zumkeller, Oct 06 2003
Numbers n such that phi(2n) = phi(n), where phi is Euler's totient (A000010). - Lekraj Beedassy, Aug 27 2004
Pi*sqrt(2)/4 = Sum_{n>=0} (-1)^floor(n/2)/(2n+1) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 ... [since periodic f(x)=x over -Pi < x < Pi = 2(sin(x)/1 - sin(2x)/2 + sin(3x)/3 - ...) using x = Pi/4 (Maor)]. - Gerald McGarvey, Feb 04 2005
For n > 1, numbers having 2 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
a(n) = shortest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1.
First differences of squares (A000290). - Lekraj Beedassy, Jul 15 2006
The odd numbers are the solution to the simplest recursion arising when assuming that the algorithm "merge sort" could merge in constant unit time, i.e., T(1):= 1, T(n):= T(floor(n/2)) + T(ceiling(n/2)) + 1. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 14 2006
2n-5 counts the permutations in S_n which have zero occurrences of the pattern 312 and one occurrence of the pattern 123. - David Hoek (david.hok(AT)telia.com), Feb 28 2007
For n > 0: number of divisors of (n-1)th power of any squarefree semiprime: a(n) = A000005(A001248(k)^(n-1)); a(n) = A000005(A000302(n-1)) = A000005(A001019(n-1)) = A000005(A009969(n-1)) = A000005(A087752(n-1)). - Reinhard Zumkeller, Mar 04 2007
For n > 2, a(n-1) is the least integer not the sum of < n n-gonal numbers (0 allowed). - Jonathan Sondow, Jul 01 2007
A134451(a(n)) = abs(A134452(a(n))) = 1; union of A134453 and A134454. - Reinhard Zumkeller, Oct 27 2007
Numbers n such that sigma(2n) = 3*sigma(n). - Farideh Firoozbakht, Feb 26 2008
a(n) = A139391(A016825(n)) = A006370(A016825(n)). - Reinhard Zumkeller, Apr 17 2008
Number of divisors of 4^(n-1) for n > 0. - J. Lowell, Aug 30 2008
Equals INVERT transform of A078050 (signed - cf. comments); and row sums of triangle A144106. - Gary W. Adamson, Sep 11 2008
Odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular number(3*n+1). - Pierre CAMI, Sep 27 2008
A000035(a(n))=1, A059841(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
Multiplicative closure of A065091. - Reinhard Zumkeller, Oct 14 2008
a(n) is also the maximum number of triangles that n+2 points in the same plane can determine. 3 points determine max 1 triangle; 4 points can give 3 triangles; 5 points can give 5; 6 points can give 7 etc. - Carmine Suriano, Jun 08 2009
Binomial transform of A130706, inverse binomial transform of A001787(without the initial 0). - Philippe Deléham, Sep 17 2009
Also the 3-rough numbers: positive integers that have no prime factors less than 3. - Michael B. Porter, Oct 08 2009
Or n without 2 as prime factor. - Juri-Stepan Gerasimov, Nov 19 2009
Given an L(2,1) labeling l of a graph G, let k be the maximum label assigned by l. The minimum k possible over all L(2,1) labelings of G is denoted by lambda(G). For n > 0, this sequence gives lambda(K_{n+1}) where K_{n+1} is the complete graph on n+1 vertices. - K.V.Iyer, Dec 19 2009
A176271 = odd numbers seen as a triangle read by rows: a(n) = A176271(A002024(n+1), A002260(n+1)). - Reinhard Zumkeller, Apr 13 2010
For n >= 1, a(n-1) = numbers k such that arithmetic mean of the first k positive integers is an integer. A040001(a(n-1)) = 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179084 and A179085. - Reinhard Zumkeller, Jun 28 2010
For n>0, continued fraction [1,1,n] = (n+1)/a(n); e.g., [1,1,7] = 8/15. - Gary W. Adamson, Jul 15 2010
Numbers that are the sum of two sequential integers. - Dominick Cancilla, Aug 09 2010
Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h and n in A000027), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 4). Also a(n)^2 - 1 == 0 (mod 8). - Bruno Berselli, Nov 17 2010
A004767 = a(a(n)). - Reinhard Zumkeller, Jun 27 2011
A001227(a(n)) = A000005(a(n)); A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012
a(n) is the minimum number of tosses of a fair coin needed so that the probability of more than n heads is at least 1/2. In fact, Sum_{k=n+1..2n+1} Pr(k heads|2n+1 tosses) = 1/2. - Dennis P. Walsh, Apr 04 2012
A007814(a(n)) = 0; A037227(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2012
1/N (i.e., 1/1, 1/2, 1/3, ...) = Sum_{j=1,3,5,...,infinity} k^j, where k is the infinite set of constants 1/exp.ArcSinh(N/2) = convergents to barover(N). The convergent to barover(1) or [1,1,1,...] = 1/phi = 0.6180339..., whereas c.f. barover(2) converges to 0.414213..., and so on. Thus, with k = 1/phi we obtain 1 = k^1 + k^3 + k^5 + ..., and with k = 0.414213... = (sqrt(2) - 1) we get 1/2 = k^1 + k^3 + k^5 + .... Likewise, with the convergent to barover(3) = 0.302775... = k, we get 1/3 = k^1 + k^3 + k^5 + ..., etc. - Gary W. Adamson, Jul 01 2012
Conjecture on primes with one coach (A216371) relating to the odd integers: iff an integer is in A216371 (primes with one coach either of the form 4q-1 or 4q+1, (q > 0)); the top row of its coach is composed of a permutation of the first q odd integers. Example: prime 19 (q = 5), has 5 terms in each row of its coach: 19: [1, 9, 5, 7, 3] ... [1, 1, 1, 2, 4]. This is interpreted: (19 - 1) = (2^1 * 9), (19 - 9) = (2^1 * 5), (19 - 5) = (2^1 - 7), (19 - 7) = (2^2 * 3), (19 - 3) = (2^4 * 1). - Gary W. Adamson, Sep 09 2012
A005408 is the numerator 2n-1 of the term (1/m^2 - 1/n^2) = (2n-1)/(mn)^2, n = m+1, m > 0 in the Rydberg formula, while A035287 is the denominator (mn)^2. So the quotient a(A005408)/a(A035287) simulates the Hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013
This sequence has unique factorization. The primitive elements are the odd primes (A065091). (Each term of the sequence can be expressed as a product of terms of the sequence. Primitive elements have only the trivial factorization. If the products of terms of the sequence are always in the sequence, and there is a unique factorization of each element into primitive elements, we say that the sequence has unique factorization. So, e.g., the composite numbers do not have unique factorization, because for example 36 = 4*9 = 6*6 has two distinct factorizations.) - Franklin T. Adams-Watters, Sep 28 2013
These are also numbers k such that (k^k+1)/(k+1) is an integer. - Derek Orr, May 22 2014
a(n-1) gives the number of distinct sums in the direct sum {1,2,3,..,n} + {1,2,3,..,n}. For example, {1} + {1} has only one possible sum so a(0) = 1. {1,2} + {1,2} has three distinct possible sums {2,3,4} so a(1) = 3. {1,2,3} + {1,2,3} has 5 distinct possible sums {2,3,4,5,6} so a(2) = 5. - Derek Orr, Nov 22 2014
The number of partitions of 4*n into at most 2 parts. - Colin Barker, Mar 31 2015
a(n) is representable as a sum of two but no fewer consecutive nonnegative integers, e.g., 1 = 0 + 1, 3 = 1 + 2, 5 = 2 + 3, etc. (see A138591). - Martin Renner, Mar 14 2016
Unique solution a( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the number of maximal and maximum cliques in the n-centipede graph. - Eric W. Weisstein, Dec 01 2017
Lexicographically earliest sequence of distinct positive integers such that the average of any number of consecutive terms is always an integer. (For opposite property see A042963.) - Ivan Neretin, Dec 21 2017
Maximum number of non-intersecting line segments between vertices of a convex (n+2)-gon. - Christoph B. Kassir, Oct 21 2022
a(n) is the number of parking functions of size n+1 avoiding the patterns 123, 132, and 231. - Lara Pudwell, Apr 10 2023

