A171898 Forward van Eck transform of A181391.

1, 2, 6, 2, 2, 5, 1, 6, 42, 5, 2, 4, 5, 9, 14, 3, 9, 3, 15, 2, 4, 6, 17, 3, 6, 32, 56, 5, 3, 131, 5, 11, 5, 3, 20, 6, 2, 8, 15, 31, 170, 3, 31, 18, 3, 3, 33, 5, 1, 11, 46, 56, 4, 37, 152, 307, 3, 7, 92, 4, 7, 62, 52, 3, 42, 3, 6, 2, 19, 6, 8, 3, 9, 3, 650, 2, 23, 8, 223, 7, 206, 3, 21, 25, 5, 8

Offset

1,2

Comments

Given a sequence a, the forward van Eck transform b is defined as follows: If a(n) also appears again in a later position, let a(m) be the next occurrence, and set b(n)=m-n; otherwise b(n)=0.

This is a permutation of the positive terms in A181391, where each term m > 0 from that sequence is shifted backwards m+1 positions. - Jan Ritsema van Eck, Aug 16 2019

The backwards van Eck transform searches backwards for a repeated value: if a(n) also has appeared in earlier positions, a(m)=a(n) with m<n, then b(n) is the minimum n-m. - R. J. Mathar, Jun 24 2021

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

From Jan Ritsema van Eck, Aug 16 2019: (Start)

A181391(i+a(i)+1) = a(i) for any i, a(i)>0.

Conversely, a(j-A(181391(j)-1) = A181391(j) for any j, A181391(j)>0. (End)

Maple

ECKf:=proc(a) local b, i, m, n;
if whattype(a) <> list then RETURN([]); fi:
b:=[];
for n from 1 to nops(a)-1 do
# does a(n) appear again?
m:=0;
for i from n+1 to nops(a) do
if (a[i]=a[n]) then m:=i-n; break; fi
od:
b:=[op(b), m];
od:
b:=[op(b), 0];
RETURN(b);
end:

Mathematica

terms = 100;
m = 14 terms; (* Increase m until no zero appears in the output *)
ClearAll[b, last]; b[_] = 0; last[_] = -1; last[0] = 2; nxt = 1;
Do[hist = last[nxt]; b[n] = nxt; last[nxt] = n; nxt = 0; If[hist > 0, nxt = n - hist], {n, 3, m}];
A181391 = Array[b, m];
ECKf[a_List] := Module[{b = {}, i, m, n}, For[n = 1, n <= Length[a]-1, n++, m = 0; For[i = n+1, i <= Length[a], i++, If[a[[i]] == a[[n]], m = i-n; Break[]]]; b = Append[b, m]]; b = Append[b, 0]; Return[b]];
ECKf[A181391][[;; terms]] (* Jean-François Alcover, Oct 30 2020, after Maple *)

Crossrefs

Cf. A181391 (van Eck's sequence), A171899, A171942.

Sequence in context: A062539 A064136 A347238 * A330541 A320575 A110218

Adjacent sequences: A171895 A171896 A171897 * A171899 A171900 A171901

Keyword

nonn

Author

N. J. A. Sloane, Oct 22 2010

Status

approved