cp's OEIS Frontend

What is the asymptotic behavior of the sequences defined by the recursive map x -> a(x)? Do these sequences increase without bound or do they enter a repeating cycle?

For example, the trajectory of 11 under the above map starts 11, 2047, 23, 8388607, 47, 140737488355327, 2351, s, 4703, t, ..., where s is a 708-digit number and t is a 1416-digit number. t has no prime factor less than 2^64 (cf. GIMPS link).

Since Mersenne numbers of the form 2^x - 1 consist entirely of 1's when written in binary, this continued fraction is nothing but 1's if written in binary.

Binary continued fraction: 1/(1+1/(11+1/(111+1/(1111+1/(11111+1/(111111+1/...

2^0 = 1 is the only odd power of 2.

Number of subsets of an n-set.

There are 2^(n-1) compositions (ordered partitions) of n (see for example Riordan). This is the unlabeled analog of the preferential labelings sequence A000670.

This is also the number of weakly unimodal permutations of 1..n + 1, that is, permutations with exactly one local maximum. E.g., a(4) = 16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - Jon Perry, Jul 27 2003 [Proof: see next line! See also A087783.]

Proof: n must appear somewhere and there are 2^(n-1) possible choices for the subset that precedes it. These must appear in increasing order and the rest must follow n in decreasing order. QED. - N. J. A. Sloane, Oct 26 2003

a(n+1) is the smallest number that is not the sum of any number of (distinct) earlier terms.

Same as Pisot sequences E(1, 2), L(1, 2), P(1, 2), T(1, 2). See A008776 for definitions of Pisot sequences.

With initial 1 omitted, same as Pisot sequences E(2, 4), L(2, 4), P(2, 4), T(2, 4). - David W. Wilson

Not the sum of two or more consecutive numbers. - Lekraj Beedassy, May 14 2004

Least deficient or near-perfect numbers (i.e., n such that sigma(n) = A000203(n) = 2n - 1). - Lekraj Beedassy, Jun 03 2004. [Comment from Max Alekseyev, Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2.]

Almost-perfect numbers referred to as least deficient or slightly defective (Singh 1997) numbers. Does "near-perfect numbers" refer to both almost-perfect numbers (sigma(n) = 2n - 1) and quasi-perfect numbers (sigma(n) = 2n + 1)? There are no known quasi-perfect or least abundant or slightly excessive (Singh 1997) numbers.

The sum of the numbers in the n-th row of Pascal's triangle; the sum of the coefficients of x in the expansion of (x+1)^n.

The Collatz conjecture (the hailstone sequence will eventually reach the number 1, regardless of which positive integer is chosen initially) may be restated as (the hailstone sequence will eventually reach a power of 2, regardless of which positive integer is chosen initially).

The only hailstone sequence which doesn't rebound (except "on the ground"). - Alexandre Wajnberg, Jan 29 2005

With p(n) as the number of integer partitions of n, p(i) is the number of parts of the i-th partition of n, d(i) is the number of different parts of the i-th partition of n, m(i,j) is the multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i = 1..p(n)} (p(i)! / (Product_{j=1..d(i)} m(i,j)!)). - Thomas Wieder, May 18 2005

The number of binary relations on an n-element set that are both symmetric and antisymmetric. Also the number of binary relations on an n-element set that are symmetric, antisymmetric and transitive.

The first differences are the sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005

a(n) is the largest number with shortest addition chain involving n additions. - David W. Wilson, Apr 23 2006

Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - Giovanni Teofilatto, Aug 06 2006

For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n} -> {1, 2} such that for a fixed x in {1, 2, ..., n} and a fixed y in {1, 2} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007

Let P(A) be the power set of an n-element set A. Then a(n) is the number of pairs of elements {x,y} of P(A) for which x = y. - Ross La Haye, Jan 09 2008

a(n) is the number of permutations on [n+1] such that every initial segment is an interval of integers. Example: a(3) counts 1234, 2134, 2314, 2341, 3214, 3241, 3421, 4321. The map "p -> ascents of p" is a bijection from these permutations to subsets of [n]. An ascent of a permutation p is a position i such that p(i) < p(i+1). The permutations shown map to 123, 23, 13, 12, 3, 2, 1 and the empty set respectively. - David Callan, Jul 25 2008

2^(n-1) is the largest number having n divisors (in the sense of A077569); A005179(n) is the smallest. - T. D. Noe, Sep 02 2008

a(n) appears to match the number of divisors of the modified primorials (excluding 2, 3 and 5). Very limited range examined, PARI example shown. - Bill McEachen, Oct 29 2008

Successive k such that phi(k)/k = 1/2, where phi is Euler's totient function. - Artur Jasinski, Nov 07 2008

A classical transform consists (for general a(n)) in swapping a(2n) and a(2n+1); examples for Jacobsthal A001045 and successive differences: A092808, A094359, A140505. a(n) = A000079 leads to 2, 1, 8, 4, 32, 16, ... = A135520. - Paul Curtz, Jan 05 2009

This is also the (L)-sieve transform of {2, 4, 6, 8, ..., 2n, ...} = A005843. (See A152009 for the definition of the (L)-sieve transform.) - John W. Layman, Jan 23 2009

a(n) = a(n-1)-th even natural number (A005843) for n > 1. - Jaroslav Krizek, Apr 25 2009

For n >= 0, a(n) is the number of leaves in a complete binary tree of height n. For n > 0, a(n) is the number of nodes in an n-cube. - K.V.Iyer, May 04 2009

Permutations of n+1 elements where no element is more than one position right of its original place. For example, there are 4 such permutations of three elements: 123, 132, 213, and 312. The 8 such permutations of four elements are 1234, 1243, 1324, 1423, 2134, 2143, 3124, and 4123. - Joerg Arndt, Jun 24 2009

Catalan transform of A099087. - R. J. Mathar, Jun 29 2009

a(n) written in base 2: 1,10,100,1000,10000,..., i.e., (n+1) times 1, n times 0 (A011557(n)). - Jaroslav Krizek, Aug 02 2009

Or, phi(n) is equal to the number of perfect partitions of n. - Juri-Stepan Gerasimov, Oct 10 2009