			G.f. = q + 3*q^3 + 5*q^5 + 7*q^7 + 9*q^9 + 11*q^11 + 13*q^13 + 15*q^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
T. Dantzig, The Language of Science, 4th Edition (1954) page 276.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.1 Terminology, p. 264.
D. Hök, Parvisa mönster i permutationer [Swedish], (2007).
E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 203-205.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
K. Ciesielski and Z. Pogoda, On ordering the natural numbers, or the Sharkovski theorem, Amer. Math. Monthly, 115 (No. 2, 2008), 158-165.
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See p. 35.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, theorem 2.5, k=4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 935
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Bridges.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Franck Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
Michael Somos, Rational Function Multiplicative Coefficients
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, The modular forms database
Leo Tavares, Illustration: Triangular Sides
Eric Weisstein's World of Mathematics, Centipede Graph
Eric Weisstein's World of Mathematics, Davenport-Schinzel Sequence
Eric Weisstein's World of Mathematics, Gnomonic Number
Eric Weisstein's World of Mathematics, Inverse Cotangent,
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cotangent
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Tangent
Eric Weisstein's World of Mathematics, Inverse Tangent
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Nexus Number
Eric Weisstein's World of Mathematics, Odd Number
Eric Weisstein's World of Mathematics, Pythagorean Triple
Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000027, A005843, A065091.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A128200, A000290, A078050, A144106, A109613, A167875.
Cf. A001651 (n=1 or 2 mod 3), A047209 (n=1 or 4 mod 5).
Cf. A003558, A216371, A179480 (relating to the Coach theorem).
Cf. A000754 (boustrophedon transform).

List([0..100],n->2*n+1); # Muniru A Asiru, Oct 16 2018

a005408 n = (+ 1) . (* 2)
a005408_list = [1, 3 ..]  -- Reinhard Zumkeller, Feb 11 2012, Jun 28 2011

Magma
```
[ 2*n+1 : n in [0..100]];
```

A005408 := n->2*n+1;
A005408:=(1+z)/(z-1)^2; # Simon Plouffe in his 1992 dissertation

Table[2 n - 1, {n, 1, 50}] (* Stefan Steinerberger, Apr 01 2006 *)
Range[1, 131, 2] (* Harvey P. Dale, Apr 26 2011 *)
2 Range[0, 20] + 1 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{2, -1}, {1, 3}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

makelist(2*n+1, n, 0, 30); /* Martin Ettl, Dec 11 2012 */

PARI
```
{a(n) = 2*n + 1}
```

first(n) = Vec((1 + x)/(1 - x)^2 + O(x^n)) \\ Iain Fox, Dec 29 2017

a=lambda n: 2*n+1 # Indranil Ghosh, Jan 04 2017

[2*n+1 for n in range(100)] # G. C. Greubel, Nov 28 2018

a(n) = 2*n + 1. a(-1 - n) = -a(n). a(n+1) = a(n) + 2.
G.f.: (1 + x) / (1 - x)^2.
E.g.f.: (1 + 2*x) * exp(x).
G.f. with interpolated zeros: (x^3+x)/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*(exp(x)+exp(-x))/2. - Geoffrey Critzer, Aug 25 2012
a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005
Euler transform of length 2 sequence [3, -1]. - Michael Somos, Mar 30 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 2*u) * (1 - 2*u + 16*v) - (u - 4*v)^2 * (1 + 2*u + 2*u^2). - Michael Somos, Mar 30 2007
a(n) = b(2*n + 1) where b(n) = n if n is odd is multiplicative. [This seems to say that A000027 is multiplicative? - R. J. Mathar, Sep 23 2011]
From Hieronymus Fischer, May 25 2007: (Start)
a(n) = (n+1)^2 - n^2.
G.f. g(x) = Sum_{k>=0} x^floor(sqrt(k)) = Sum_{k>=0} x^A000196(k). (End)
a(0) = 1, a(1) = 3, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = A000330(A016777(n))/A000217(A016777(n)). - Pierre CAMI, Sep 27 2008
a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n) = A005843(n) + 1. - Jaroslav Krizek, Sep 05 2009
a(n) = (n - 1) + n (sum of two sequential integers). - Dominick Cancilla, Aug 09 2010
a(n) = 4*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. - Bruno Berselli, Nov 17 2010
n*a(2n+1)^2+1 = (n+1)*a(2n)^2; e.g., 3*15^2+1 = 4*13^2. - Charlie Marion, Dec 31 2010
arctanh(x) = Sum_{n>=0} x^(2n+1)/a(n). - R. J. Mathar, Sep 23 2011
a(n) = det(f(i-j+1))A113311(n);%20for%20n%20%3C%200%20we%20have%20f(n)=0.%20-%20_Mircea%20Merca">{1<=i,j<=n}, where f(n) = A113311(n); for n < 0 we have f(n)=0. - _Mircea Merca, Jun 23 2012
G.f.: Q(0), where Q(k) = 1 + 2*(k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013
a(n) = floor(sqrt(2*A000384(n+1))). - Ivan N. Ianakiev, Jun 17 2013
a(n) = 3*A000330(n)/A000217(n), n > 0. - Ivan N. Ianakiev, Jul 12 2013
a(n) = Product_{k=1..2*n} 2*sin(Pi*k/(2*n+1)) = Product_{k=1..n} (2*sin(Pi*k/(2*n+1)))^2, n >= 0 (undefined product = 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
Noting that as n -> infinity, sqrt(n^2 + n) -> n + 1/2, let f(n) = n + 1/2 - sqrt(n^2 + n). Then for n > 0, a(n) = round(1/f(n))/4. - Richard R. Forberg, Feb 16 2014
a(n) = Sum_{k=0..n+1} binomial(2*n+1,2*k)*4^(k)*bernoulli(2*k). - Vladimir Kruchinin, Feb 24 2015
a(n) = Sum_{k=0..n} binomial(6*n+3, 6*k)*Bernoulli(6*k). - Michel Marcus, Jan 11 2016
a(n) = A000225(n+1) - A005803(n+1). - Miquel Cerda, Nov 25 2016
O.g.f.: Sum_{n >= 1} phi(2*n-1)*x^(n-1)/(1 - x^(2*n-1)), where phi(n) is the Euler totient function A000010. - Peter Bala, Mar 22 2019
Sum_{n>=0} 1/a(n)^2 = Pi^2/8 = A111003. - Bernard Schott, Dec 10 2020
Sum_{n >= 1} (-1)^n/(a(n)*a(n+1)) = Pi/4 - 1/2 = 1/(3 + (1*3)/(4 + (3*5)/(4 + ... + (4*n^2 - 1)/(4 + ... )))). Cf. A016754. - Peter Bala, Mar 28 2024
a(n) = A055112(n)/oblong(n) = A193218(n+1)/Hex number(n). Compare to the Sep 27 2008 comment by Pierre CAMI. - Klaus Purath, Apr 23 2024
a(k*m) = k*a(m) - (k-1). - Ya-Ping Lu, Jun 25 2024
a(n) = A000217(a(n))/n for n > 0. - Stefano Spezia, Feb 15 2025

