A181391 -id:A181391 - cp's OEIS frontend

A171865 Positions of 0's in A181391.

1, 2, 4, 6, 11, 13, 18, 21, 25, 31, 36, 42, 45, 48, 53, 57, 60, 64, 67, 73, 82, 85, 90, 93, 99, 102, 106, 111, 117, 122, 126, 130, 135, 139, 143, 149, 153, 157, 163, 169, 174, 180, 184, 188, 197, 200, 203, 209, 213, 216, 226, 229, 233, 237, 243, 247, 252, 255, 258, 264, 270

Offset: 1

N. J. A. Sloane, Oct 17 2010

nonn

A181391(a(n)) = 0; the sequence is infinite, proof by Jan Ritsema van Eck in comments in A181391.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A181391, A171868.

import Data.List (elemIndices)
a171865 n = a171865_list !! (n-1)
a171865_list = map (+ 1) $ elemIndices 0 a181391_list
-- Reinhard Zumkeller, Oct 31 2011

A181391_list, A171865_list = [0, 0], [1,2]
for n in range(1, 10**4):
    for m in range(n-1, -1, -1):
        if A181391_list[m] == A181391_list[n]:
            A181391_list.append(n-m)
            break
    else:
        A181391_list.append(0)
        A171865_list.append(n+2) # Chai Wah Wu, Jan 02 2015

A171911 Van Eck's sequence (cf. A181391), but starting with a(1) = 1.

Original entry on oeis.org

1, 0, 0, 1, 3, 0, 3, 2, 0, 3, 3, 1, 8, 0, 5, 0, 2, 9, 0, 3, 9, 3, 2, 6, 0, 6, 2, 4, 0, 4, 2, 4, 2, 2, 1, 23, 0, 8, 25, 0, 3, 19, 0, 3, 3, 1, 11, 0, 5, 34, 0, 3, 7, 0, 3, 3, 1, 11, 11, 1, 3, 5, 13, 0, 10, 0, 2, 33, 0, 3, 9, 50, 0, 4, 42, 0, 3, 7, 25, 40, 0, 5, 20, 0, 3, 8, 48, 0, 4, 15

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

After the initial value, the sequence is extended by a(n+1) = min { k > 0: a(n-k) = a(n) } or 0 if no such k exists, i.e., if a(n) did not appear earlier.

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Cf. A181391, A171912, A171913, A171914, A171915, A171916, A171917, A171918 (same but starting with 0, 2, ..., 8).

t = {1};
Do[
d = Quiet[Check[Position[t, Last[t]][[-2]][[1]], 0], Part::partw];
If[d == 0, x = 0, x = Length[t] - d];
AppendTo[t, x], 100]
t (* Horst H. Manninger, Aug 03 2020 *)

A171911_vec(N,a=1,i=Map())={vector(N,n,a=if(n>1,iferr(n-mapget(i,a),E,0)+mapput(i,a,n),a))} \\ M. F. Hasler, Jun 11 2019

A171911_list, l = [1, 0], 0
for n in range(1, 10**4):
    for m in range(n-1, -1, -1):
        if A171911_list[m] == l:
            l = n-m
            break
    else:   # break did not occur
        l = 0
    A171911_list.append(l) # Chai Wah Wu, Jan 02 2015

Edited by M. F. Hasler, Jun 11 2019

A171898 Forward van Eck transform of A181391.

Original entry on oeis.org

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

N. J. A. Sloane, Oct 22 2010

nonn

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 mR. J. Mathar, Jun 24 2021

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

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

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:

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 *)

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-A181391(j)-1) = A181391(j) for any j, A181391(j)>0. (End)

A171918 Van Eck sequence (cf. A181391) starting with a(1) = 8.

Original entry on oeis.org

8, 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3, 0, 3, 2, 9, 0, 4, 9, 3, 6, 14, 0, 6, 3, 5, 15, 0, 5, 3, 5, 2, 17, 0, 6, 11, 0, 3, 8, 47, 0, 4, 24, 0, 3, 7, 0, 3, 3, 1, 49, 0, 5, 22, 0, 3, 7, 11, 22, 5, 7, 4, 20, 0, 9, 46, 0, 3, 12, 0, 3, 3, 1, 23, 0, 5, 16, 0, 3, 7, 19, 0, 4

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

A van Eck sequence is defined recursively by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty. - M. F. Hasler, Jun 12 2019

Cf. A181391, A171911, ..., A171917 (same but starting with 0, 1, ..., 7).

A171918_vec(N,a=8,i=Map())={vector(N,n,a=if(n>1,iferr(n-mapget(i,a),E,0)+mapput(i,a,n),a))} \\ M. F. Hasler, Jun 11 2019

from itertools import count, islice
def A171918gen():
    yield 8
    b, bdict = 8, {8:(1,)}
    for n in count(2):
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 0
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
        yield b
A171918_list = list(islice(A171918gen(),30)) # Chai Wah Wu, Dec 21 2021

a(n+1) = A181391(n) up to the first occurrence of a(1) = 8 in A181391. - M. F. Hasler, Jun 15 2019

Name edited and cross-references added by M. F. Hasler, Jun 12 2019

A171862 Index of first occurrence of n in A181391, or 0 if n never appears.

