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A182321 Number of iterations of A025581(n) required to reach 0. 1
0, 1, 2, 1, 3, 2, 1, 2, 3, 2, 1, 4, 2, 3, 2, 1, 3, 4, 2, 3, 2, 1, 2, 3, 4, 2, 3, 2, 1, 3, 2, 3, 4, 2, 3, 2, 1, 4, 3, 2, 3, 4, 2, 3, 2, 1, 3, 4, 3, 2, 3, 4, 2, 3, 2, 1, 2, 3, 4, 3, 2, 3, 4, 2, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Following the notation in the link, for n >= 0, let n = (0 + 1 + 2 + ... + f(n)) - g(n) be the representation of n with f(n) and g(n) minimal such that 0 <= g(n) <= f(n). Then f(n) = A002024(n) = round(sqrt(2n)), and g(n) = A025581(n) = f(n)*(f(n)+1)/2 - n.
With this notation, a(n) is the number of iterations of g(n) needed to reach 0.
The sequence a(n) is essentially the function phi(n) of the link.
The sequence a(n) has a high degree of fractal-like symmetry. Consider, for instance, the sequence in the triangular array (read left to right then top to bottom, with the term for a(0) on top):
0
1
2 1
3 2 1
2 3 2 1
Then the rows of this triangle (read from right to left) are simply 1+a(n).
a(n) is related to the recurrence between A186053 and A182298.
For n >= 1, a(n) is the number of terms in the minimal alternating triangular-number representation of n+1, defined at A255974. - Clark Kimberling, Apr 10 2015
LINKS
Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, arXiv:1202.1331v1 [math.CO]
FORMULA
The Devlin link shows a(n) < log_2(log_2(n/2)) + 2.
EXAMPLE
g(8) = 2, g(2) = 1, g(1) = 0. Therefore a(8) = 3.
MAPLE
# With this code, the n-th term of the sequence is given by a call to a(n)
f:=n->round(sqrt(2*n)): g:=n->f(n)*(f(n)+1)/2-n:
a:=proc(n) option remember:
if n < 1 then return 0: fi: return 1 + a(g(n)):
end proc:
MATHEMATICA
(* This program computes the sequence as the number of terms in the minimal alternative triangular-number representation of n+1. *)
b[n_] := n (n + 1)/2; bb = Table[b[n], {n, 0, 1000}];
s[n_] := Table[b[n], {k, 1, n}];
h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[100]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
Join[{0}, Rest[Table[Length[r[n]], {n, 0, 100}]]] (* A182321 for n >= 1 *)
(* Clark Kimberling, Apr 10 2015 *)
CROSSREFS
Sequence in context: A023510 A005678 A296976 * A285731 A114905 A200651
KEYWORD
nonn
AUTHOR
Patrick Devlin, Apr 24 2012
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)