A141827 -id:A141827 - cp's OEIS frontend

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

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

			2.8702221569733963308194588658111996012403192622809957012...

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

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.

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

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

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 *)

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)

A141828 a(n) = (n^4*a(n-1)-1)/(n-1) for n >= 2, with a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 79, 3199, 272981, 42653281, 11055730435, 4424134795739, 2588750874763849, 2123099311165701661, 2358999234628557401111, 3453810779419670890966615, 6510747302004208690462157149, 15496121141045183700690805861049

Offset: 0

Peter Bala, Jul 09 2008

For related recurrences of the form a(n) = (n^k*a(n-1)-1)/(n-1) see A001339, A007808 (both k = 2) and A141827 (k = 3). a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.

Harvey P. Dale, Table of n, a(n) for n = 0..181

Cf. A001339, A007808, A141827.

a := n -> n!^3*add((n-k+1)*(k^2+k+1)/k!^3, k = 0..n): seq(a(n), n = 0..16);

nxt[{n_,a_}]:={n+1,((n+1)^4*a-1)/n}; Join[{1},NestList[nxt,{1,5},15][[All,2]]] (* Harvey P. Dale, Mar 12 2017 *)

Sum_{n>=0} a(n)*x^n/n!^3 = (1/(1-x)^2)*Sum_{n>=0} (n^2+n+1)*x^n/n!^3.
a(n) = n!^3*Sum_{k=0..n} (n-k+1)*(k^2+k+1)/k!^3.
a(n) = n*n!^3*(5 - Sum_{k=2..n} 1/(k!^3*k*(k-1))) for n > 0. [corrected by Jason Yuen, Jan 31 2025]
Congruence property: a(n) == (1+n+n^2+n^3) (mod n^4).
The recurrence a(n) = (n^3+n^2+n+2)*a(n-1) - (n-1)^3*a(n-2), n >= 2, shows that a(n) is always a positive integer. The sequence b(n) := n*n!^3 also satisfies the same recurrence with b(0) = 0, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/(n*n!^3) = 5 - 1^3/(16 - 2^3/(41 - 3^3/(86 -...-(n-1)^3/(n^3+n^2+n+2)))), for n >= 1. a(n)*b(n+1) - b(n)*a(n+1) = n!^3.
Limit_{n->oo} a(n)/(n*n!^3) = Sum_{n>=0} (n^2+n+1)/n!^3 = 4.9367223378... .
Limit_{n->oo} a(n)/(n*n!^3) = 1 + Sum_{n>=0} 1/(Product_{k=0..n} A008620(k)).

cp's OEIS Frontend

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

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