Incorrect comment and example removed by Joerg Arndt, Mar 11 2010

Peripheral comments deleted by N. J. A. Sloane, May 09 2022

A005843 The nonnegative even numbers: a(n) = 2n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

-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
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
Number of neighbors in von Neumann neighborhood. - Dmitry Zaitsev, Nov 30 2015
Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - Martin Renner, Jan 26 2020
Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - Bernard Schott, Dec 10 2020
Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n) gives the y-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The x-value is given by 2*n^2 + 1 (see Tattersall). - Stefano Spezia, Jul 24 2025

			G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
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 [math.CO], 2012, J. Int. Seq. 15 (2012) # 12.6.2
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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Wikipedia, Alkane
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009
Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
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).

a005843 = (* 2)
a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

Magma
```
[ 2*n : n in [0..100]];
```

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

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

PARI
```
A005843(n) = 2*n
```

def a(n): return 2*n # Martin Gergov, Oct 20 2022

R
```
seq(0,200,2)
```

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*sinh(x). - 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_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
From Ilya Gutkovskiy, Aug 19 2016: (Start)
Convolution of A007395 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
From Bernard Schott, Dec 10 2020: (Start)
Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)

A002194 Decimal expansion of sqrt(3).

Original entry on oeis.org

1, 7, 3, 2, 0, 5, 0, 8, 0, 7, 5, 6, 8, 8, 7, 7, 2, 9, 3, 5, 2, 7, 4, 4, 6, 3, 4, 1, 5, 0, 5, 8, 7, 2, 3, 6, 6, 9, 4, 2, 8, 0, 5, 2, 5, 3, 8, 1, 0, 3, 8, 0, 6, 2, 8, 0, 5, 5, 8, 0, 6, 9, 7, 9, 4, 5, 1, 9, 3, 3, 0, 1, 6, 9, 0, 8, 8, 0, 0, 0, 3, 7, 0, 8, 1, 1, 4, 6, 1, 8, 6, 7, 5, 7, 2, 4, 8, 5, 7, 5, 6, 7, 5, 6, 2, 6, 1, 4, 1, 4, 1, 5, 4

Offset: 1

N. J. A. Sloane

"The square root of 3, the 2nd number, after root 2, to be proved irrational, by Theodorus."
Length of a diagonal between any vertex of the unit cube and the one corresponding (opposite) vertex not part of the three faces meeting at the original vertex. (Diagonal is hypotenuse of a triangle with sides 1 and sqrt(2)). Hence the diameter of the sphere circumscribed around the unit cube; the ratio of the diameter of any sphere to the edge length of its inscribed cube. - Rick L. Shepherd, Jun 09 2005
The square root of 3 is the length of the minimal Y-shaped (symmetrical) network linking three points unit distance apart. - Lekraj Beedassy, Apr 12 2006
Continued fraction expansion is 1 followed by {1, 2} repeated. - Harry J. Smith, Jun 01 2009
Also, tan(Pi/3) = 2 sin(Pi/3). - M. F. Hasler, Oct 27 2011
Surface of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
This is the case n=6 of Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)) = 2*cos(Pi/n), therefore sqrt(3) = A175379*A203145/(A073005*A073006). - Bruno Berselli, Dec 13 2012
Ratio of base length to leg length in the isosceles "vampire" triangle, that is, the only isosceles triangle without reflection triangle. The product of cosines of the internal angles of a triangle with sides 1, 1 and sqrt(3) and all similar triangles is -3/8. Hence its reflection triangle is degenerate. See the link below. - Martin Janecke, May 09 2013
Half of the surface of regular octahedron with unit edge (A010469), and one fifth that of a regular icosahedron with unit edge (i.e., 2*A010527). - Stanislav Sykora, Nov 30 2013
Diameter of a sphere whose surface area equals 3*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018
Sometimes called Theodorus's constant, after the ancient Greek mathematician Theodorus of Cyrene (5th century BC). - Amiram Eldar, Apr 02 2022
For any triangle ABC, cotan(A) + cotan(B) + cotan(C) >= sqrt(3); equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 13 2022

			1.73205080756887729352744634150587236694280525381038062805580697945193...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 184.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §12.4 Theorems and Formulas (Solid Geometry), pp. 84, 450.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Madeleine Bonsma-Fisher and Kent Bonsma-Fisher, How big a table do you need for your jigsaw puzzle?, arXiv:2312.04588 [math.HO], 2023.
M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., Vol. 22, No. 101 (1968), pp. 234-235.
Kiran S. Kedlaya, A < B, (1999) Problem 6.4, p. 6.
Jason Kimberley, Index of expansions of sqrt(d) in base b.
Robert J. Nemiroff and Jerry Bonnell, The first 1 million digits of the square root of 3.
Simon Plouffe, Plouffe's Inverter, The square root of 3 to 10 million digits.
Horace S. Uhler, Approximations exceeding 1300 decimals for sqrt 3, 1/sqrt 3, sin(pi/3) and distribution of digits in them, Proc. Nat. Acad. Sci. U. S. A., Vol. 37, No. 7 (1951), pp. 443-447.
Eric Weisstein's World of Mathematics, Reflection Triangle.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Theodorus's Constant.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2

Cf. A040001 (continued fraction), A220335.

Cf. A010469 (double), A010527 (half), A131595 (surface of regular dodecahedron).

SetDefaultRealField(RealField(100)); Sqrt(3); // G. C. Greubel, Aug 21 2018

evalf(sqrt(3), 100); # Michal Paulovic, Feb 24 2023

Mathematica
```
RealDigits[Sqrt[3], 10, 100][[1]]
```

default(realprecision, 20080); x=(sqrt(3)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002194.txt", n, " ", d));  \\ Harry J. Smith, Jun 01 2009

Equals Sum_{k>=0} binomial(2*k,k)/6^k = Sum_{k>=0} binomial(2*k,k) * k/6^k. - Amiram Eldar, Aug 03 2020
sqrt(3) = 1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*2) + 1/(2*3*4*2*8) + 1/(2*3*4*2*8*14) + 1/(2*3*4*2*8*14*2) + 1/(2*3*4*2*8*14*2*98) + 1/(2*3*4*2*8*14*2*98*194) + .... (Define F(n) = (n-1)*sqrt(n^2 - 1) - (n^2 - n - 1). Show F(n) = 1/2 + 1/(2*(n+1)) +  1/(2*(n+1)*(2*n)) + 1/(2*(n+1)*(2*n))*F(2*n^2 - 1) for n >= 0; then iterate this identity at n = 2. See A220335.) - Peter Bala, Mar 18 2022
Equals i^(1/3) + i^(-1/3). - Gary W. Adamson, Jul 06 2022
Equals Product_{n>=1} 3^(1/3^n). - Michal Paulovic, Feb 24 2023
Equals Product_{n>=0} ((6*n + 2)*(6*n + 4))/((6*n + 1)*(6*n + 5)). - Antonio Graciá Llorente, Feb 22 2024
Equals tan(Pi/3) = A010527/(1/2). - R. J. Mathar, Aug 31 2025

