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 A112368 #19 Dec 30 2023 08:13:17
%S A112368 1,3,11,59,443,4283,50363,695483,11017403,196811963,3912703163,
%T A112368 85662309563,2047652863163,53059407256763,1481388530277563,
%U A112368 44331262220901563,1415527220320869563,48036189795719781563,1726380042510080613563,65503446445655792229563,2616586102571484256869563
%N A112368 a(n) = Sum_{i=0..n} 2^i*i!.
%C A112368 a(n) is divisible by 73 for all n >= 72, hence this sequence contains only a finite number of primes. - _Giovanni Resta_, Mar 11 2017
%C A112368 Partial sums of A000165. One less than A004400. one more than A112369. - _Michael Somos_, Sep 27 2017
%H A112368 Seiichi Manyama, <a href="/A112368/b112368.txt">Table of n, a(n) for n = 0..403</a>
%F A112368 0 = +a(n)*(+2*a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(-a(n+1) + a(n+2) - a(n+3)) + a(n+2)*(a(n+2)) for all n>=0. - _Michael Somos_, Sep 27 2017
%e A112368 G.f. = 1 + 3*x + 11*x^2 + 59*x^3 + 443*x^4 + 4283*x^5 + 50363*x^6 + 695483*x^7 + ...
%t A112368 s = 1; lst = {s}; Do[s += n!!; AppendTo[lst, s], {n, 2, 38, 2}]; lst (* _Zerinvary Lajos_, Jul 13 2009 *)
%t A112368 a[ n_] := Sum[ 2^k k!, {k, 0, n}]; (* _Michael Somos_, Sep 27 2017 *)
%o A112368 (PARI) {a(n) = sum(k=0, n, 2^k * k!)}; /* _Michael Somos_, Sep 27 2017 */
%Y A112368 Cf. A000165, A004400, A112369.
%K A112368 nonn
%O A112368 0,2
%A A112368 _N. J. A. Sloane_, Dec 02 2005