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

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, 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, 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

Offset: 1

Author

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

Keywords

• cons
• easy
• nonn

Examples

			2.8702221569733963308194588658111996012403192622809957012...
		

References

• Engel, F. "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Links

• F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
• Eric Weisstein's World of Mathematics, Engel Expansion.

Crossrefs

Cf. A006784, A064648, A101689, A001405, A070910, A096789, A141827.

Programs

Maple

evalf(BesselI(0, 2) + BesselI(1, 2) - 1, 100); # Peter Bala, Jul 02 2016

Mathematica

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 *)
RealDigits[3 - HypergeometricPFQ[{1, 1}, {3, 3, 3}, 1]/8, 10, 100][[1]] (* Vaclav Kotesovec, Jul 03 2016 *)

Formula

From Peter Bala, Jul 01 2016: (Start)

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!.

Alternative series representations:

c = 3 - Sum_{n >= 2} 1/(n*(n - 1)*n!^2);

c = 1 + Sum_{n >= 1} (n + 2)/(n!*(n + 1)!);

c = 5/3 + 1/3*Sum_{n >= 2} (n + 1)*(n + 2)/n!^2;

c = A070910 + A096789 - 1.

Continued fraction: c = 3 - 1/(8 - 4/(14 - 9/(32 - ... - (n-1)^2/(n^2 + n + 2 - ...)))). See comments in A141827. (End)