More terms from Robert G. Wilson v, Dec 07 2000

A001651 Numbers not divisible by 3.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104

Offset: 1

N. J. A. Sloane

Inverse binomial transform of A084858. - Benoit Cloitre, Jun 12 2003
Earliest monotonic sequence starting with (1,2) and satisfying the condition: "a(n)+a(n-1) is not in the sequence." - Benoit Cloitre, Mar 25 2004. [The numbers of the form a(n)+a(n-1) form precisely the complement with respect to the positive integers. - David W. Wilson, Feb 18 2012]
a(1) = 1; a(n) is least number which is relatively prime to the sum of all the previous terms. - Amarnath Murthy, Jun 18 2001
For n > 3, numbers having 3 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
Also numbers n such that (n+1)*(n+2)/6 = A000292(n)/n is an integer. - Ctibor O. Zizka, Oct 15 2010
Notice the property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 3). - Bruno Berselli, Nov 17 2010
A001651 mod 9 gives A141425. - Paul Curtz, Dec 31 2010. (Correct for the modified offset 1. - M. F. Hasler, Apr 07 2015)
The set of natural numbers (1, 2, 3, ...), sequence A000027; represents the numbers of ordered compositions of n using terms in the signed set: (1, 2, -4, -5, 7, 8, -10, -11, 13, 14, ...). This follows from (1, 2, 3, ...) being the INVERT transform of A011655, signed and beginning: (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013
Union of A047239 and A047257. - Wesley Ivan Hurt, Dec 19 2013
Numbers whose sum of digits (and digital root) is != 0 (mod 3). - Joerg Arndt, Aug 29 2014
The number of partitions of 3*(n-1) into at most 2 parts. - Colin Barker, Apr 22 2015
a(n) is the number of partitions of 3*n into two distinct parts. - L. Edson Jeffery, Jan 14 2017
Conjectured (and like even easily proved) to be the graph bandwidth of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Numbers k such that Fibonacci(k) mod 4 = 1 or 3. Equivalently, sequence lists the indices of the odd Fibonacci numbers (see A014437). - Bruno Berselli, Oct 17 2017
Minimum value of n_3 such that the "rectangular spiral pattern" is the optimal solution for Ripà's n_1 X n_2 x n_3 Dots Problem, for any n_1 = n_2. For example, if n_1 = n_2 = 5, n_3 = floor((3/2)*(n_1 - 1)) + 1 = a(5). - Marco Ripà, Jul 23 2018
For n >= 54, a(n) = sat(n, P_n), the minimum number of edges in a P_n-saturated graph on n vertices, where P_n is the n-vertex path (see Dudek, Katona, and Wojda, 2003; Frick and Singleton, 2005). - Danny Rorabaugh, Nov 07 2017
From Roger Ford, May 09 2021: (Start)
a(n) is the smallest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
      /\   The top arch has a length of 3. /\   The top arch has a length of 3.
     /  \  Both bottom arches have a      //\\  The middle arch has a length of 2.
    //\/\\ length of 1.                  ///\\\ The bottom arch has a length of 1.
  Example: a(6) = 8     /\      /\
                       //\\ /\ //\\ /\   2 + 1 + 1 + 2 + 1 + 1 = 8. (End)
This is the lexicographically earliest increasing sequence of positive integers such that no polynomial of degree d can be fitted to d+2 consecutive terms (equivalently, such that no iterated difference is zero). - Pontus von Brömssen, Dec 26 2021

			G.f.: x + 2*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 10*x^7 + 11*x^8 + 13*x^9 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., Vol. 11, No. 4 (1973), pp. 337-386.
Aneta Dudek, Gyula Y. Katona, and A.Pawel Wojda, m_Path Cover Saturated Graphs, Electronic Notes in Discrete Math., Vol. 13 (April 2003), pp. 41-44.
Marietjie Frick and Joy Singleton, Lower Bound for the Size of Maximal Nontraceable Graphs, Electron. J. Combin., 12#R32 (2005), 9 pp.
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications, Vol. 1, No. 1 (2021), Article #S1H1.
G. Ledin, Jr., Is Eratosthenes out?, Fib. Quart., Vol. 6, No. 4 (1968), pp. 261-265.
Gerard P. Michon, Counting Polyhedra.
Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly, Vol. 109, No. 6 (2002), pp. 559-564, Ex. 2.2.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem , Notes on Number Theory and Discrete Mathematics, Vol. 20, No. 1 (2014), pp. 59-71.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph, Graph Bandwidth, and RATS Sequence.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Differs from A059564 after 35 = a(24) = A059564(24).
Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A073010, A191967, A225227, A004526, A248897.
Cf. A000027, A000217, A000292, A000982, A001477, A008619, A014437, A040001, A047239, A047257, A077043, A084858, A113801, A141425, A215879.

Filtered([0..110],n->n mod 3<>0); # Muniru A Asiru, Jul 24 2018

a001651 = (`div` 2) . (subtract 1) . (* 3)
a001651_list = filter ((/= 0) . (`mod` 3)) [1..]
-- Reinhard Zumkeller, Jul 07 2012, Aug 23 2011

[3*(2*n-1)/4-(-1)^n/4: n in [1..80]]; // Vincenzo Librandi, Jun 07 2011

A001651 := n -> 3*floor(n/2) - (-1)^n; # Corrected by M. F. Hasler, Apr 07 2015
A001651:=(1+z+z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation
a[1]:=1:a[2]:=2:for n from 3 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=1..69); # Zerinvary Lajos, Mar 16 2008, offset corrected by M. F. Hasler, Apr 07 2015

Select[Table[n,{n,200}],Mod[#,3]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
Drop[Range[200 + 1], {1, -1, 3}] - 1 (* József Konczer, May 24 2016 *)
Floor[(3 Range[70] - 1)/2] (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Series[(x^2 + x + 1)/((x - 1)^2 (x + 1)), {x, 0, 70}],
  x] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 70] (* Robert G. Wilson v, Jul 25 2018 *)

