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A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

%I A130820 #23 Jun 08 2025 01:13:21
%S A130820 2,8,7,0,2,2,2,1,5,6,9,7,3,3,9,6,3,3,0,8,1,9,4,5,8,8,6,5,8,1,1,1,9,9,
%T A130820 6,0,1,2,4,0,3,1,9,2,6,2,2,8,0,9,9,5,7,0,1,2,0,3,1,2,7,7,3,6,2,7,2,8,
%U A130820 5,0,3,8,0,7,6,8,0,3,7,5,2,7,8,4,5,6,3,9,2,3,6,1,5,0,7,1,4,8,2,4
%N A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...
%D A130820 Engel, F. "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.
%H A130820 F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H A130820 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.
%F A130820 From _Peter Bala_, Jul 01 2016: (Start)
%F A130820 Constant c = 1/1 + 1/(1*1) + 1/(1*1*2) + 1/(1*1*2*2) + 1/(1*1*2*2*3) + 1/(1*1*2*2*3*3) + ... = Sum_{n >= 1} binomial(n,floor(n/2))/n!.
%F A130820 Alternative series representations:
%F A130820 c = 3 - Sum_{n >= 2} 1/(n*(n - 1)*n!^2);
%F A130820 c = 1 + Sum_{n >= 1} (n + 2)/(n!*(n + 1)!);
%F A130820 c = 5/3 + 1/3*Sum_{n >= 2} (n + 1)*(n + 2)/n!^2;
%F A130820 c = A070910 + A096789 - 1.
%F A130820 Continued fraction: c = 3 - 1/(8 - 4/(14 - 9/(32 - ... - (n-1)^2/(n^2 + n + 2 - ...)))). See comments in A141827. (End)
%e A130820 2.8702221569733963308194588658111996012403192622809957012...
%p A130820 evalf(BesselI(0, 2) + BesselI(1, 2) - 1, 100); # _Peter Bala_, Jul 02 2016
%t A130820 First@ RealDigits@ N[Sum[1/Product[Ceiling[r/2], {r, n}], {n, 1000}], 100] (* Original program amended to generate output by _Michael De Vlieger_, Jul 03 2016 *)
%t A130820 RealDigits[3 - HypergeometricPFQ[{1, 1}, {3, 3, 3}, 1]/8, 10, 100][[1]] (* _Vaclav Kotesovec_, Jul 03 2016 *)
%Y A130820 Cf. A006784, A064648, A101689, A001405, A070910, A096789, A141827.
%K A130820 cons,easy,nonn
%O A130820 1,1
%A A130820 Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007