cp's OEIS Frontend

Binomial transform is A085351.

a(n) mod 10 = period 4:repeat 1,5,3,1 = A132400. - Paul Curtz, Nov 13 2009

Also the second differences of A001047.

Equals sum of "terms added" to current row of the triangle version of A038573 to get the next row. a(3) = 32 sum of (3, 7, 7, 15) = terms appended to row 2 of the triangle in A038573. - Gary W. Adamson, Jun 04 2009

a(n+1) is the sum of the elements in the n-th row of triangle pertaining to A036561. - Amarnath Murthy, Jan 02 2002

Number of 2 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002

With offset 1, partial sums of A027649. - Paul Barry, Jun 24 2003

Number of distinct lines through the origin in the n-dimensional lattice of side length 2. A049691 has the values for the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003

a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of the harmonic sequence 1/(n+1). - Paul Barry, Apr 19 2005

a(n+1) is the sum of n-th row of A036561. - Reinhard Zumkeller, May 14 2006

The sequence gives the sum of the lengths of the segments in Cantor's dust generating sequence up to the i-th step. Measurement unit = length of the segment of i-th step. - Giorgio Balzarotti, Nov 18 2006

Let T be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xTy if x is a proper subset of y. Then a(n) = |T|. - Ross La Haye, Dec 22 2006

From Alexander Adamchuk, Jan 04 2007: (Start)

a(n) is prime for n in A057468.

p divides a(p) - 1 for prime p.

Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.

Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.

Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073, which includes all Carmichael numbers A002997.

Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074.

5 divides a(2n).

5^2 divides a(2*5n).

5^3 divides a(2*5^2n).

5^4 divides a(2*5^3n).

7^2 divides a(6*7n).

13 divides a(4n).

13^2 divides a(4*13n).

19 divides a(3n).

19^2 divides a(3*19n).

23^2 divides a(11n).

23^3 divides a(11*23n).

23^4 divides a(11*23^2n).

29 divides a(7n).

p divides a((p-1)n) for prime p>3.

p divides a((p-1)/2) for prime p in A097934. Also primes p such that 6 is a square mod p, except {2,3}, A038876(n).

p^(k+1) divides a(p^k*(p-1)/2*n) for prime p in A097934.

p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.

Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).

There are no more such exceptions for primes p up to 600000. (End)

a(n) divides a(q*(n+1)-1), for all q integer. Leonardo Sarasua, Apr 15 2024

Final digits of terms follow sequence 1,5,9,5. - Enoch Haga, Nov 26 2007

This is also the second column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below. - Wolfdieter Lang, Oct 08 2011

Partial sums give A000392. - Jon Perry, Apr 05 2014

For n >= 1, this is also row 2 of A281890: when consecutive positive integers are written as a product of primes in nondecreasing order, "3" occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 17 2017

a(n) is the number of ternary sequences of length n which include the digit 2. For example, a(2)=5 since the sequences are 02,20,12,21,22. - Enrique Navarrete, Apr 05 2021

a(n-1) is the number of ways we can form disjoint unions of two nonempty subsets of [n] such that the union contains n. For example, for n = 3, a(2) = 5 since the disjoint unions are {1}U{3}, {1}U{2,3}, {2}U{3}, {2}U{1,3}, and {1,2}U{3}. Cf. A000392 if we drop the requirement that the union contains n. - Enrique Navarrete, Aug 24 2021

Configures as a composite Koch Snowflake Fractal (see illustration in links) based on the five-fold division of the Cantor Square/Cantor Dust Fractal of (9^n-4^n)/5 see my illustration in (A016153). - John Elias, Oct 13 2021

Number of pairs (A,B) where B is a subset of {1,2,...,n} and A is a proper subset of B. - Jianing Song, Jun 18 2022

From Manfred Boergens, Mar 29 2023: (Start)

With regard to the comments by Ross La Haye and Jianing Song: Omitting "proper" gives A000244.

Number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty subset of B. For nonempty proper subsets see a(n+1) in A028243. (End)

a(n) is the number of n-digit numbers whose smallest decimal digit is 7. - Stefano Spezia, Nov 15 2023

a(n-1) is the number of all possible player-reduced binary games observed by each player in an nx2 game assuming the individual strategies of k < n - 1 players are fixed and the remaining n - k - 1 player will play as one, either maintaining their status quo strategies or jointly adopting an alternative strategy. - Ambrosio Valencia-Romero, Apr 11 2024

If the binary expansion of n is 1^b 0^c 1^d 0^e ..., then a(n) is the number whose binary expansion is 1^(b-1) 0^(c-1) 1^(d-1) 0^(e-1) .... Leading 0's are omitted, and if the result is the empty string, here we set a(n) = 0. See A319419 for a version which represents the empty string by -1.

Lenormand refers to this operation as planing ("raboter") the runs (or blocks) of the binary expansion.