{a(n) = n + (n-1)\2}; /* Michael Somos, Jan 15 2011 */

x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ Altug Alkan, Oct 22 2015

print([k for k in range(1, 105) if k%3]) # Michael S. Branicky, Sep 06 2021

def A001651(n): return (n<<1)-(n>>1)-1 # Chai Wah Wu, Mar 05 2024

a(n) = 3 + a(n-2) for n > 2.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
a(2*n+1) = 3*n+1, a(2*n) = 3*n-1.
G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - Michael Somos, Jun 08 2000
a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article).
a(n) = floor((3*n-1)/2). [Corrected by Gary Detlefs]
a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - Benoit Cloitre, Aug 17 2002
a(n+1) = 1 + n - n mod 2 + (n + n mod 2)/2. - Reinhard Zumkeller, Dec 17 2002
a(1) = 1, a(n+1) = a(n) + (a(n) mod 3). - Reinhard Zumkeller, Mar 23 2003
a(1) = 1, a(n) = 3*(n-1) - a(n-1). - Benoit Cloitre, Apr 12 2003
a(n) = 3*(2*n-1)/4 - (-1)^n/4. - Benoit Cloitre, Jun 12 2003
Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003
Partial sums of A040001. a(n) = A032766(n-1)+1. - Paul Barry, Sep 02 2003
a(n) = T(n, 1) = T(n, n-1), where T is the array in A026386. - Emeric Deutsch, Feb 18 2004
a(n) = sqrt(3*A001082(n)+1). - Zak Seidov, Dec 12 2007
a(n) = A077043(n) - A077043(n-1). - Reinhard Zumkeller, Dec 28 2007
a(n) = A001477(n-1) + A008619(n-1). - Yosu Yurramendi, Aug 10 2008
Euler transform of length 3 sequence [2, 1, -1]. - Michael Somos, Sep 06 2008
A011655(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
a(n) = n - 1 + ceiling(n/2). - Michael Somos, Jan 15 2011
a(n) = 3*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i), for n>1. - Bruno Berselli, Nov 17 2010
a(n) = 3*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
A215879(a(n)) > 0. - Reinhard Zumkeller, Dec 28 2012 [More precisely, A215879 is the characteristic function of A001651. - M. F. Hasler, Apr 07 2015]
a(n) = 2n - 1 - floor(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = (3n - 2 + (n mod 2)) / 2. - Wesley Ivan Hurt, Mar 31 2014
a(n) = A000217(n) - A000982(n-1). - Bui Quang Tuan, Mar 28 2015
1/1^3 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + ... = 4 Pi^3/(3 sqrt(3)). - M. F. Hasler, Mar 29 2015
E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - Ilya Gutkovskiy, May 24 2016
a(n) = a(n+k-1) + a(n-k) - a(n-1) for n > k >= 0. - Bob Selcoe, Feb 03 2017
a(n) = -a(1-n) for all n in Z. - Michael Somos, Jul 31 2018
a(n) = n + A004526(n-1). - David James Sycamore, Sep 06 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Dec 04 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*Pi/(3*sqrt(3)) (A248897). (End)

This is a list, so the offset should be 1. I corrected this and adjusted some of the comments and formulas. Other lines probably also need to be adjusted. - N. J. A. Sloane, Jan 01 2011

Offset of pre-2011 formulas verified or corrected by M. F. Hasler, Apr 07-18 2015 and by Danny Rorabaugh, Oct 23 2015

A000034 Period 2: repeat [1, 2]; a(n) = 1 + (n mod 2).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

Also continued fraction for (sqrt(3)+1)/2 (cf. A040001) and base-3 digital root of n+1 (cf. A007089, A010888). - Henry Bottomley, Jul 05 2001
The sequence 1,-2,-1,2,1,-2,-1,2,... with g.f. (1-2x)/(1+x^2) has a(n) = cos(Pi*n/2)-2*sin(Pi*n/2). - Paul Barry, Oct 18 2004
Hankel transform is [1,-3,0,0,0,0,0,0,0,...]. - Philippe Deléham, Mar 29 2007
4/33 = 0.121212... - Eric Desbiaux, Nov 03 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1) = charpoly(A,2). - Milan Janjic, Jan 24 2010
First differences of A032766. - Tom Edgar, Jul 17 2014
Denominator of the harmonic mean of the first n triangular numbers. - Colin Barker, Nov 13 2014
This is the lexicographically earliest sequence of positive integers such that no polynomial of degree d can be fitted to d+2 consecutive terms (equivalently, such that no iterated difference is zero). - Pontus von Brömssen, Dec 26 2021 [See A300002 for the case where not only consecutive terms are considered. - Pontus von Brömssen, Jan 03 2023]
Number of maximum antichains in the power set of {1,2,...,n} partially ordered by set inclusion. For even n, there is a unique maximum antichain formed by all subsets of size n/2; for odd n, there are two maximum antichains, one formed by all subsets of size (n-1)/2 and the other formed by all subsets of size (n+1)/2. See the David Guichard link below for a proof. - Jianing Song, Jun 19 2022

Jozsef Beck, Combinatorial Games, Cambridge University Press, 2008.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 545 pages 73 and 260, Ellipses, Paris 2004.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Table of n, a(n) for n = 0..9999
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
David Guichard, Sperner's Theorem
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 383
Agustín Moreno Cañadas, Hernán Giraldo and Robinson Julian Serna Vanegas, Some integer partitions induced by orbits of Dynkin type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 12 (2017), pp. 2745-2766.
Wikipedia, Collatz conjecture
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000035, A003945 (binomial transf), A007089, A010693, A010704, A010888, A032766, A040001, A123344, A134451, A300002.

Cf. sequences listed in Comments section of A283393.

List([0..120],n->1+(n mod 2)); # Muniru A Asiru, Feb 01 2019

a000034 = (+ 1) . (`mod` 2)
a000034_list = cycle [1,2]
-- Reinhard Zumkeller, Jul 03 2012, Dec 02 2011 and corrected by James Spahlinger, Oct 08 2012

[1+(n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 11 2014

(1+2*x)/(1-x^2);
A000034 := proc(n) op((n mod 2)+1,[1,2]) ; end proc: # R. J. Mathar, Feb 05 2011

a[n_] := If[OddQ[n], 2, 1]; Table[a[n], {n, 0, 90}] (* Stefan Steinerberger, Apr 17 2006 *)
Nest[ Flatten[# /. {    0 -> {1}, 1 -> {2}, 2 -> {1, 2, 1}  }] &, {1}, 8] (* Robert G. Wilson v, May 20 2014 *)

PARI
```
a(n)=1+n%2
```

a(n)=1+bittest(n,0) \\ M. F. Hasler, Jan 13 2012

def A000034(n): return 1 + (n & 1) # Chai Wah Wu, May 25 2022

G.f.: (1+2*x)/(1-x^2).
a(n) = 2^((1-(-1)^n)/2) = 2^(ceiling(n/2) - floor(n/2)). - Paul Barry, Jun 03 2003
a(n) = (3-(-1)^n)/2; a(n) = 1 + (n mod 2) = 3-a(n-1) = a(n-2) = a(-n).
a(n) = gcd(n-1, n+1). - Paul Barry, Sep 16 2004
Binomial transform of A123344, inverse binomial transform of A003945. - Philippe Deléham, Jun 04 2007
a(n) = A134451(n+1). - Reinhard Zumkeller, Oct 27 2007
a(n) = if(n=0,1,if(mod(a(n-1),2)=0,a(n-1)/2,(3*a(n-1)+1)/2)). See Collatz conjecture. - Paul Barry, Mar 31 2008
a(n) = 2^n (mod 3). - Vincenzo Librandi, Feb 05 2011
a(n) = A000035(n) + 1. - M. F. Hasler, Jan 13 2012
a(n) = abs(sin(n*Pi/2) - 2*cos(n*Pi/2)). - Mohammad K. Azarian, Mar 12 2012
a(n) = A010704(n) / 3. - Reinhard Zumkeller, Jul 03 2012
a(n) = floor((4/33)*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = floor((5/8)*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013
a(n) = floor((n+1)*3/2) - floor((n)*3/2). - Hailey R. Olafson, Jul 23 2014
a(n) = denominator(n/2). - Wesley Ivan Hurt, Sep 11 2014
Dirichlet g.f.: zeta(s)*(1 + 1/2^s). - Mats Granvik, Jul 18 2016
E.g.f.: 2*sinh(x) + cosh(x). - Ilya Gutkovskiy, Jul 18 2016
a(n) = A010693(n) - 1. - Filip Zaludek, Oct 29 2016
a(n) = n + 1 - 2*floor(n/2). - Lorenzo Sauras Altuzarra, Jun 28 2019
Limit_{n->oo} (1/n)*Sum_{k=1..n} a(k) = 3/2 (De Koninck reference). - Bernard Schott, Nov 09 2021

