The number of triangles (of all sizes, including holes) in Sierpiński's triangle after n inscriptions. - Lee Reeves, May 10 2004
The sequence is not only related to Sierpiński's triangle, but also to "Floret's cube" and the quaternion factor space Q X Q / {(1,1), (-1,-1)}. It can be written as a_n = ves((A+1)x)^n) as described at the Math Forum Discussions link. -
Creighton Dement, Jul 28 2004
Relation to C(n) = Collatz function iteration using only odd steps: If we look for record subsequences where C(n) > n, this subsequence starts at 2^n - 1 and stops at the local maximum of 2*3^n - 1. Examples: [3,5], [7,11,17], [15,23,35,53], ..., [127,191,287,431,647,971,1457]. - Lambert Klasen, Mar 11 2005
Group the natural numbers so that the (2n-1)-th group sum is a multiple of the (2n)-th group containing one term. (1,2),(3),(4,5,6,7,8,9,10,11),(12),(13,14,15,16,17,18,19,...,38),(39),(40,41,...,118,119),(120), (121,122,123,...) ... a(n) = {the sum of the terms of (2n-1)-th group}/{the term of (2n)th group}. The first term of the odd numbered group is given by
A003462. The only term of even numbered group is given by
A029858. -
Amarnath Murthy, Aug 01 2005
a(n)+1 =
A008776(n); it appears that this gives the number of terms in the (n+1)-th "gap" of numbers missing in
A171884. -
M. F. Hasler, May 09 2013
Sum of n-th row of triangle of powers of 3: 1; 1 3 1; 1 3 9 3 1; 1 3 9 27 9 3 1; ... -
Philippe Deléham, Feb 23 2014
For n >= 3, also the number of dominating sets in the n-helm graph. -
Eric W. Weisstein, May 28 2017
The number of elements of length <= n in the free group on two generators. -
Anton Mellit, Aug 10 2017
In general, a first order inhomogeneous recurrence of the form s(0) = a, s(n) = m*s(n-1) + k, n>0, will have a closed form of a*m^n + ((m^n-1)/(m-1))*k. -
Gary Detlefs, Jun 07 2024