These are the 2-smooth numbers, positive integers with no prime factors greater than 2. - Michael B. Porter, Oct 04 2009

A064614(a(n)) = A000244(n) and A064614(m) < A000244(n) for m < a(n). - Reinhard Zumkeller, Feb 08 2010

a(n) is the largest number m such that the number of steps of iterations of {r - (largest divisor d < r)} needed to reach 1 starting at r = m is equal to n. Example (a(5) = 32): 32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4; 4 - 2 = 2; 2 - 1 = 1; number 32 has 5 steps and is the largest such number. See A105017, A064097, A175125. - Jaroslav Krizek, Feb 15 2010

a(n) is the smallest proper multiple of a(n-1). - Dominick Cancilla, Aug 09 2010

The powers-of-2 triangle T(n, k), n >= 0 and 0 <= k <= n, begins with: {1}; {2, 4}; {8, 16, 32}; {64, 128, 256, 512}; ... . The first left hand diagonal T(n, 0) = A006125(n + 1), the first right hand diagonal T(n, n) = A036442(n + 1) and the center diagonal T(2*n, n) = A053765(n + 1). Some triangle sums, see A180662, are: Row1(n) = A122743(n), Row2(n) = A181174(n), Fi1(n) = A181175(n), Fi2(2*n) = A181175(2*n) and Fi2(2*n + 1) = 2*A181175(2*n + 1). - Johannes W. Meijer, Oct 10 2010

Records in the number of prime factors. - Juri-Stepan Gerasimov, Mar 12 2011

Row sums of A152538. - Gary W. Adamson, Dec 10 2008

A078719(a(n)) = 1; A006667(a(n)) = 0. - Reinhard Zumkeller, Oct 08 2011

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 2-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Equals A001405 convolved with its right-shifted variant: (1 + 2x + 4x^2 + ...) = (1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + ...) * (1 + x + x^2 + 2x^3 + 3x^4 + 6x^5 + ...). - Gary W. Adamson, Nov 23 2011

The number of odd-sized subsets of an n+1-set. For example, there are 2^3 odd-sized subsets of {1, 2, 3, 4}, namely {1}, {2}, {3}, {4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, and {2, 3, 4}. Also, note that 2^n = Sum_{k=1..floor((n+1)/2)} C(n+1, 2k-1). - Dennis P. Walsh, Dec 15 2011

a(n) is the number of 1's in any row of Pascal's triangle (mod 2) whose row number has exactly n 1's in its binary expansion (see A007318 and A047999). (The result of putting together A001316 and A000120.) - Marcus Jaiclin, Jan 31 2012

A204455(k) = 1 if and only if k is in this sequence. - Wolfdieter Lang, Feb 04 2012

For n>=1 apparently the number of distinct finite languages over a unary alphabet, whose minimum regular expression has alphabetic width n (verified up to n=17), see the Gruber/Lee/Shallit link. - Hermann Gruber, May 09 2012

First differences of A000225. - Omar E. Pol, Feb 19 2013

This is the lexicographically earliest sequence which contains no arithmetic progression of length 3. - Daniel E. Frohardt, Apr 03 2013

a(n-2) is the number of bipartitions of {1..n} (i.e., set partitions into two parts) such that 1 and 2 are not in the same subset. - Jon Perry, May 19 2013

Numbers n such that the n-th cyclotomic polynomial has a root mod 2; numbers n such that the n-th cyclotomic polynomial has an even number of odd coefficients. - Eric M. Schmidt, Jul 31 2013

More is known now about non-power-of-2 "Almost Perfect Numbers" as described in Dagal. - Jonathan Vos Post, Sep 01 2013

Number of symmetric Ferrers diagrams that fit into an n X n box. - Graham H. Hawkes, Oct 18 2013

Numbers n such that sigma(2n) = 2n + sigma(n). - Jahangeer Kholdi, Nov 23 2013

a(1), ..., a(floor(n/2)) are all values of permanent on set of square (0,1)-matrices of order n>=2 with row and column sums 2. - Vladimir Shevelev, Nov 26 2013

Numbers whose base-2 expansion has exactly one bit set to 1, and thus has base-2 sum of digits equal to one. - Stanislav Sykora, Nov 29 2013

A072219(a(n)) = 1. - Reinhard Zumkeller, Feb 20 2014

a(n) is the largest number k such that (k^n-2)/(k-2) is an integer (for n > 1); (k^a(n)+1)/(k+1) is never an integer (for k > 1 and n > 0). - Derek Orr, May 22 2014

If x = A083420(n), y = a(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014

The mini-sequence b(n) = least number k > 0 such that 2^k ends in n identical digits is given by {1, 18, 39}. The repeating digits are {2, 4, 8} respectively. Note that these are consecutive powers of 2 (2^1, 2^2, 2^3), and these are the only powers of 2 (2^k, k > 0) that are only one digit. Further, this sequence is finite. The number of n-digit endings for a power of 2 with n or more digits id 4*5^(n-1). Thus, for b(4) to exist, one only needs to check exponents up to 4*5^3 = 500. Since b(4) does not exist, it is clear that no other number will exist. - Derek Orr, Jun 14 2014

The least number k > 0 such that 2^k ends in n consecutive decreasing digits is a 3-number sequence given by {1, 5, 25}. The consecutive decreasing digits are {2, 32, 432}. There are 100 different 3-digit endings for 2^k. There are no k-values such that 2^k ends in '987', '876', '765', '654', '543', '321', or '210'. The k-values for which 2^k ends in '432' are given by 25 mod 100. For k = 25 + 100*x, the digit immediately before the run of '432' is {4, 6, 8, 0, 2, 4, 6, 8, 0, 2, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus, we see the digit before '432' will never be a 5. So, this sequence is complete. - Derek Orr, Jul 03 2014

a(n) is the number of permutations of length n avoiding both 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014

Numbers n such that sigma(n) = sigma(2n) - phi(4n). - Farideh Firoozbakht, Aug 14 2014