Better definition from M. F. Hasler, Jan 13 2012

A000325 a(n) = 2^n - n.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556, 2097131, 4194282, 8388585, 16777192, 33554407, 67108838, 134217701, 268435428, 536870883, 1073741794, 2147483617

Offset: 0

Rosario Salamone (Rosario.Salamone(AT)risc.uni-linz.ac.at)

Number of permutations of degree n with at most one fall; called "Grassmannian permutations" by Lascoux and Schützenberger. - Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de)
Number of different permutations of a deck of n cards that can be produced by a single shuffle. [DeSario]
Number of Dyck paths of semilength n having at most one long ascent (i.e., ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U = (1, 1) and D = (1, -1). Also number of ordered trees with n edges having at most one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004
Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.
Number of 1342-avoiding circular permutations on [n+1].
2^n - n is the number of ways to partition {1, 2, ..., n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005
if b(0) = x and b(n) = b(n-1) + b(n-1)^2*x^(n-2) for n > 0, then b(n) is a polynomial of degree a(n). - Michael Somos, Nov 04 2006
The chromatic invariant of the Mobius ladder graph M_n for n >= 2. - Jonathan Vos Post, Aug 29 2008
Dimension sequence of the dual alternative operad (i.e., associative and satisfying the identity xyz + yxz + zxy + xzy + yzx + zyx = 0) over the field of characteristic 3. - Pasha Zusmanovich, Jun 09 2009
An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 26 and 176, lead to this sequence (without the first leading 1). For the central square these vectors lead to the companion sequence A168604. - Johannes W. Meijer, Aug 15 2010
a(n+1) is also the number of order-preserving and order-decreasing partial isometries (of an n-chain). - Abdullahi Umar, Jan 13 2011
A040001(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
A130103(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
The number of labeled graphs with n vertices whose vertex set can be partitioned into a clique and a set of isolated points. - Alex J. Best, Nov 20 2012
For n > 0, a(n) is a B_2 sequence. - Thomas Ordowski, Sep 23 2014
See coefficients of the linear terms of the polynomials of the table on p. 10 of the Getzler link. - Tom Copeland, Mar 24 2016
Consider n points lying on a circle, then for n>=2 a(n-1) is the maximum number of ways to connect two points with non-intersecting chords. - Anton Zakharov, Dec 31 2016
Also the number of cliques in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
From Eric M. Schmidt, Jul 17 2017: (Start)
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k). [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i), e(j), e(k) pairwise distinct. [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(j) >= e(k) and e(i) != e(k) pairwise distinct. [Martinez and Savage, 2.7]
(End)
Number of F-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are F-equivalent iff the positions of pattern F are identical in these paths. -  Sergey Kirgizov, Apr 08 2018
From Gus Wiseman, Feb 10 2019: (Start)
Also the number of connected partitions of an n-cycle. For example, the a(1) = 1 through a(4) = 12 connected partitions are:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{1}{23}}    {{1}{234}}
                   {{12}{3}}    {{12}{34}}
                   {{13}{2}}    {{123}{4}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{2}{34}}
                                {{1}{23}{4}}
                                {{12}{3}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
(End)
Number of subsets of n-set without the single-element subsets. - Yuchun Ji, Jul 16 2019
For every prime p, there are infinitely many terms of this sequence that are divisible by p (see IMO Compendium link and Doob reference). Corresponding indices n are: for p = 2, even numbers A299174; for p = 3, A047257; for p = 5, A349767. - Bernard Schott, Dec 10 2021
Primes are in A081296 and corresponding indices in A048744. - Bernard Schott, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 58*x^6 + 121*x^7 + ...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024.
Cory B. H. Ball, The Apprentices' Tower of Hanoi, Electronic Theses and Dissertations, East Tennessee State University, Paper 2512, 2015.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178;
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
David Callan, Pattern avoidance in circular permutations, arXiv:math/0210014 [math.CO], 2002.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Robert DeSario and LeRoy Wenstrom, Invertible shuffles, Problem 10931, Amer. Math. Monthly, 111 (No. 2, 2004), 169-170.
Askar Dzhumadil'daev and Pasha Zusmanovich, The alternative operad is not Koszul, arXiv:0906.1272 [math.RA], 2009-2013.
Marty Getz, Dixon Jones and Ken Dutch, Partitioning by Arithmetic Progressions: Problem 11005, American Math. Monthly, Vol. 112, 2005, p. 89. (The published solution is incomplete. Letting d be the common difference of the arithmetic progressions, the solver's expression q_1(n,d)=2^(n-d) must be summed over all d=1,...,n and duplicate partitions must be removed.)
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018.
The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
R. Kehinde, S. O. Makanjuola and A. Umar, On the semigroup of order-decreasing partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
Alain Lascoux and Marcel-Paul Schützenberger, Schubert polynomials and the Littlewood Richardson rule, Letters in Math. Physics 10 (1985) 111-124.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Moebius Ladder
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
Index to sequences related to Olympiads.

Column 1 of triangle A008518.
Row sum of triangles A184049 and A184050.
Cf. A000108, A002064, A133116, A160692, A005803, A006127, A186947.
Cf. A001610, A066982, A323950, A323951.
Cf. A047257, A299174, A349767.
Cf.  A048744, A081296.

a000325 n = 2 ^ n - n
a000325_list = zipWith (-) a000079_list [0..]
-- Reinhard Zumkeller, Jul 17 2012

[2^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

A000325 := proc(n) option remember; if n <=1 then n+1 else 2*A000325(n-1)+n-1; fi; end;
g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[2^n - n, {n, 0, 39}] (* Alonso del Arte, Sep 15 2014 *)
LinearRecurrence[{4, -5, 2}, {1, 2, 5}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)

{a(n) = 2^n - n}; /* Michael Somos, Nov 04 2006 */

def A000325(n): return (1<Chai Wah Wu, Jan 11 2023

a(n+1) = 2*a(n) + n - 1, a(0) = 1. - Reinhard Zumkeller, Apr 12 2003
Binomial transform of 1, 0, 1, 1, 1, .... The sequence starting 1, 2, 5, ... has a(n) = 1 + n + 2*Sum_{k=2..n} binomial(n, k) = 2^(n+1) - n - 1. This is the binomial transform of 1, 1, 2, 2, 2, 2, .... a(n) = 1 + Sum_{k=2..n} C(n, k). - Paul Barry, Jun 06 2003
G.f.: (1-3x+3x^2)/((1-2x)*(1-x)^2). - Emeric Deutsch, Feb 22 2004
A107907(a(n+2)) = A000051(n+2) for n > 0. - Reinhard Zumkeller, May 28 2005
a(n+1) = sum of n-th row of the triangle in A109128. - Reinhard Zumkeller, Jun 20 2005
Row sums of triangle A133116. - Gary W. Adamson, Sep 14 2007
G.f.: 1 / (1 - x / (1 - x / ( 1 - x / (1 + x / (1 - 2*x))))). - Michael Somos, May 12 2012
First difference is A000225. PSUM transform is A084634. - Michael Somos, May 12 2012
a(n) = [x^n](B(x)^n-B(x)^(n-1)), n>0, a(0)=1, where B(x) = (1+2*x+sqrt(1+4*x^2))/2. - Vladimir Kruchinin, Mar 07 2014
E.g.f.: (exp(x) - x)*exp(x). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A125128(n) - A000225(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = 2*A125128(n) - A095151(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = A079583(n-1) - A000225(n-1). - Miquel Cerda, Aug 15 2016
a(n)^2 - 4*a(n-1)^2 = (n-2)*(a(n)+2*a(n-1)). - Yuchun Ji, Jul 13 2018
a(n) = 2^(-n) * A186947(n) = 2^n * A002064(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(2^n) = (2^a(n) - 1)*2^n. - Lorenzo Sauras Altuzarra, Feb 01 2022

A083329 a(0) = 1; for n > 0, a(n) = 3*2^(n-1) - 1.

