A086714 -id:A086714 - cp's OEIS frontend

A251794 Decimal expansion of a constant related to A086714.

1, 2, 8, 0, 4, 9, 7, 8, 0, 8, 5, 4, 1, 6, 5, 7, 0, 6, 6, 6, 8, 5, 3, 2, 3, 4, 6, 0, 2, 0, 9, 0, 8, 9, 2, 7, 8, 7, 8, 2, 0, 4, 0, 1, 9, 6, 5, 2, 2, 9, 5, 4, 8, 9, 1, 3, 5, 8, 2, 4, 6, 1, 0, 2, 6, 4, 3, 2, 0, 1, 8, 5, 7, 4, 7, 0, 1, 9, 2, 0, 5, 3, 7, 9, 3, 7, 2, 1, 1, 4, 2, 6, 9, 9, 4, 5, 6, 6, 5, 3, 4, 0, 2, 6, 8

Offset: 1

Vaclav Kotesovec, Dec 09 2014

			1.2804978085416570666853234602090892787820401965229548913582461026432...

Cf. A086714, A251702.

exact = 32; terms = 200; b = ConstantArray[0, terms]; b[[1]] = N[Log[4], 100]; Do[b[[n]] = b[[n - 1]] + If[n > exact, b[[n - 1]], Log[Exp[b[[n - 1]]] - 1]] - Log[2], {n, 2, terms}]; Do[Print[Exp[b[[n]]/2^n]], {n, 1, Length[b]}] (* after Jon E. Schoenfield *)

Equals limit n->infinity A086714(n)^(1/2^n).

A013589 a(n+1) = a(n)*(a(n)+1)/2.

Original entry on oeis.org

4, 10, 55, 1540, 1186570, 703974775735, 247790242435923759782980, 30700002123226936025189367747945843590228731690, 471245065183069190019624521318729501785547966685566656416776916445985874756815447988119493895

Offset: 0

Carl Bellinger (carlb(AT)ctron.com)

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..11
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000217, A007501, A050542, A050536, A050548, A050909.

Cf. A086714.

a013589 n = a013589_list !! n
a013589_list = iterate a000217 4  -- Reinhard Zumkeller, Apr 10 2014

f := proc(n) options remember; if n = 0 then 4 else 1/2*f(n-1)*(f(n-1)+1) fi end

a=4;lst={a};Do[b=(a+1)*a/2;AppendTo[lst,b];a=b,{n,1,9}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 16 2010 *)
NestList[# (#+1)/2&,4,10] (* Harvey P. Dale, Sep 20 2012 *)

a(n) ~ 2 * c^(2^n), where c = 2.29524458756644851072790131811443639562180382439442714497... . - Vaclav Kotesovec, Dec 17 2014

One more term from Vladimir Joseph Stephan Orlovsky, Mar 16 2010

A251702 a(1)=5, a(n) = a(n-1)(a(n-1)-1)(a(n-1)-2)/6.

Original entry on oeis.org

5, 10, 120, 280840, 3691654113991480, 8385167839605753859676710992399730619003333960

Offset: 1

Frank M Jackson, Dec 07 2014

nonn

In general, sequence a(n) = binomial(a(n-1),k) is asymptotic to (k!)^(1/(k-1)) * c^(k^n), where the constant c is dependent on k and a(1). For big a(1), c asymptotically approaches (a(1)/(k!)^(1/(k-1)))^(1/k). - Vaclav Kotesovec, Dec 09 2014

			a(2) = a(1)*(a(1)-1)*(a(1)-2)/6 = 5*4*3/6 = 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..8

Cf. A086714, A251792, A129440, A086714.

RecurrenceTable[{a[1] == 5, a[n] == a[n - 1](a[n - 1] - 1)(a[n - 1] - 2)/6}, a[n], {n, 10}]

Limit_{n->oo} a(n)^(1/3^n) = 1.1546796279605837888382808629570944052320556413... (see A251792).
a(n) ~ sqrt(6) * A251792^(3^n). - Vaclav Kotesovec, Dec 09 2014
a(n) = binomial(a(n-1),3) for n >= 1. - Shel Kaphan, Feb 06 2023

A251792 Decimal expansion of a constant related to A251702.

Original entry on oeis.org

1, 1, 5, 4, 6, 7, 9, 6, 2, 7, 9, 6, 0, 5, 8, 3, 7, 8, 8, 8, 3, 8, 2, 8, 0, 8, 6, 2, 9, 5, 7, 0, 9, 4, 4, 0, 5, 2, 3, 2, 0, 5, 5, 6, 4, 1, 3, 0, 0, 0, 5, 9, 3, 1, 4, 2, 7, 9, 8, 4, 5, 3, 0, 2, 2, 3, 8, 5, 7, 7, 9, 1, 0, 4, 1, 1, 6, 4, 1, 9, 2, 5, 7, 9, 7, 3, 6, 8, 9, 1, 4, 9, 5, 4, 6, 1, 2, 6, 9, 6, 2, 7, 5, 3, 3

Offset: 1

Vaclav Kotesovec and Jon E. Schoenfield, Dec 09 2014

			1.1546796279605837888382808629570944052320556413000593142798453022385779...

Cf. A086714, A251702.

nMax:=160; nExactMax:=20; DP:=100; R:=RealField(DP); SetDefaultRealField(R); logA:=[Log(5.0)]; for n in [2..nMax] do logAprev:=logA[n-1]; if n le nExactMax then Aprev:=Exp(logAprev); logA[n]:=logAprev + Log(Aprev-1) + Log(Aprev-2) - Log(6); else logA[n]:=3*logAprev - Log(6); end if; t:=Exp((1/3^n)*logA[n]); n, ChangePrecision(t,72); end for; // Jon E. Schoenfield, Dec 09 2014

exact = 20; terms = 200; b = ConstantArray[0, terms]; b[[1]] = N[Log[5], 100]; Do[b[[n]] = b[[n - 1]] + If[n > exact, b[[n - 1]], Log[Exp[b[[n - 1]]] - 1]] + If[n > exact, b[[n - 1]], Log[Exp[b[[n - 1]]] - 2]] - Log[6], {n, 2, terms}]; Do[Print[Exp[b[[n]]/3^n]], {n, 1, Length[b]}] (* after Jon E. Schoenfield *)

Equals lim_{n->infinity} A251702(n)^(1/3^n).

cp's OEIS Frontend

A251794 Decimal expansion of a constant related to A086714.

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A013589 a(n+1) = a(n)*(a(n)+1)/2.

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A251702 a(1)=5, a(n) = a(n-1)*(a(n-1)-1)*(a(n-1)-2)/6.

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A251792 Decimal expansion of a constant related to A251702.

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A251702 a(1)=5, a(n) = a(n-1)(a(n-1)-1)(a(n-1)-2)/6.