A248939 -id:A248939 - cp's OEIS frontend

A248973 Table read by rows: row n contains the partial sums of the wrecker ball sequences starting with n, cf. A248939.

0, 1, 1, 2, 3, 2, -2, -2, 3, 5, 5, 4, 7, 8, 6, 8, 5, -4, -20, -28, -45, -52, -70, -76, -69, -48, -42, -52, -79, -124, -150, -196, -221, -268, -292, -292, 5, 9, 11, 10, 13, 11, 3, -12, -19, -35, -41, -58, -63, -55, -33, -26, -35, -61, -105, -130, -175, -199, -245, -268, -267, -241, -241

Offset: 0

Reinhard Zumkeller, Oct 20 2014

A228474(n) + 1 = length of row n;
row n = partial sums of row n in A248939;
T(n,A228474(n)) = A248961(n).

Reinhard Zumkeller, Rows n = 0..16 of triangle, flattened
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A228474 (row lengths - 1), A248961 (right edge).

a248973 n k = a248973_tabf !! n !! k
a248973_row n = a248973_tabf !! n
a248973_tabf = map (scanl1 (+)) a248939_tabf

T(n,0) = n; T(n,k) = T(n,k-1) + A248939(n,k) for k=1..A228474(n).

A228474 Number of steps required to reach zero in the wrecker ball sequence starting with n: On the k-th step (k = 1, 2, 3, ...) move a distance of k in the direction of zero. If the result has occurred before, move a distance of k away from zero instead. Set a(n) = -1 if 0 is never reached.

Original entry on oeis.org

0, 1, 4, 2, 24, 26, 3, 1725, 12, 14, 4, 26, 123, 125, 15, 5, 119, 781802, 20, 22, 132896, 6, 51, 29, 31, 1220793, 23, 25, 7, 429, 8869123, 532009, 532007, 532009, 532011, 26, 8, 94, 213355, 213353, 248, 33, 31, 33, 1000, 9, 144, 110, 112, 82, 84, 210, 60, 34

Offset: 0

Gordon Hamilton, Aug 23 2013

This is a Recamán-like sequence (cf. A005132).
The n-th triangular number A000217(n) has a(A000217(n)) = n.
a(n) + 1 = length of row n in tables A248939 and A248973. - Reinhard Zumkeller, Oct 20 2014
Hans Havermann, running code from Hugo van der Sanden, has found that a(11281) is 3285983871526. - N. J. A. Sloane, Mar 22 2019
If a(n) != -1 then floor((a(n)-1)/2)+n is odd. - Robert Gerbicz, Mar 28 2019

			a(2) = 4 because 2 -> 1 -> -1 -> -4 -> 0.
See A248940 for the full 1725-term trajectory of 7. See A248941 for a bigger example, which shows the start of the 701802-term trajectory of 17. - _N. J. A. Sloane_, Mar 07 2019

Jon E. Schoenfield, Table of n, a(n) for n = 0..10000
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013. [The sequence is mentioned about 4.5 minutes in to the video. The video begins by discussing A005132. - _N. J. A. Sloane_, Apr 25 2019]
Hans Havermann, Table of n, a(n) for n = 0..36617
Hans Havermann, Log-plot of terms through n = 36617
Hans Havermann, Sharp peaks and high plateaus An overview of large knowns and unknowns up to index 10^6.
Index entries for sequences related to Recamán's sequence

Cf. A248939 (rows = the full sequences), A248961 (row sums), A248973 (partial sums per row), A248952 (min per row), A248953 (max per row).
Cf. also A248940 (row 7), A248941 (row 17), A248942 (row 20).
Cf. also A000217, A005132 (Recamán).

a228474 = subtract 1 . length . a248939_row  -- Reinhard Zumkeller, Oct 20 2014
(C++) #include 
  int A228474(long n) { int c=0, s; for(std::map seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { seen[n]=true; ++c; } return c; } // M. F. Hasler, Mar 18 2019

function A228474(n)
    k, position, beenhere = 0, n, [n]
    while position != 0
        k += 1
        step = position > 0 ? k : -k
        position += (position - step) in beenhere ? step : -step
        push!(beenhere, position)
    end
    return length(beenhere) - 1
end
println([A228474(n) for n in 0:16]) # Peter Luschny, Mar 24 2019

# To compute at most the first M steps of the trajectory of n:
f:=proc(n) local M,i,j,traj,h;
  M:=200; traj:=[n]; h:=n; s:=1;
  for i from 1 to M do j:=h-s*i;
    if member(j,traj,'p') then s:=-s; fi;
    h:=h-s*i; traj:=[op(traj),h];
    if h=0 then return("steps, trajectory =", i, traj); fi;
    s:=sign(h);
  od;
  lprint("trajectory so far = ", traj); error("Need to increase M");
end;  # N. J. A. Sloane, Mar 07 2019

{0}~Join~Array[-1 + Length@ NestWhile[Append[#1, If[FreeQ[#1, #3], #3, Sign[#1[[-1]] ] (Abs[#1[[-1]] ] + #2)]] & @@ {#1, #2, Sign[#1[[-1]] ] (Abs[#1[[-1]] ] - #2)} & @@ {#, Length@ #} &, {#}, Last@ # != 0 &] &, 16] (* Michael De Vlieger, Mar 27 2019 *)

a(n)={my(M=Map(),k=0); while(n, k++; mapput(M,n,1); my(t=if(n>0, -k, +k)); n+=if(mapisdefined(M,n+t),-t,t)); k} \\ Charles R Greathouse IV, Aug 18 2014, revised Andrew Howroyd, Feb 28 2018 [Warning: requires latest PARI. - N. J. A. Sloane, Mar 09 2019]

More terms from Jon E. Schoenfield, Jan 10 2014
Escape clause in definition added by N. J. A. Sloane, Mar 07 2019
Edited by M. F. Hasler, Mar 18 2019

A248952 Smallest term in wrecker ball sequence starting with n.