Original entry on oeis.org

1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943

Offset: 0

Paul Barry, Apr 27 2003

Apart from leading term (which should really be 3/2), same as A055010.
Binomial transform of A040001. Inverse binomial transform of A053156.
a(n) = A105728(n+1,2). - Reinhard Zumkeller, Apr 18 2005
Row sums of triangle A133567. - Gary W. Adamson, Sep 16 2007
Row sums of triangle A135226. - Gary W. Adamson, Nov 23 2007
a(n) = number of partitions Pi of [n+1] (in standard increasing form) such that the permutation Flatten[Pi] avoids the patterns 2-1-3 and 3-1-2. Example: a(3)=11 counts all 15 partitions of [4] except 13/24, 13/2/4 which contain a 2-1-3 and 14/23, 14/2/3 which contain a 3-1-2. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. - David Callan, Jul 22 2008
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 42, 138, 162, 168, lead to this sequence. For the central square these vectors lead to the companion sequence A003945. - Johannes W. Meijer, Aug 15 2010
The binary representation of a(n) has n+1 digits, where all digits are 1's except digit n-1. For example: a(4) = 23 = 10111 (2). - Omar E. Pol, Dec 02 2012
Row sums of triangle A209561. - Reinhard Zumkeller, Dec 26 2012
If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, A162909/A162910, A071766/A229742, A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n >= 0), and the fractions, term by term, are summed at each level n, then the resulting sequence of integers is a(n) + 1/2, apart from leading term (which should be 1/2). - Yosu Yurramendi, May 23 2015
For n >= 2, A083329(n) in binary representation is a string [101..1], also 10 followed with (n-1) 1's. For n >= 3, A036563(n) in binary representation is a string [1..101], also (n-2) 1's followed with 01. Thus A083329(n) is a reflection of the binary representation of A036563(n+1). Example: A083329(5) = 101111 in binary, A036563(6) = 111101 in binary. - Ctibor O. Zizka, Nov 06 2018
For n > 0, a(n) is the minimum number of turns in (n+1)-dimensional Euclidean space needed to visit all 2^(n+1) vertices of the (n+1)-cube (e.g., {0,1}^(n+1)) and return to the starting point, moving along straight-line segments between turns (turns may occur elsewhere in R^(n+1)). - Marco Ripà, Aug 14 2025

			a(0) = (3*2^0 - 2 + 0^0)/2 = 2/2 = 1 (use 0^0=1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Essentially the same as A055010 and A052940.
Cf. A000225, A052955, A133567, A135226.
Cf. A007505 (primes).
Cf. A266550 (independence number of the n-Mycielski graph).

a083329 n = a083329_list !! n
a083329_list = 1 : iterate ((+ 1) . (* 2)) 2
-- Reinhard Zumkeller, Dec 26 2012, Feb 22 2012

[1] cat [3*2^(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Jan 01 2016

seq(ceil((2^i+2^(i+1)-2)/2), i=0..31); # Zerinvary Lajos, Oct 02 2007

a[1] = 2; a[n_] := 2a[n - 1] + 1; Table[ a[n], {n, 31}] (* Robert G. Wilson v, May 04 2004 *)
Join[{1}, LinearRecurrence[{3, -2}, {2, 5}, 40]] (* Vincenzo Librandi, Jan 01 2016 *)

a(n)=(3*2^n-2+0^n)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3*2^n - 2 + 0^n)/2.
G.f.: (1-x+x^2)/((1-x)*(1-2*x)). [corrected by Martin Griffiths, Dec 01 2009]
E.g.f.: (3*exp(2*x) - 2*exp(x) + exp(0))/2.
a(0) = 1, a(n) = sum of all previous terms + n. - Amarnath Murthy, Jun 20 2004
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=5. - Philippe Deléham, Nov 29 2013
From Bob Selcoe, Apr 25 2014: (Start)
a(n) = (...((((((1)+1)*2+1)*2+1)*2+1)*2+1)...), with n+1 1's, n >= 0.
a(n) = 2*a(n-1) + 1, n >= 2.
a(n) = 2^n + 2^(n-1) - 1, n >= 2. (End)
a(n) = A086893(n) + A061547(n+1), n > 0. - Yosu Yurramendi, Jan 16 2017

A020760 Decimal expansion of 1/sqrt(3).

Original entry on oeis.org

5, 7, 7, 3, 5, 0, 2, 6, 9, 1, 8, 9, 6, 2, 5, 7, 6, 4, 5, 0, 9, 1, 4, 8, 7, 8, 0, 5, 0, 1, 9, 5, 7, 4, 5, 5, 6, 4, 7, 6, 0, 1, 7, 5, 1, 2, 7, 0, 1, 2, 6, 8, 7, 6, 0, 1, 8, 6, 0, 2, 3, 2, 6, 4, 8, 3, 9, 7, 7, 6, 7, 2, 3, 0, 2, 9, 3, 3, 3, 4, 5, 6, 9, 3, 7, 1, 5, 3, 9, 5, 5, 8, 5, 7, 4, 9, 5, 2, 5

Offset: 0

N. J. A. Sloane

If the sides of a triangle form an arithmetic progression in the ratio 1:1+d:1+2d then when d=1/sqrt(3) it uniquely maximizes the area of the triangle. This triangle has approximate internal angles 25.588 degs, 42.941 degs, 111.471 degs. - Frank M Jackson, Jun 15 2011
When a cylinder is completely enclosed by a sphere, it occupies a fraction f of the sphere volume. The value of f has a trivial lower bound of 0, and an upper bound which is this constant. It is achieved iff the cylinder diameter is sqrt(2) times its height, and the sphere is circumscribed to it. A similar constant can be associated with any n-dimensional geometric shape. For 3D cuboids it is A165952. - Stanislav Sykora, Mar 07 2016
The ratio between the thickness and diameter of a dynamically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on the dynamic of rigid body (Yong and Mahadevan, 2011). See A020765 for a simplified geometrical solution. - Amiram Eldar, Sep 01 2020
The coefficient of variation (relative standard deviation) of natural numbers: Limit_{n->oo} sqrt((n-1)/(3*n+3)) = 1/sqrt(3). - Michal Paulovic, Mar 21 2023

			0.577350269189625764509148780501957455647601751270126876018602326....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.17, pp. 495, 531.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ee Hou Yong and L. Mahadevan, Probability, geometry, and dynamics in the toss of a thick coin, American Journal of Physics, Vol. 79, No. 12 (2011), pp. 1195-1201, preprint, arXiv:1008.4559 [physics.class-ph], 2010-2011.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (sqrt(3)), A010701 (1/3).

