cp's OEIS Frontend

From Peter M. Chema, Mar 02 2017: (Start)

Number of rods (line segments) required to make a Sierpinski tetrahedron of side length 2^n.

Also equals the number of balls (vertices) in a Sierpinski tetrahedron of side length 2^n+1 minus the number of balls in a Sierpinski tetrahedron of side length 2^n (the first difference in the tetrix numbers). See formula. (End)

Equivalently, the number of edges in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017

These numbers a(n) together with the 13 numbers from A337217 give the positive integers m represented uniquely by the ternary form x^2 + y^2 + 2*z^2, with integers 0 <= x <= y and 0 <= z. This is theorem 2.1 of Kaplansky, p. 87 with proof on p. 90. - Wolfdieter Lang, Aug 20 2020

a(n) is also the domination number of the (n+3)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 13 2021

This sequence gives Theorem 2.2. of Kaplansky, p. 88, with a proof on p. 90.

This sequence is composed of two finite ones and an infinite one: (i) 2*A337217 = {2, 6, 10, 14, 22, 30, 42, 46, 58, 70, 78, 142, 190}, the even members of A094739, (ii) {1, 5, 13, 21, 37, 93, 133, 253}, the 1 (mod 4) members of A094739, and (iii) A002001(k+1) = 4^k*3, for integer k >= 0. Beginning with a(26) = 768 only the powers 4^k*3, for k >= 4 appear.

See eq. (2.2), (2,4), p. 87, of Kaplansky for the two finite sequences with 13 and 8 members, respectively.

The positive integers which have no such solution (x, y, z) are given by 4^k*(7+8*m) = A002001(k+1)*A004771(m), for k >= 0 and m >= 0. See Kaplansky, p. 88. The other missing positive integers have more than 1 solution.