This is a B_2 sequence: for i < j, differences a(j) - a(i) are all distinct. Here 2*a(n) < a(n+1) + 1, so a(n) - a(0) < a(n+1) - a(n). - Thomas Ordowski, Sep 23 2014

a(n) counts n-walks (closed) on the graph G(1-vertex; 1-loop, 1-loop). - David Neil McGrath, Dec 11 2014

a(n-1) counts walks (closed) on the graph G(1-vertex; 1-loop, 2-loop, 3-loop, 4-loop, ...). - David Neil McGrath, Jan 01 2015

b(0) = 4; b(n+1) is the smallest number not in the sequence such that b(n+1) - Prod_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n) = a(n) for n > 2. - Derek Orr, Jan 15 2015

a(n) counts the permutations of length n+2 whose first element is 2 such that the permutation has exactly one descent. - Ran Pan, Apr 17 2015

a(0)-a(30) appear, with a(26)-a(30) in error, in tablet M 08613 (see CDLI link) from the Old Babylonian period (c. 1900-1600 BC). - Charles R Greathouse IV, Sep 03 2015

Subsequence of A028982 (the squares or twice squares sequence). - Timothy L. Tiffin, Jul 18 2016

A000120(a(n)) = 1. A000265(a(n)) = 1. A000593(a(n)) = 1. - Juri-Stepan Gerasimov, Aug 16 2016

Number of monotone maps f : [0..n] -> [0..n] which are order-increasing (i <= f(i)) and idempotent (f(f(i)) = f(i)). In other words, monads on the n-th ordinal (seen as a posetal category). Any monad f determines a subset of [0..n] that contains n, by considering its set of monad algebras = fixed points { i | f(i) = i }. Conversely, any subset S of [0..n] containing n determines a monad on [0..n], by the function i |-> min { j | i <= j, j in S }. - Noam Zeilberger, Dec 11 2016

Consider n points lying on a circle. Then for n>=2 a(n-2) gives the number of ways to connect two adjacent points with nonintersecting chords. - Anton Zakharov, Dec 31 2016

Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017

Also the number of independent vertex sets and vertex covers in the n-empty graph. - Eric W. Weisstein, Sep 21 2017

Also the number of maximum cliques in the n-halved cube graph for n > 4. - Eric W. Weisstein, Dec 04 2017

Number of pairs of compositions of n corresponding to a seaweed algebra of index n-1. - Nick Mayers, Jun 25 2018

The multiplicative group of integers modulo a(n) is cyclic if and only if n = 0, 1, 2. For n >= 3, it is a product of two cyclic groups. - Jianing Song, Jun 27 2018

k^n is the determinant of n X n matrix M_(i, j) = binomial(k + i + j - 2, j) - binomial(i+j-2, j), in this case k=2. - Tony Foster III, May 12 2019

Solutions to the equation Phi(2n + 2*Phi(2n)) = 2n. - M. Farrokhi D. G., Jan 03 2020

a(n-1) is the number of subsets of {1,2,...,n} which have an element that is the size of the set. For example, for n = 4, a(3) = 8 and the subsets are {1}, {1,2}, {2,3}, {2,4}, {1,2,3}, {1,3,4}, {2,3,4}, {1,2,3,4}. - Enrique Navarrete, Nov 21 2020

a(n) is the number of self-inverse (n+1)-order permutations with 231-avoiding. E.g., a(3) = 8: [1234, 1243, 1324, 1432, 2134, 2143, 3214, 4321]. - Yuchun Ji, Feb 26 2021

For any fixed k > 0, a(n) is the number of ways to tile a strip of length n+1 with tiles of length 1, 2, ... k, where the tile of length k can be black or white, with the restriction that the first tile cannot be black. - Greg Dresden and Bora Bursalı, Aug 31 2023

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

R_n is a string of n 1's.

Base-4 representation of Jacobsthal bisection sequence A002450. E.g., a(4)= 1111 because A002450(4)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1*(4^3) + 1*(4^2) + 1*(4^1) + 1. - Paul Barry, Mar 12 2004

Except for the first two terms, these numbers cannot be perfect squares, because x^2 != 11 (mod 100). - Zak Seidov, Dec 05 2008

For n >= 0: a(n) = (A000225(n) written in base 2). - Jaroslav Krizek, Jul 27 2009, edited by M. F. Hasler, Jul 03 2020

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010

Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - Bernard Schott, Dec 24 2012

Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - Bernard Schott, Dec 24 2012

The terms 0 and 1 are the only squares in this sequence, as a(n) == 3 (mod 4) for n>=2. - Nehul Yadav, Sep 26 2013

For n>=2 the multiplicative order of 10 modulo the a(n) is n. - Robert G. Wilson v, Aug 20 2014

The above is a special case of the statement that the order of z modulo (z^n-1)/(z-1) is n, here for z=10. - Joerg Arndt, Aug 21 2014

From Peter Bala, Sep 20 2015: (Start)

Let d be a divisor of a(n). Let m*d be any multiple of d. Split the decimal expansion of m*d into 2 blocks of contiguous digits a and b, so we have m*d = 10^k*a + b for some k, where 0 <= k < number of decimal digits of m*d. Then d divides a^n - (-b)^n (see McGough). For example, 271 divides a(5) and we find 2^5 + 71^5 = 11*73*271*8291 and 27^5 + 1^5 = 2^2*7*31*61*271 are both divisible by 271. Similarly, 4*271 = 1084 and 10^5 + 84^5 = 2^5*31*47*271*331 while 108^5 + 4^5 = 2^12*7*31*61*271 are again both divisible by 271. (End)

Starting with the second term this sequence is the binary representation of the n-th iteration of the Rule 220 and 252 elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016

If p > 5 is a prime, then p divides a(p-1). - Thomas Ordowski, Apr 10 2016

0, 1 and 11 are only terms that are of the form x^2 + y^2 + z^2 where x, y, z are integers. In other words, a(n) is a member of A004215 for all n > 2. - Altug Alkan, May 08 2016

Except for the initial terms, the binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 17 2017

The term "repunit" was coined by Albert H. Beiler in 1964. - Amiram Eldar, Nov 13 2020

q-integers for q = 10. - John Keith, Apr 12 2021

Binomial transform of A001019 with leading zero. - Jules Beauchamp, Jan 04 2022

Same as Pisot sequence L(2,3).