Cf. A002193, A047235, A165952, A195696, A040001.

evalf(1/sqrt(3)); # Michal Paulovic, Mar 21 2023

RealDigits[N[1/Sqrt[3],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

\\ Works in v2.15.0; n = 100 decimal places
my(n=100); digits(floor(10^n/quadgen(12))) \\ Michal Paulovic, Mar 21 2023

Equals 1/A002194. - Michel Marcus, Oct 12 2014
Equals cosine of the magic angle: cos(A195696). - Stanislav Sykora, Mar 07 2016
Equals square root of A010701. - Michel Marcus, Mar 07 2016
Equals 1 + Sum_{k>=0} -(4*k+1)^(-1/2) + (4*k+3)^(-1/2) + (4*k+5)^(-1/2) - (4*k+7)^(-1/2). - Gerry Martens, Nov 22 2022
Equals (1/2)*(2 - zeta(1/2,1/4) + zeta(1/2,3/4) + zeta(1/2,5/4) - zeta(1/2,7/4)). - Gerry Martens, Nov 22 2022
Has periodic continued fraction expansion [0, 1; 1, 2] (A040001). - Michal Paulovic, Mar 21 2023
Equals Product_{k>=1} (1 + (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024
Equals tan(Pi/6) = (1/2)/A010527. - R. J. Mathar, Aug 31 2025

A004396 One even number followed by two odd numbers.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47

Offset: 0

N. J. A. Sloane

Maximal number of points on a triangular grid of edge length n-1 with no 2 points on same row, column, or diagonal. See Problem 252 in The Inquisitive Problem Solver. - R. K. Guy [Comment revised by N. J. A. Sloane, Jul 01 2016]
See also Problem C2 of 2009 International Mathematical Olympiad. - Ruediger Jehn, Oct 19 2021
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(3).
Starting at 3, 3, ..., gives maximal number of acute angles in an n-gon. - Takenov Nurdin (takenov_vert(AT)e-mail.ru), Mar 04 2003
Let b(1) = b(2) = 1, b(k) = b(k-1)+( b(k-2) reduced (mod 2)); then a(n) = b(n-1). - Benoit Cloitre, Aug 14 2002
(1+x+x^2+x^3 ) / ( (1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for Sigma_4.
For n > 6, maximum number of knight moves to reach any square from the corner of an (n-2) X (n-2) chessboard. Likewise for n > 6, the maximum number of knight moves to reach any square from the middle of an (2n-5) X (2n-5) chessboard. - Ralf Stephan, Sep 15 2004
A transform of the Jacobsthal numbers A001045 under the mapping of g.f.s g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005
For n >= 1; a(n) = number of successive terms of A040001 that add to n; or length of n-th term of A028359. - Jaroslav Krizek, Mar 28 2010
For n > 0: a(n) = length of n-th row in A082870. - Reinhard Zumkeller, Apr 13 2014
Also the independence number of the n-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
In a game of basketball points can be accumulated by making field goals (two or three points) or free throws (one point). a(n) is the number of different ways to score n-1 points. For example, a score of 4 can be achieved in 3 different ways, with 2 shots (3+1 or 2+2), 3 shots (2+1+1) or 4 shots (1+1+1+1), so a(5) = 3. - Ivan N. Ianakiev, Mar 31 2025

			G.f. = x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + ...

J. Kurschak, Hungarian Mathematical Olympiads, 1976, Mir, Moscow.
Paul Vanderlind, Richard K. Guy, and Loren C. Larson, The Inquisitive Problem Solver, MAA, 2002. See Problem 252.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 246.
Art of Problem Solving Forum, Ordered triples choosing. - _Joel B. Lewis_, May 21 2009
J. Choi and N. Pippenger, Counting the Angels and Devils in Escher's Circle Limit IV, arXiv preprint arXiv:1310.1357 [math.CO], 2013.
C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes, Discrete Math., 9 (1974), 391-400 (see proof of Theorem 1).
Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See Eq. (4).
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Independence Number.
50th International Mathematical Olympiad 2009, Problem Shortlist with Solutions.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index to sequences related to Olympiads.

Cf. A001045, A002620, A004523, A004773, A006369, A008620, A032766, A040001, A082870, A092200, A096777.

a004396 n = a004396_list !! n
a004396_list = 0 : 1 : 1 : map (+ 2) a004396_list
-- Reinhard Zumkeller, Nov 06 2012

[(Floor(n/3) + Ceiling(n/3)): n in [0..70]]; // Vincenzo Librandi, Aug 07 2011

A004396:=n->floor((2*n + 1)/3); seq(A004396(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Table[Floor[(2 n + 1)/3], {n, 0, 75}]
With[{n = 50}, Riffle[Range[0, n], Range[1, n, 2], {3, -1, 3}]] (* Harvey P. Dale, May 14 2015 *)
CoefficientList[Series[(x + x^3)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, Oct 27 2016 *)
a[ n_] := Quotient[2 n + 1, 3]; (* Michael Somos, Oct 23 2017 *)
a[ n_] := Sign[n] SeriesCoefficient[ (x + x^3) / ((1 - x) (1 - x^3)), {x, 0, Abs@n}]; (* Michael Somos, Oct 23 2017 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
f[-1]=0; f[n_]:=Length[Union[Plus@@@FrobeniusSolve[{1,2,3},n]]]; f/@Range[-1,100] (* Ivan N. Ianakiev, Mar 31 2025 *)

a(n)=2*n\/3 \\ Charles R Greathouse IV, Apr 17 2012

def a(n) : return( dimension_cusp_forms( Gamma0(3), 2*n+4) ); # Michael Somos, Jul 03 2014

G.f.: (x+x^3)/((1-x)*(1-x^3)).
a(n) = floor( (2*n + 1)/3 ).
a(n) = a(n-1) + (1/2)*((-1)^floor((4*n+2)/3) + 1), a(0) = 0. - Mario Catalani (mario.catalani(AT)unito.it), Oct 20 2003
a(n) = 2n/3 - cos(2*Pi*n/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*n/3 + Pi/3)/9. - Paul Barry, Mar 18 2004
a(n) = A096777(n+1) - A096777(n) for n > 0. - Reinhard Zumkeller, Jul 09 2004
From Paul Barry, Jan 16 2005: (Start)
G.f.: x*(1+x^2)/(1-x-x^3+x^4).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
a(n) = Sum_{k = 0..n} binomial(n-k-1, k)*(-1)^k*A001045(n-2k). (End)
a(n) = (A006369(n) - (A006369(n) mod 2) * (-1)^(n mod 3)) / (1 + A006369(n) mod 2). - Reinhard Zumkeller, Jan 23 2005
a(n) = A004773(n) - A004523(n). - Reinhard Zumkeller, Aug 29 2005
a(n) = floor(n/3) + ceiling(n/3). - Jonathan Vos Post, Mar 19 2006
a(n+1) = A008620(2n). - Philippe Deléham, Dec 14 2006
a(A032766(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor((2*n^2+4*n+2)/(3*n+4)). - Gary Detlefs, Jul 13 2010
Euler transform of length 4 sequence [1, 1, 1, -1]. - Michael Somos, Jul 03 2014
a(n) = n - floor((n+1)/3). - Wesley Ivan Hurt, Sep 17 2015
a(n) = A092200(n) - floor((n+5)/3). - Filip Zaludek, Oct 27 2016
a(n) = -a(-n) for all n in Z. - Michael Somos, Oct 30 2016
E.g.f.: (2/9)*(3*exp(x)*x + sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 20 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2. - Amiram Eldar, Sep 29 2022

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A184418 Convolution square of A040001.

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A005408 The odd numbers: a(n) = 2*n + 1.

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A005843 The nonnegative even numbers: a(n) = 2n.

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A002194 Decimal expansion of sqrt(3).

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A001651 Numbers not divisible by 3.

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