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COMMENTS
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a(n) = 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) = A083323(n)-1 = A056182(n)/2 = (A002783(n)-1)/2 = (A003063(n+2)-A003063(n+1))/2. - Ralf Stephan, Jan 12 2004
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) = A112626(n, 1). - Ross La Haye, Jan 11 2006
a(n+1) = sums 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 = {2,3,5,17,29,31,53,59,101,277,647,1061,2381,...} = A057468(n) Numbers n such that 3^n - 2^n is prime.
p divides a(p) - 1 for prime p.
Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071 = {2,6,42,294,15918,122010,7588770,61144062,...}.
Numbers n such that n divides 3^n - 2^n - 1 are listed in A127072 = {1,2,3,4,5,7,8,9,11,13,16,17,19,23,27,29,31,32,37,41,43,45,47,49,53,59,61,64,67,71,73,79,81,83,89,97,...}.
Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073(n) = {45,245,405,561,637,639,833,891,...}, which includes all Carmichael numbers A002997(n) = {561,1105,1729,2465,2821,6601,8911,10585,15841,29341,...}.
Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074(n) = {1,2,3,4,7,49,179,619,17807,...}.
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 = {5,19,23,29,43,47,53,...} = A097936(n) Primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2). 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 = {5,19,23,29,43,47,53,...} = A097936(n).
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)
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
a(n) = sum_{k=0..2} Stirling1(2,k)*(k+1)^n = c_2^{(-n)}, poly-Cauchy numbers. - Takao Komatsu, Mar 28 2013
a(n) = A227048(n,A098294(n)). - Reinhard Zumkeller, Jun 30 2013
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
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