Length of the continued fraction for Sum_{k=0..n} 1/3^(2^k). - Benoit Cloitre, Nov 12 2003

See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - Philippe Deléham, Feb 20 2004

From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg, May 31 2005

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)* charpoly(A,3). - Milan Janjic, Jan 27 2010

First differences of A006127. - Reinhard Zumkeller, Apr 14 2011

The odd prime numbers in this sequence form A019434, the Fermat primes. - David W. Wilson, Nov 16 2011

Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012

Is the mentioned Pisano period lengths (see above) the same as A007733? - Omar E. Pol, Aug 10 2012

Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012

For n >= 1, a(n) is the total length of the segments of the Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014

Frénicle de Bessy (1657) proved that a(3) = 9 is the only square in this sequence. - Charles R Greathouse IV, May 13 2014

a(n) is the number of distinct possible sums made with at most two elements in {1,...,a(n-1)} for n > 0. - Derek Orr, Dec 13 2014

For n > 0, given any set of a(n) lattice points in R^n, there exist 2 distinct members in this set whose midpoint is also a lattice point. - Melvin Peralta, Jan 28 2017

Also the number of independent vertex sets, irredundant sets, and vertex covers in the (n+1)-star graph. - Eric W. Weisstein, Aug 04 and Sep 21 2017

Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - Eric W. Weisstein, Dec 31 2017

Conjecture: For any integer n >= 0, a(n) is the permanent of the (n+1) X (n+1) matrix with M(j, k) = -floor((j - k - 1)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

When n > 0 is written as k*2^j with k odd then k = A000265(n) and j = A007814(n), so: when n is written as k*2^j - 1 with k odd then k = A000265(n+1) and j = A007814(n+1), when n > 1 is written as k*2^j + 1 with k odd then k = A000265(n-1) and j = A007814(n-1).

Also denominator of 2^n/n (numerator is A075101(n)). - Reinhard Zumkeller, Sep 01 2002

Slope of line connecting (o, a(o)) where o = (2^k)(n-1) + 1 is 2^k and (by design) starts at (1, 1). - Josh Locker (joshlocker(AT)macfora.com), Apr 17 2004

Numerator of n/2^(n-1). - Alexander Adamchuk, Feb 11 2005

From Marco Matosic, Jun 29 2005: (Start)

"The sequence can be arranged in a table:

1

1 3 1

1 5 3 7 1

1 9 5 11 3 13 7 15 1

1 17 9 19 5 21 11 23 3 25 13 27 7 29 15 31 1

Every new row is the previous row interspaced with the continuation of the odd numbers.

Except for the ones; the terms (t) in each column are t+t+/-s = t_+1. Starting from the center column of threes and working to the left the values of s are given by A000265 and working to the right by A000265." (End)

This is a fractal sequence. The odd-numbered elements give the odd natural numbers. If these elements are removed, the original sequence is recovered. - Kerry Mitchell, Dec 07 2005

2k + 1 is the k-th and largest of the subsequence of k terms separating two successive equal entries in a(n). - Lekraj Beedassy, Dec 30 2005

It's not difficult to show that the sum of the first 2^n terms is (4^n + 2)/3. - Nick Hobson, Jan 14 2005

In the table, for each row, (sum of terms between 3 and 1) - (sum of terms between 1 and 3) = A020988. - Eric Desbiaux, May 27 2009

This sequence appears in the analysis of A160469 and A156769, which resemble the numerator and denominator of the Taylor series for tan(x). - Johannes W. Meijer, May 24 2009

Indices n such that a(n) divides 2^n - 1 are listed in A068563. - Max Alekseyev, Aug 25 2013

From Alexander R. Povolotsky, Dec 17 2014: (Start)

With regard to the tabular presentation described in the comment by Marco Matosic: in his drawing, starting with the 3rd row, the first term in the row, which is equal to 1 (or, alternatively the last term in the row, which is also equal to 1), is not in the actual sequence and is added to the drawing as a fictitious term (for the sake of symmetry); an actual A000265(n) could be considered to be a(j,k) (where j >= 1 is the row number and k>=1 is the column subscript), such that a(j,1) = 1:

1

1 3

1 5 3 7

1 9 5 11 3 13 7 15

1 17 9 19 5 21 11 23 3 25 13 27 7 29 15 31

and so on ... .

The relationship between k and j for each row is 1 <= k <= 2^(j-1). In this corrected tabular representation, Marco's notion that "every new row is the previous row interspaced with the continuation of the odd numbers" remains true. (End)

Partitions natural numbers to the same equivalence classes as A064989. That is, for all i, j: a(i) = a(j) <=> A064989(i) = A064989(j). There are dozens of other such sequences (like A003602) for which this also holds: In general, all sequences for which a(2n) = a(n) and the odd bisection is injective. - Antti Karttunen, Apr 15 2017

From Paul Curtz, Feb 19 2019: (Start)

This sequence is the truncated triangle:

1, 1;

3, 1, 5;

3, 7, 1, 9;

5, 11, 3, 13, 7;

15, 1, 17, 9, 19, 5;

21, 11, 23, 3, 25, 13, 27;

7, 29, 15, 31, 1, 33, 17, 35;

...

The first column is A069834. The second column is A213671. The main diagonal is A236999. The first upper diagonal is A125650 without 0.

c(n) = ((n*(n+1)/2))/A069834 = 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 8, 8, 1, 1, ... for n > 0. n*(n+1)/2 is the rank of A069834. (End)

As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019

a(n) is also the map n -> A026741(n) applied at least A007814(n) times. - Federico Provvedi, Dec 14 2021

Don Knuth points out (personal communication) that Jacobsthal may never have seen the actual values of this sequence. However, Horadam uses the name "Jacobsthal sequence", such an important sequence needs a name, and there is a law that says the name for something should never be that of its discoverer. - N. J. A. Sloane, Dec 26 2020

Number of ways to tile a 3 X (n-1) rectangle with 1 X 1 and 2 X 2 square tiles.