Original entry on oeis.org

1, 3, 5, 20, 17, 12, 10, 66, 47, 24, 148, 44, 173, 121, 30, 35, 138, 41, 63, 89, 56, 105, 324, 101, 215, 110, 257, 142, 320, 225, 471, 72, 59, 81, 183, 539, 134, 92, 263, 168, 129, 548, 52, 199, 179, 228, 98, 297, 621, 670, 357, 156, 116, 1486, 475

Offset: 0

N. J. A. Sloane, Oct 17 2010

nonn

N. J. A. Sloane and Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (terms 1..1594 from N. J. A. Sloane)

Cf. A181391, A171863, A171864.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a171862 n = 1 + fromJust (elemIndex n a181391_list)
-- Reinhard Zumkeller, Oct 31 2011

nmax = 100;
seq[m_] := seq[m] = Module[{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}];
 A171862 = Array[b, m];
 a[n_] := FirstPosition[A171862, n];
 Table[a[n], {n, 0, nmax}] /. Missing["NotFound"] -> 0 // Flatten
];
seq[m = nmax];
seq[m = 2 m];
While[Print["m = ", m]; seq[m] != seq[m/2], m = 2 m;];
seq[m] (* Jean-François Alcover, Aug 02 2018, borrowing some Maple code from N. J. A. Sloane *)

Offset corrected by Reinhard Zumkeller, Oct 31 2011

A171863 Numbers that take a record number of steps to appear in A181391.

Original entry on oeis.org

0, 1, 2, 3, 7, 10, 12, 22, 30, 35, 41, 48, 49, 53, 74, 77, 81, 127, 161, 205, 251, 349, 379, 560, 604, 725, 726, 820, 1037, 1095, 1112, 1145, 1239, 1594, 1673, 1908, 2895, 4391, 4551, 5342, 5404, 5410, 8756, 11054, 12081, 13923, 16635, 17598, 18088, 31960, 38099

Offset: 1

N. J. A. Sloane, Oct 17 2010

nonn

Records in A171862 (positions).

Benjamin Chaffin, Table of n, a(n) for n = 1..215

Cf. A181391, A171862, A171864.

a(34) onwards from Benjamin Chaffin, Sep 11 2019

A171951 Positions of 1's in A181391.

Original entry on oeis.org

3, 9, 51, 97, 133, 146, 160, 172, 191, 206, 240, 261, 273, 313, 342, 362, 396, 415, 440, 470, 483, 502, 516, 536, 565, 593, 615, 640, 660, 677, 693, 724, 750, 784, 796, 807, 832, 853, 902, 949, 963, 985, 1009, 1038, 1052, 1073, 1095, 1113, 1177

Offset: 1

N. J. A. Sloane, Oct 29 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..2762

Cf. A181391, A171865, A171951-A171956.

A171956 Positions of 6's in A181391.

Original entry on oeis.org

10, 15, 29, 32, 43, 74, 77, 100, 118, 150, 164, 170, 171, 181, 193, 210, 221, 244, 265, 271, 272, 305, 311, 312, 317, 332, 346, 419, 444, 453, 457, 459, 463, 465, 467, 485, 506, 547, 597, 607, 613, 614, 619, 644, 653, 664, 697, 707, 717, 719

Offset: 1

N. J. A. Sloane, Oct 29 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..5453

Cf. A181391, A171865, A171951-A171956.

A171864 Positions in A181391 where the terms listed in A171863 appear.

Original entry on oeis.org

1, 3, 5, 20, 66, 148, 173, 324, 471, 539, 548, 621, 670, 1486, 2109, 2693, 3735, 7574, 7619, 7886, 13191, 15116, 24862, 27717, 29981, 31679, 45110, 59597, 60876, 65607, 78847, 82959, 96998, 102840, 199861, 293758, 368318, 371583, 382472, 440644, 523875, 841338

Offset: 1

N. J. A. Sloane, Oct 17 2010

nonn

Records in A171862 (values).

Benjamin Chaffin, Table of n, a(n) for n = 1..215

Cf. A181391, A171862, A171863.

a(34) onwards from Benjamin Chaffin, Sep 11 2019

A171866 Records in A181391 (values).

Original entry on oeis.org

0, 1, 2, 6, 9, 14, 15, 17, 42, 56, 62, 92, 131, 152, 170, 206, 223, 307, 319, 331, 367, 368, 415, 508, 650, 705, 832, 1024, 1086, 1302, 1411, 1582, 1605, 1760, 1800, 1833, 2021, 2028, 2238, 2287, 2368, 2944, 3151, 3323, 3420, 3451, 3815, 4029, 4164, 4175, 4466

Offset: 1

N. J. A. Sloane, Oct 17 2010

nonn

Benjamin Chaffin, Table of n, a(n) for n = 1..13808 (first 345 terms from N. J. A. Sloane)

Cf. A181391 (Van Eck's sequence).

cp's OEIS Frontend

A171865 Positions of 0's in A181391.

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A171911 Van Eck's sequence (cf. A181391), but starting with a(1) = 1.

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A171898 Forward van Eck transform of A181391.

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A171918 Van Eck sequence (cf. A181391) starting with a(1) = 8.

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A171862 Index of first occurrence of n in A181391, or 0 if n never appears.

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A171863 Numbers that take a record number of steps to appear in A181391.

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A171951 Positions of 1's in A181391.

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A171956 Positions of 6's in A181391.

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A171864 Positions in A181391 where the terms listed in A171863 appear.

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