0,2

The number of triangles (of all sizes, including holes) in Sierpinski's triangle after n inscriptions. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004

The sequence is not only related to Sierpinski's triangle, but also to "Floret's cube" and the quaternion factor space Q X Q / {(1,1), (-1,-1)} as described at www.crowdog.de It can be written as a_n = ves((A+1)x)^n) as described at http://mathforum.org/discuss/sci.math/t/622432 - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 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 (lambert.klasen(AT)gmx.net), 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 (amarnath_murthy(AT)yahoo.com), Aug 01 2005

C. Dement, The Math Forum.

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+1) for n=1, 2, 3...

a(0)=1, a(n) = a(n-1) + 3^n + 3^(n-1) - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004

a(n) = (3^n + 3^(n+1) - 2)/2. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jul 31 2004

(1, 5, 17, 53, 161, ...) = Ternary (1, 12, 122, 1222, 12222, ...) - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 02 2005

Row sums of triangle A134347. Also, binomial transform of A046055: (1, 4, 8, 16, 32, 64,...); and double binomial transform of A010684: (1, 3, 1, 3, 1, 3,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2007

a(n)=T(2,n), array T given by A048471.

Cf. A003462, A029858. A column of A119725.

Cf. A134347, A046055, A010684.

Adjacent sequences: A048470 A048471 A048472 this_sequence A048474 A048475 A048476

Sequence in context: A088210 A135344 A027028 this_sequence A097160 A079363 A034346

nonn

Clark Kimberling (ck6(AT)evansville.edu)

Better description from Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 27 2001