Also, number of ways to tile a 2 X (n-1) rectangle with 1 X 2 dominoes and 2 X 2 squares. - Toby Gottfried, Nov 02 2008

Also a(n) counts each of the following four things: n-ary quasigroups of order 3 with automorphism group of order 3, n-ary quasigroups of order 3 with automorphism group of order 6, (n-1)-ary quasigroups of order 3 with automorphism group of order 2 and (n-2)-ary quasigroups of order 3. See the McKay-Wanless (2008) paper. - Ian Wanless, Apr 28 2008

Also the number of ways to tie a necktie using n + 2 turns. So three turns make an "oriental", four make a "four in hand" and for 5 turns there are 3 methods: "Kelvin", "Nicky" and "Pratt". The formula also arises from a special random walk on a triangular grid with side conditions (see Fink and Mao, 1999). - arne.ring(AT)epost.de, Mar 18 2001

Also the number of compositions of n + 1 ending with an odd part (a(2) = 3 because 3, 21, 111 are the only compositions of 3 ending with an odd part). Also the number of compositions of n + 2 ending with an even part (a(2) = 3 because 4, 22, 112 are the only compositions of 4 ending with an even part). - Emeric Deutsch, May 08 2001

Arises in study of sorting by merge insertions and in analysis of a method for computing GCDs - see Knuth reference.

Number of perfect matchings of a 2 X n grid upon replacing unit squares with tetrahedra (C_4 to K_4):

o----o----o----o...

| \/ | \/ | \/ |

| /\ | /\ | /\ |

o----o----o----o... - Roberto E. Martinez II, Jan 07 2002

Also the numerators of the reduced fractions in the alternating sum 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 + ... - Joshua Zucker, Feb 07 2002

Also, if A(n), B(n), C(n) are the angles of the n-orthic triangle of ABC then A(1) = Pi - 2*A, A(n) = s(n)*Pi + (-2)^n*A where s(n) = (-1)^(n-1) * a(n) [1-orthic triangle = the orthic triangle of ABC, n-orthic triangle = the orthic triangle of the (n-1)-orthic triangle]. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 05 2002

Also the number of words of length n+1 in the two letters s and t that reduce to the identity 1 by using the relations sss = 1, tt = 1 and stst = 1. The generators s and t and the three stated relations generate the group S3. - John W. Layman, Jun 14 2002

Sums of pairs of consecutive terms give all powers of 2 in increasing order. - Amarnath Murthy, Aug 15 2002

Excess clockwise moves (over counterclockwise) needed to move a tower of size n to the clockwise peg is -(-1)^n*(2^n - (-1)^n)/3; a(n) is its unsigned version. - Wouter Meeussen, Sep 01 2002

Also the absolute value of the number represented in base -2 by the string of n 1's, the negabinary repunit. The Mersenne numbers (A000225 and its subsequences) are the binary repunits. - Rick L. Shepherd, Sep 16 2002

Note that 3*a(n) + (-1)^n = 2^n is significant for Pascal's triangle A007318. It arises from a Jacobsthal decomposition of Pascal's triangle illustrated by 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = (7 + 35 + 1) + (1 + 35 + 7) + (21 + 21) = 43 + 43 + 42 = 3a(7) - 1; 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = (1 + 56 + 28) + (28 + 56 + 1) + (8 + 70 + 8) = 85 + 85 + 86 = 3a(8)+1. - Paul Barry, Feb 20 2003

Number of positive integers requiring exactly n signed bits in the nonadjacent form representation.

Equivalently, number of length-(n-1) words with letters {0, 1, 2} where no two consecutive letters are nonzero, see example and fxtbook link. - Joerg Arndt, Nov 10 2012

Counts walks between adjacent vertices of a triangle. - Paul Barry, Nov 17 2003

Every amphichiral rational knot written in Conway notation is a palindromic sequence of numbers, not beginning or ending with 1. For example, for 4 <= n <= 12, the amphichiral rational knots are: 2 2, 2 1 1 2, 4 4, 3 1 1 3, 2 2 2 2, 4 1 1 4, 3 1 1 1 1 3, 2 3 3 2, 2 1 2 2 1 2, 2 1 1 1 1 1 1 2, 6 6, 5 1 1 5, 4 2 2 4, 3 3 3 3, 2 4 4 2, 3 2 1 1 2 3, 3 1 2 2 1 3, 2 2 2 2 2 2, 2 2 1 1 1 1 2 2, 2 1 2 1 1 2 1 2, 2 1 1 1 1 1 1 1 1 2. For the number of amphichiral rational knots for n=2*k (k=1, 2, 3, ...), we obtain the sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, ... - Slavik Jablan, Dec 26 2003

a(n+2) counts the binary sequences of total length n made up of codewords from C = {0, 10, 11}. - Paul Barry, Jan 23 2004

Number of permutations with no fixed points avoiding 231 and 132.

The n-th entry (n > 1) of the sequence is equal to the 2,2-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004

a(n) is the number of Motzkin (n+1)-sequences whose flatsteps all occur at level 1 and whose height is less than or equal to 2. For example, a(4) = 5 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD. - David Callan, Dec 09 2004

a(n+1) gives row sums of A059260. - Paul Barry, Jan 26 2005

If (m + n) is odd, then 3*(a(m) + a(n)) is always of the form a^2 + 2*b^2, where a and b both equal powers of 2; consequently every factor of (a(m) + a(n)) is always of the form a^2 + 2*b^2. - Matthew Vandermast, Jul 12 2003

Number of "0,0" in f_{n+1}, where f_0 = "1" and f_{n+1} = a sequence formed by changing all "1"s in f_n to "1,0" and all "0"s in f_n to "0,1". - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006

All prime Jacobsthal numbers A049883[n] = {3, 5, 11, 43, 683, 2731, 43691, ...} have prime indices except for a(4) = 5. All prime Jacobsthal numbers with prime indices (all but a(4) = 5) are of the form (2^p + 1)/3 - the Wagstaff primes A000979[n]. Indices of prime Jacobsthal numbers are listed in A107036[n] = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}. For n>1 A107036[n] = A000978[n] Numbers n such that (2^n + 1)/3 is prime. - Alexander Adamchuk, Oct 03 2006

