This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362706 #26 Jul 09 2023 08:34:00
%S A362706 0,0,0,1,1,2,3,6,6,7,9,13,16,21,27,36,36,37,40,45,50,57,66,78,85,94,
%T A362706 106,121,136,154,175,200,200,201,205,211,219,229,242,258,271,286,305,
%U A362706 327,351,378,409,444,463,484,510,539,571,606,646,690,729,771,819
%N A362706 Number of squares formed by first n vertices of the infinite-dimensional hypercube.
%C A362706 We can take the coordinates of a vertex to represent a binary number, so we define the n-th point to have coordinates represented by the binary expansion of n-1.
%C A362706 Let d(m) = a(m+1) - a(m) be the shifted first differences of a(n), so that d(m) represents the additional squares introduced by the (m+1)-th vertex. Then d(0) = d(2^x) = 0; when m = 2^x + 2^y, x > y, d(m) = A115990(x - 1, x - y - 1); generally, d(m) = sum d(k) for all k formed by selecting two 1's from the binary expansion of m. Thus d(7) = d(3) + d(5) + d(6).
%C A362706 a(n) is a lower bound for an infinite-dimensional extension of A051602. _Peter Munn_ notes that it is not an upper bound: for example, the vertices of a regular {k-1}-simplex duplicated at unit distance in any orthogonal direction gives T_k squares from 2k+2 points, which exceeds a(n) at 6, 10 and 12 points.
%F A362706 a(2^k) = A345340(k).
%e A362706 The 6 points (0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1) give the squares (0,0,0), (1,0,0), (0,1,0), (1,1,0) and (0,0,0), (1,0,0), (0,0,1), (1,0,1). So a(6) = 2.
%Y A362706 Cf. A345340, A115990, A051602.
%K A362706 nonn
%O A362706 1,6
%A A362706 _Hugo van der Sanden_, Jun 22 2023