A175046 expands the runs in a similar way, and a(A175046(n)) = A001477(n). - Andrew Weimholt, Sep 08 2018. In other words, this is a left inverse to A175046: A318921 o A175046 = A001477 = id on [0..oo). - M. F. Hasler, Sep 10 2018

Conjecture: For n in the range 2^k, ..., 2^(k+1)-1, the total value of a(n) appears to be 2*3^(k-1) - 2^(k-1) (see A027649), and so the average value of a(n) appears to be (3/2)^(k-1) - 1/2. - N. J. A. Sloane, Sep 25 2018

The above conjecture was proved by Doron Zeilberger on Nov 16 2018 (see link) and independently by Chai Wah Wu on Nov 18 2018 (see below). - N. J. A. Sloane, Nov 20 2018

From Chai Wah Wu, Nov 18 2018: (Start)

Conjecture is correct for k > 0. Proof: looking at the least significant 2 bits of n, it is easy to see that a(4n) = 2a(2n), a(4n+1) = a(2n), a(4n+2) = a(2n+1) and a(4n+3) = 2a(2n+1)+1. Define f(k) = Sum_{i=2^k..2^(k+1)-1} a(i), i.e. the sum ranges over all numbers with a (k+1)-bit binary expansion. Thus f(0) = a(1) = 0 and f(1) = a(2)+a(3) = 1. By summing over the recurrence relations for a(n), we get f(k+2) = Sum_{i=2^k..2^(k+1)-1} (f(4i) + f(4i+1) + f(4i+2) + f(4i+3)) = Sum_{i=2^k..2^(k+1)-1} (3a(2i) + 3a(2i+1) + 1) = 3*f(k+1) + 2^k. Solving this first order recurrence relation with the initial condition f(1) = 1 shows that f(k) = 2*3^(k-1)-2^(k-1) for k > 0.

(End)

Essentially the same sequence as A001316, which has much more information, and also A159913. - N. J. A. Sloane, Jun 05 2009

Smallest number with same number of 1's in its binary expansion as n.

Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v, Jan 24 2006

From Gary W. Adamson, Jun 04 2009: (Start)

As an infinite string, 2^n terms per row starting with "1": (1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31;...)

Row sums of that triangle = A027649: (1, 4, 14, 46, 454, ...); where the next row sum = current term of A027649 + next term in finite difference row of A027649, i.e., (1, 3, 10, 32, 100, 308, ...) = A053581. (End)

From Omar E. Pol, Jan 24 2016: (Start)

Partial sums give A267700.

a(n) is also the number of cells turned ON at n-th generation of the cellular automaton of A267700 in a 90-degree sector on the square grid.

a(n) is also the number of Y-toothpicks added at n-th generation of the structure of A267700 in a 120-degree sector on the triangular grid. (End)

Row sums of A090971. - Nikolaos Pantelidis, Nov 23 2022

B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructible from its row and column sums. [Brewbaker]

B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k }. [Launois]

T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan, Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)

T(n,k) is also the number of acyclic orientations of the complete bipartite graph K_{n,k}. - Vincent Pilaud, Sep 15 2020

When indexed as a triangular array, T(n,k) is the number of permutations of [n] in which 1 is in position k and the excedance entries are precisely the entries to the left of 1. See link. - David Callan, Dec 12 2021

T(n,k) is also the number of max-closed relations between an ordered n-element set and an ordered k-element set (see the paper by Jeavons and Cooper 1995). - Don Knuth, Feb 12 2024

A130321^2 = A130322. Binomial transform of A130321 = triangle A027649. A007318^2 = A038207 = A007318(n,k) * A130321(n,k); i.e., the square of Pascal's triangle = dot product of Pascal's triangle rows and A130321 rows: A007318^2 = (1; 2,1; 4,4,1; 8,12,6,1;...), where row 3, (8,12,6,1) = (1,3,3,1) dot (8,4,2,1).

Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A130321 is the reverse reluctant sequence of sequence of power of 2 (A000079). - Boris Putievskiy, Dec 13 2012

From Wolfdieter Lang, Jan 10 2015: (Start)

This is the Riordan array (1/(1-2*x), x).

Row sums give A000225(n+1) = 2^(n+1) - 1.

Alternating row sums give A001045(n+1).

The inverse Riordan array is (1-2*x, x) = A251635. (End)

Let V be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xVy if x is a proper subset of y or y is a proper subset of x. Then a(n) = |V|. - Ross La Haye, Dec 22 2006

With regard to the comment by Ross La Haye: For nonempty subsets see a(n+1) in A260217. - If "proper" is omitted see A027649. - For nonempty subsets with "proper" omitted see A091344. - Manfred Boergens, Sep 04 2023

It appears that a(n) is the number of permutations p of 1,..,(n+2) such that max[p(i+1)-p(i)] is 2. For example, for n=1, the permutations (1,3,2) and (2,1,3) and no others have the desired property, so a(1)=2. This approach gives values in agreement with all listed terms. [John W. Layman, Nov 09 2011]

In the terdragon curve, a(n-1) is the number of enclosed unit triangles in expansion level n. - Kevin Ryde, Oct 20 2020

a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).

a(n,1) = a(n-1,1) + a(n-1,3) for n > 0; a(n,1) = A001047(n) = 2^(2n) - A083324(n); a(n,2) = A000244(n) = 2^(2n) - A005061(n); a(n,3) = 2a(n-1,4) for n > 0; a(n,3) = A027649(n); a(n,4) = A083313(n+1); a(n,5) = A084171(n+1).

Sum[a(n-k,k), {k,0,n}] = A098703(n+1), antidiagonal sums.

Let R, S and T be binary relations on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x, xSy if x is a subset of y and xTy if x is a proper subset of y. Then a(n,3) = |R|, a(n,2) = |S| and a(n,1) = |T|. Note that a binary relation W on P(A) can be defined also such that for every element x, y of P(A) xWy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. A090802(n,1) = |W|. Also, a(n,0) = |P(A)|.

Starting with offset 1 = binomial transform of A068293: (1, 6, 18, 42, 90, ...) and double binomial transform of (1, 5, 7, 5, 7, 5, ...). - Gary W. Adamson, Jan 13 2009

Number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a subset of the other. - For the case that one of these subsets is a proper subset of the other see a(n+1) in A260217. - If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). - Manfred Boergens, Aug 02 2023