Correspondence: a(n) = b(n)*2^(n-1), where b(n) is the sequence of the arithmetic means of previous two terms defined by b(n) = 1/2*(b(n-1) + b(n-2)) with initial values b(0) = 0, b(1) = 1; the g.f. for b(n) is B(x) := x/(1-(x^1+x^2)/2), so the g.f. A(x) for a(n) satisfies A(x) = B(2*x)/2. Because b(n) converges to the limit lim (1-x)*B(x) = 1/3*(b(0) + 2*b(1)) = 2/3 (for x --> 1), it follows that a(n)/2^(n-1) also converges to 2/3 (see also A103770). - Hieronymus Fischer, Feb 04 2006

Inverse: floor(log_2(a(n))) = n - 2 for n >= 2. Also: log_2(a(n) + a(n-1)) = n - 1 for n >= 1 (see also A130249). Characterization: x is a Jacobsthal number if and only if there is a power of 4 (= c) such that x is a root of p(x) = 9*x*(x-c) + (c-1)*(2*c+1) (see also the indicator sequence A105348). - Hieronymus Fischer, May 17 2007

This sequence counts the odd coefficients in the expansion of (1 + x + x^2)^(2^n - 1), n >= 0. - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Oct 18 2007, Jan 08 2008

2^(n+1) = 2*A005578(n) + 2*a(n) + 2*A000975(n-1). Let A005578(n), a(n), A000975(n-1) = triangle (a, b, c). Then ((S-c), (S-b), (S-a)) = (A005578(n-1), a(n-1), A000975(n-2)). Example: (a, b, c) = (11, 11, 10) = (A005578(5), a(5), A000975(4)). Then ((S-c), (S-b), (S-a)) = (6, 5, 5) = (A005578(4), a(4), A000975(3)). - Gary W. Adamson, Dec 24 2007

Sequence is identical to the absolute values of its inverse binomial transform. A similar result holds for [0,A001045*2^n]. - Paul Curtz, Jan 17 2008

From a(2) on (i.e., 1, 3, 5, 11, 21, ...) also: least odd number such that the subsets of {a(2), ..., a(n)} sum to 2^(n-1) different values, cf. A138000 and A064934. It is interesting to note the pattern of numbers occurring (or not occurring) as such a sum (A003158). - M. F. Hasler, Apr 09 2008

a(n) is the term (5, 1) of n-th power of the 5 X 5 matrix shown in A121231. - Gary W. Adamson, Oct 03 2008

A147612(a(n)) = 1. - Reinhard Zumkeller, Nov 08 2008

a(n+1) = Sum(A153778(i): 2^n <= i < 2^(n+1)). - Reinhard Zumkeller, Jan 01 2009

It appears that a(n) is also the number of integers between 2^n and 2^(n+1) that are divisible by 3 with no remainder. - John Fossaceca (john(AT)fossaceca.net), Jan 31 2009

Number of pairs of consecutive odious (or evil) numbers between 2^(n+1) and 2^(n+2), inclusive. - T. D. Noe, Feb 05 2009

Equals eigensequence of triangle A156319. - Gary W. Adamson, Feb 07 2009

A three-dimensional interpretation of a(n+1) is that it gives the number of ways of filling a 2 X 2 X n hole with 1 X 2 X 2 bricks. - Martin Griffiths, Mar 28 2009

Starting with offset 1 = INVERTi transform of A002605: (1, 2, 6, 16, 44, ...). - Gary W. Adamson, May 12 2009

Convolved with (1, 2, 2, 2, ...) = A000225: (1, 3, 7, 15, 31, ...). - Gary W. Adamson, May 23 2009

The product of a pair of successive terms is always a triangular number. - Giuseppe Ottonello, Jun 14 2009

Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := -2, A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^(n-1)*det(A). - Milan Janjic, Jan 26 2010

Let R denote the irreducible representation of the symmetric group S_3 of dimension 2, and let s and t denote respectively the sign and trivial irreducible representations of dimension 1. The decomposition of R^n into irreducible representations consists of a(n) copies of R and a(n-1) copies of each of s and t. - Andrew Rupinski, Mar 12 2010

As a fraction: 1/88 = 0.0113636363... or 1/9898 = 0.00010103051121... - Mark Dols, May 18 2010

Starting with "1" = the INVERT transform of (1, 0, 2, 0, 4, 0, 8, ...); e.g., a(7) = 43 = (1, 1, 1, 3, 5, 11, 21) dot (8, 0, 4, 0, 2, 0, 1) = (8 + 4 + 10 + 21) = 43. - Gary W. Adamson, Oct 28 2010

Rule 28 elementary cellular automaton (A266508) generates this sequence. - Paul Muljadi, Jan 27 2011

This is a divisibility sequence. - Michael Somos, Feb 06 2011

From L. Edson Jeffery, Apr 04 2011: (Start)

Let U be the unit-primitive matrix (see [Jeffery])

U = U_(6,2) =

(0 0 1)

(0 2 0)

(2 0 1).

Then a(n+1) = (Trace(U^n))/3, a(n+1) = ((U^n){3, 3})/3, a(n) = ((U^n){1, 3})/3 and a(n) = ((U^(n+1))_{1, 1})/2. (End)

The sequence emerges in using iterated deletion of strictly dominated strategies to establish the best-response solution to the Cournot duopoly problem as a strictly dominant strategy. The best response of firm 1 to firm 2's chosen quantity is given by q*1 = 1/2*(a - c - q_2), where a is reservation price, c is marginal cost, and q_2 is firm two's chosen quantity. Given that q_2 is in [o, a - c], q*_1 must be in [o, 1/2*(a - c)]. Since costs are symmetric, we know q_2 is in [0, 1/2*(a - c)]. Then we know q*_1 is in [1/4*(a - c), 1/2*(a - c)]. Continuing in this way, the sequence of bounds we get (factoring out a - c) is {1/2, 1/4, 3/8, 5/16, ...}; the numerators are the Jacobsthal numbers. - _Michael Chirico, Sep 10 2011

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 3*a(n-1) equals the number of 3-colored compositions of n with all parts greater than or equal to 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

This sequence is connected with the Collatz problem. We consider the array T(i, j) where the i-th row gives the parity trajectory of i, for example for i = 6, the infinite trajectory is 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1... and T(6, j) = [0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, ..., 1, 0, 0, 1, ...]. Now, we consider the sum of the digits "1" of each column. We obtain the sequence a(n) = Sum_{k = 1..2^n} T(k, n) = Sum {k = 1..2^n} digits "1" of the n-th column. Because a(n) + a(n+1) = 2^n, then a(n+1) = Number of digits "0" among the 2^n elements of the n-th column. - _Michel Lagneau, Jan 11 2012

