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 A342379 #16 Mar 12 2021 04:05:01
%S A342379 0,0,0,0,0,5,21,63,162,381,847,1815,3796,7813,15913,32191,64838,
%T A342379 130237,261155,523127,1047224,2095589,4192509,8386559,16774890,
%U A342379 33551805,67105911,134214423,268431772,536866821,1073737297,2147478655,4294961806,8589928573,17179862603
%N A342379 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^3/6 - x^2/2 - x - 1).
%C A342379 a(n) is the number of binary strings of length n that contain at least four 0's but not all digits are 0.
%C A342379 a(n) is also the number of proper subsets with at least four elements of an n-element set.
%H A342379 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).
%F A342379 a(n) = 2^n - Sum_{i={0..3,n}} binomial(n,i).
%F A342379 G.f.: x^5*(2*x^3-7*x^2+9*x-5)/((2*x-1)*(x-1)^4). - _Alois P. Heinz_, Mar 09 2021
%e A342379 a(7) = 63 since the strings are the 35 permutations of 0000111, the 21 permutations of 0000011, and the 7 permutations of 0000001.
%Y A342379 Cf. A342352, A002663.
%K A342379 nonn,easy
%O A342379 0,6
%A A342379 _Enrique Navarrete_, Mar 09 2021