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Decimal expansion of a constant related to A251702.
+20
1
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
FORMULA
Equals lim_{n->infinity} A251702(n)^(1/3^n).
EXAMPLE
1.1546796279605837888382808629570944052320556413000593142798453022385779...
MATHEMATICA
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 *)
PROG
(Magma) 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
a(1) = 4, a(n) = a(n-1)*(a(n-1) - 1)/2.
+10
5
4, 6, 15, 105, 5460, 14903070, 111050740260915, 6166133456248548335768188155, 19010600900133834176644234577571914951562754277857057935
COMMENTS
The next two terms, a(10) and a(11), have 111 and 221 digits. - Harvey P. Dale, Jun 10 2011 Interpretation through plane geometry: Start with the a(n)-sided regular polygon, connect all the vertices to create a figure having a(n+1)= A000217(a(n)-1) edges. Repeat to obtain this sequence. - T. D. Noe, May 13 2016 Let y(1) = x1+x2+x3+x4, and define y(n+1) as the plethysm e2[y(n)], where e2 represents the second elementary symmetric function. Then a(n) is y(n) evaluated at x1=x2=x3=x4=1. - Per W. Alexandersson, Jun 06 2020 Each term is the number of coordinate planes in Euclidean space of the dimensionality of the previous term. - Shel Kaphan, Feb 06 2023
FORMULA
Limit_{n->oo} a(n)^(1/2^n) = 1.280497808541657066685323460209089278782... (see A251794). - Vaclav Kotesovec, Feb 15 2014, updated Dec 09 2014
EXAMPLE
a(2) = a(1)*(a(1)-1)/2 = 4*3/2 = 6.
MATHEMATICA
RecurrenceTable[{a[1]==4, a[n]==(a[n-1](a[n-1]-1))/2}, a[n], {n, 10}] (* Harvey P. Dale, Jun 10 2011 *)
PROG
(PARI) v=vector(10, i, (i==1)*4); for(i=2, 10, v[i]=v[i-1]*(v[i-1]-1)/2); v
(PARI) a086714(upto)={my(a217(n)=n*(n+1)/2, a=4); for(k=1, upto, print1(a, ", "); a=a217(a-1))};
a(0)=0, a(1)=1, a(2)=5 and for n>2: a(n) = a(n-1)*(a(n-1) + 1)*(2*a(n-1) + 1)/6.
+10
3
0, 1, 5, 55, 56980, 61667666167030, 78172010815921069181209893626754427513955
FORMULA
a(n) = A000330(if n<=2 then n else a(n)). a(n) ~ sqrt(3) * c^(3^n), where c = 1.13701835838072682283814038264701129587627956851233106833915157... . - Vaclav Kotesovec, Dec 17 2014
MATHEMATICA
Flatten[{0, 1, RecurrenceTable[{a[2] == 5, a[n] == a[n-1]*(a[n-1] + 1)*(2*a[n-1] + 1)/6}, a[n], {n, 8}]}] (* Vaclav Kotesovec, Dec 17 2014 *) Join[{0, 1}, NestList[(#(#+1)(2#+1))/6&, 5, 5]] (* Harvey P. Dale, Sep 13 2022 *)
PROG
(Magma) [0, 1] cat [n le 1 select 5 else Self(n-1)*(Self(n-1)+1)*(2*Self(n-1)+1)/6: n in [1..8]]; // G. C. Greubel, Feb 06 2024 (SageMath)
if n<3: return (0, 1, 5)[n]
else: return a(n-1)*(a(n-1)+1)*(2*a(n-1)+1)/6
Decimal expansion of a constant related to A086714.
+10
1
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
FORMULA
Equals limit n->infinity A086714(n)^(1/2^n).
EXAMPLE
1.2804978085416570666853234602090892787820401965229548913582461026432...
MATHEMATICA
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 *)
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