3!*a(n-1) is apparently the trace of the n-th power of the adjacency matrix of the complete 3-graph, a 3 X 3 matrix with diagonal elements all zero and off-diagonal all ones. The off-diagonal elements for the n-th power are all equal to a(n) while each diagonal element seems to be a(n) + 1 for an even power and a(n) - 1 for an odd. These are related to the lengths of closed paths on the graph (see Delfino and Viti's paper). - Tom Copeland, Nov 06 2012

From Paul Curtz, Dec 11 2012: (Start)

2^n * a(-n) = (-1)^(n-1) * a(n), which extends the sequence to negative indices: ..., -5/16, 3/8, -1/4, 1/2, 0, 1, 1, 3, 5, ...

The "autosequence" property with respect to the binomial transform mentioned in my comment of Jan 17 2008 is still valid if the term a(-1) is added to the array of the sequence and its iterated higher-order differences in subsequent rows:

0 1/2 1/2 3/2 5/2 11/2 ...

1/2 0 1 1 3 5 ...

-1/2 1 0 2 2 6 ...

3/2 -1 2 0 4 4 ...

-5/2 3 -2 4 0 8 ...

11/2 -5 6 -4 8 0 ...

The main diagonal in this array contains 0's. (End)

Assign to a triangle T(n, 0) = 1 and T(n+1, 1) = n; T(r, c) = T(r-1, c-1) + T(r-1, c-2) + T(r-2, c-2). Then T(n+1, n) - T(n, n) = a(n). - J. M. Bergot, May 02 2013

a(n+1) counts clockwise walks on n points on a circle that take steps of length 1 and 2, return to the starting point after two full circuits, and do not duplicate any steps (USAMO 2013, problem 5). - Kiran S. Kedlaya, May 11 2013

Define an infinite square array m by m(n, 0) = m(0, n) = a(n) in top row and left column and m(i, j) = m(i, j-1) + m(i-1, j-1) otherwise, then m(n+1, n+1) = 3^(n-1). - J. M. Bergot, May 10 2013

a(n) is the number of compositions (ordered partitions) of n - 1 into one sort of 1's and two sorts of 2's. Example: the a(4) = 5 compositions of 3 are 1 + 1 + 1, 1 + 2, 1 + 2', 2 + 1 and 2' + 1. - Bob Selcoe, Jun 24 2013

Without 0, a(n)/2^n equals the probability that n will occur as a partial sum in a randomly-generated infinite sequence of 1's and 2's. The limiting ratio is 2/3. - Bob Selcoe, Jul 04 2013

Number of conjugacy classes of Z/2Z X Z/2Z in GL(2,2^(n+1)). - Jared Warner, Aug 18 2013

a(n) is the top left entry of the (n-1)-st power of the 3 X 3 matrix [1, 1, 1, 1, 0, 0, 1, 0, 0]. a(n) is the top left entry of the (n+1)-st power of any of the six 3 X 3 matrices [0, 1, 0; 1, 1, 1; 0, 1, 0], [0, 1, 1; 0, 1, 1; 1, 1, 0], [0, 0, 1; 1, 1, 1; 1, 1, 0], [0, 1, 1; 1, 0, 1; 0, 1, 1], [0, 0, 1; 0, 0, 1; 1, 1, 1] or [0, 1, 0; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

This is the only integer sequence from the family of homogeneous linear recurrence of order 2 given by a(n) = k*a(n-1) + t*a(n-2) with positive integer coefficients k and t and initial values a(0) = 0 and a(1) = 1 whose ratio a(n+1)/a(n) converges to 2 as n approaches infinity. - Felix P. Muga II, Mar 14 2014

This is the Lucas sequence U(1, -2). - Felix P. Muga II, Mar 21 2014

sqrt(a(n+1) * a(n-1)) -> a(n) + 3/4 if n is even, and -> a(n) - 3/4 if n is odd, for n >= 2. - Richard R. Forberg, Jun 24 2014

a(n+1) counts closed walks on the end vertices of P_3 containing one loop at the middle vertex. a(n-1) counts closed walks on the middle vertex of P_3 containing one loop at that vertex. - David Neil McGrath, Nov 07 2014

From César Eliud Lozada, Jan 21 2015: (Start)

Let P be a point in the plane of a triangle ABC (with sides a, b, c) and barycentric coordinates P = [x:y:z]. The Complement of P with respect to ABC is defined to be Complement(P) = [b*y + c*z : c*z + a*x : a*x + b*y].

Then, for n >= 1, Complement(Complement(...(Complement(P))..)) = (n times) =

[2*a(n-1)*a*x + (2*a(n-1) - (-1)^n)*(b*y + c*z):

2*a(n-1)*b*y + (2*a(n-1) - (-1)^n)*(c*z + a*x):

2*a(n-1)*c*z + (2*a(n-1) - (-1)^n)*(a*x + b*y)]. (End)

a(n) (n >= 2) is the number of induced hypercubes of the Fibonacci cube Gamma(n-2). See p. 513 of the Klavzar reference. Example: a(5) = 11. Indeed, the Fibonacci cube Gamma(3) is <>- (cycle C(4) with a pendant edge) and the hypercubes are: 5 vertices, 5 edges, and 1 square. - Emeric Deutsch, Apr 07 2016

If the sequence of points {P_i(x_i, y_i)} on the cubic y = a*x^3 + b*x^2 + c*x + d has the property that the segment P_i(x_i, y_i) P_i+1(x_i+1, y_i+1) is always tangent to the cubic at P_i+1(x_i+1, y_i+1) then a(n) = -2^n*a/b*(x_(n+1)-(-1/2)^n*x_1). - Michael Brozinsky, Aug 01 2016

