A216403
Number of distinct values taken by 10th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
Original entry on oeis.org
1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4763, 12452, 32711, 86239
Offset: 1
a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 10th derivative at x=1: (x^(x^(x^x))) -> 37616880; ((x^x)^(x^x)), ((x^(x^x))^x) -> 42409440; (x^((x^x)^x)) -> 77899320; (((x^x)^x)^x) -> 66712680.
Cf.
A000081 (distinct functions),
A000108 (parenthesizations),
A000012 (first derivatives),
A028310 (2nd derivatives),
A199085 (3rd derivatives),
A199205 (4th derivatives),
A199296 (5th derivatives),
A199883 (6th derivatives),
A002845,
A003018,
A003019,
A145545,
A145546,
A145547,
A145548,
A145549,
A145550,
A082499,
A196244,
A198683,
A215703,
A215840. Column k=10 of
A216368.
-
# load programs from A215703, then:
a:= n-> nops({map(f-> 10!*coeff(series(subs(x=x+1, f),
x, 11), x, 10), T(n))[]}):
seq(a(n), n=1..11);
A003006
Number of n-level ladder expressions with A001622.
Original entry on oeis.org
1, 1, 2, 3, 7, 15, 35, 81, 195, 473, 1170, 2920, 7378, 18787, 48242, 124658, 324095, 846872, 2223352, 5861011, 15508423, 41173560, 109648734
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
ClearAll[phi, t, a]; t[1] = {0}; t[n_Integer] := t[n] = DeleteDuplicates[Flatten[Table[Outer[phi^#1 + #2 &, t[k], t[n - k]], {k, n - 1}]] /. phi^k_Integer :> Fibonacci[k] phi + Fibonacci[k - 1]]; a[n_Integer] := a[n] = Length[t[n]]; Table[a[n], {n, 23}]
A297074
Number of ways of inserting parentheses in x^x^...^x (with n x's) whose result is an integer where x = sqrt(2).
Original entry on oeis.org
0, 0, 1, 1, 2, 5, 10, 23, 55
Offset: 1
With x = sqrt(2),
x = 1.414213... is not an integer, so a(1) = 0;
x^x = 1.632526... is not an integer, so a(2) = 0.
(x^x)^x = 2 is an integer, but x^(x^x) = 1.760839... is not, so a(3) = 1;
((x^x)^x)^x, (x^x)^(x^x), (x^(x^x))^x, and x^(x^(x^x)) are noninteger values, but x^((x^x)^x) = 2, so a(4) = 1;
the only ways of inserting parentheses in x^x^x^x^x that yield integer values are x^(x^((x^x)^x)) = 2 and (((x^x)^x)^x)^x = 4, so a(5) = 2.
-
With[{x = Sqrt@ 2}, Array[Count[#, ?IntegerQ] &@ Map[ToExpression@ StringReplace[ToString@ #, {"{" -> "(", "}" -> ")", "," -> "^"}] &, Groupings[#, 2] /. _Integer -> x] &, 9]] (* _Michael De Vlieger, Dec 24 2017 *)
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