With the quantum integers defined by [n+1]A000225%20are%20given%20by%20q%20=%20sqrt(2).%20Cf.%20A239473.%20-%20_Tom%20Copeland">q = (q^(n+1) - q^(-n-1)) / (q - q^(-1)), the Jacobsthal numbers are a(n+1) = (-1)^n*q^n [n+1]_q with q = i * sqrt(2) for i^2 = -1, whereas the signed Mersenne numbers A000225 are given by q = sqrt(2). Cf. A239473. - _Tom Copeland, Sep 05 2016

Every positive integer has a unique expression as a sum of Jacobsthal numbers in which the index of the smallest summand is odd, with a(1) and a(2) both allowed. See the L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr. reference. - Ira M. Gessel, Dec 31 2016. See A280049 for these expansions. - N. J. A. Sloane, Dec 31 2016

For n > 0, a(n) equals the number of ternary words of length n-1 in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

For n > 0, a(n) equals the number of orbits of the finite group PSL(2,2^n) acting on subsets of size 4 for the 2^n+1 points of the projective line. - Paul M. Bradley, Jan 31 2017

For n > 1, number of words of length n-2 over alphabet {1,2,3} such that no odd letter is followed by an odd letter. - Armend Shabani, Feb 17 2017

Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283641. - Robert Price, Mar 12 2017

Also the number of independent vertex sets and vertex covers in the 2 X (n-2) king graph. - Eric W. Weisstein, Sep 21 2017

From César Eliud Lozada, Dec 14 2017: (Start)

Let T(0) be a triangle and let T(1) be the medial triangle of T(0), T(2) the medial triangle of T(1) and, in general, T(n) the medial triangle of T(n-1). The barycentric coordinates of the first vertex of T(n) are [2*a(n-1)/a(n), 1, 1], for n > 0.

Let S(0) be a triangle and let S(1) be the antimedial triangle of S(0), S(2) the antimedial triangle of S(1) and, in general, S(n) the antimedial triangle of S(n-1). The barycentric coordinates of the first vertex of S(n) are [-a(n+1)/a(n), 1, 1], for n > 0. (End)

a(n) is also the number of derangements in S_{n+1} with empty peak set. - Isabella Huang, Apr 01 2018

For n > 0, gcd(a(n), a(n+1)) = 1. - Kengbo Lu, Jul 27 2020

Number of 2-compositions of n+1 with 1 not allowed as a part; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020

The number of Hamiltonian paths of the flower snark graph of even order 2n > 2 is 12*a(n-1). - Don Knuth, Dec 25 2020

When set S = {1, 2, ..., 2^n}, n>=0, then the largest subset T of S with the property that if x is in T, then 2*x is not in T, has a(n+1) elements. For example, for n = 4, #S = 16, a(5) = 11 with T = {1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16} (see Hassan Tarfaoui link, Concours Général 1991). - Bernard Schott, Feb 14 2022

a(n) is the number of words of length n over a binary alphabet whose position in the lexicographic order is one more than a multiple of three. a(3) = 3: aaa, abb, bba. - Alois P. Heinz, Apr 13 2022

Named by Horadam (1988) after the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Oct 02 2023

Define the sequence u(n) = (u(n-1) + u(n-2))/u(n-3) with u(0) = 0, u(1) = 1, u(2) = u(3) = -1. Then u(4*n) = -1 + (-1)^n/a(n+1), u(4*n+1) = 2 - (-1)^n/a(n+1), u(4*n+2) = u(4*n+3) = -1. For example, a(3) = 3 and u(8) = -2/3, u(9) = 5/3, u(10) = u(11) = -1. - Michael Somos, Oct 24 2023

From Miquel A. Fiol, May 25 2024: (Start)

Also, a(n) is the number of (3-color) states of a cycle (n+1)-pole C_{n+1} with n+1 terminals (or semiedges).

For instance, for n=3, the a(3)=3 states (3-coloring of the terminals) of C_4 are

a a a a a b

a a b b a b (End)

Also, with offset 1, the cogrowth sequence of the 6-element dihedral group D3. - Sean A. Irvine, Nov 04 2024

Equivalently, integers k such that 2^k - 1 is prime.

It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K.

Length of prime repunits in base 2.

The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy, Aug 21 2004

In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.

Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - Artur Jasinski, Feb 09 2007

Number of divisors of n-th even perfect number, divided by 2. Number of divisors of n-th even perfect number that are powers of 2. Number of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 24 2008

Number of divisors of n-th even superperfect number A061652(n). Numbers of divisors of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Mar 01 2008

Differences between exponents when the even perfect numbers are represented as differences of powers of 2, for example: The 5th even perfect number is 33550336 = 2^25 - 2^12 then a(5)=25-12=13 (see A135655, A133033, A090748). - Omar E. Pol, Mar 01 2008

Number of 1's in binary expansion of n-th even perfect number (see A135650). Number of 1's in binary expansion of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n) (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008

Indices of the numbers A006516 that are also even perfect numbers. - Omar E. Pol, Aug 30 2008

Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - Omar E. Pol, Aug 31 2008

The (prime) number p appears in this sequence if and only if there is no prime q<2^p-1 such that the order of 2 modulo q equals p; a special case is that if p=4k+3 is prime and also q=2p+1 is prime then the order of 2 modulo q is p so p is not a term of this sequence. - Joerg Arndt, Jan 16 2011

Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - Jaroslav Krizek, Aug 02 2013

Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k-1. Equivalently, the integers k such that A001037(k) = A011260(k). - Geoffrey Critzer, Dec 08 2019

Conjecture: for k > 1, 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k-1)^(2^k-2) == 1 (mod (2^k-1)k^2). - Thomas Ordowski, Oct 05 2023

Conjecture: for p prime, 2^p-1 is (a Mersenne) prime or p = 2^(2^m)+1 (is a Fermat number) if and only if (p-1)^(2^p-2) == 1 (mod 2^p-1). - David Barina, Nov 25 2024

Already as of Dec. 2020, all exponents up to 10^8 had been verified, implying that 74207281, 77232917 and 82589933 are indeed the next three terms. As of today, all exponents up to 130439863 have been tested at least once, see the GIMPS Milestones Report. - M. F. Hasler, Apr 11 2025

On June 23. 2025 all exponents up to 74340751 have been verified, confirming that 74207281 is the exponent of the 49th Mersenne Prime. - Rodolfo Ruiz-Huidobro, Jun 23 2025