Original entry on oeis.org

0, 0, -4, 0, -47, -46, 0, -6362, -23, -22, 0, -32, -471, -470, -29, 0, -218, -4843985, -39, -38, -657367, 0, -101, -57, -56, -7609937, -45, -44, 0, -736, -56168428, -3113136, -3113135, -3113134, -3113133, -51, 0, -190, -1213998, -1213997, -495, -62, -61, -60

Offset: 0

Reinhard Zumkeller, Oct 18 2014

sign

Starting at n, a(n) is the minimum value reached according to the following rules. On the k-th step (k = 1, 2, 3, ...) move a distance of k in the direction of zero. If the number landed on has been landed on before, move a distance of k away from zero instead. See A228474 and A248939. - David Nacin, Mar 15 2019

It is currently unproved whether all orbits are finite, and therefore unclear whether all a(n) are well defined. In particular, the orbit of n = 11281 is of unknown length, but is certainly greater than 32*10^9. - M. F. Hasler, Mar 18 2019

			a(0) = min{0} = 0;
a(1) = min{1,0} = 0;
a(2) = min{2,1,-1,-4,0} = -4;
a(3) = min{3,2,0} = 0;
a(4) = min{4,3,1,-2,2,-3,-9,-16,-8,-17,-7,-18,-6,7,21,6,-10,-27,...} = -47;
a(5) = min{5,4,2,-1,3,-2,-8,-15,-7,-16,-6,-17,-5,8,22,7,-9,-26,...} = -46;
a(6) = min{6,5,3,0} = 0;
a(7) = min{7,6,4,1,-3,2,-4,3,-5,-14,-24,-13,-1,12,-2,13,29,46,...} = -6362;
a(8) = min{8,7,5,2,-2,3,-3,4,-4,-13,-23,-12,0} = -23;
a(9) = min{9,8,6,3,-1,4,-2,5,-3,-12,-22,-11,1,14,0} = -22.

M. F. Hasler, Table of n, a(n) for n = 0..5000 (first 1000 terms from Reinhard Zumkeller), Mar 19 2019
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A228474 (main entry for wrecker ball sequences).

Cf. A248939, A248953, A000217, A014132.

import Data.IntSet (singleton, member, insert, findMin, findMax)
a248952 n = a248952_list !! n
(a248952_list, a248953_list) = unzip $
   map (\x -> minmax 1 x $ singleton x) [0..] where
   minmax _ 0 s = (findMin s, findMax s)
   minmax k x s = minmax (k + 1) y (insert y s) where
                         y = x + (if (x - j) `member` s then j else -j)
                         j = k * signum x

#This and sequences A324660-A324692 generated by manipulating this trip function
#spots - positions in order with possible repetition
#flee - positions from which we move away from zero with possible repetition
#stuck - positions from which we move to a spot already visited with possible repetition
def trip(n):
    stucklist = list()
    spotsvisited = [n]
    leavingspots = list()
    turn = 0
    forbidden = {n}
    while n != 0:
        turn += 1
        sign = n // abs(n)
        st = sign * turn
        if n - st not in forbidden:
            n = n - st
        else:
            leavingspots.append(n)
            if n + st in forbidden:
                stucklist.append(n)
            n = n + st
        spotsvisited.append(n)
        forbidden.add(n)
    return {'stuck':stucklist, 'spots':spotsvisited,
                'turns':turn, 'flee':leavingspots}
#Actual sequence
def a(n):
    d = trip(n)
    return min(d['spots'])
# David Nacin, Mar 15 2019

def A248952(n):
    return min(A248939_row(n)); # M. F. Hasler, Mar 18 2019
(C++) #include
long A248952(long n) { long c=0, s, m=0; for(std::map seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { if(nM. F. Hasler, Mar 18 2019

a(n) = smallest term in row n of triangle A248939;

a(A000217(n)) = 0; a(A014132(n)) < 0.

Edited by M. F. Hasler, Mar 18 2019

A248953 Largest term in wrecker ball sequence starting with n.

Original entry on oeis.org

0, 1, 2, 3, 21, 26, 6, 6843, 8, 14, 10, 72, 365, 366, 14, 15, 352, 4674389, 18, 22, 891114, 21, 102, 23, 31, 7856204, 26, 27, 28, 1700, 61960674, 3702823, 3702824, 3702825, 3702826, 35, 36, 370, 1047903, 1047904, 596, 41, 42, 43, 2976, 45, 341, 260, 261, 123

Offset: 0

Reinhard Zumkeller, Oct 18 2014

nonn

Starting at n, a(n) is the maximum reached according to the following rules: Unless 0 has been reached, on the k-th step (k = 1, 2, 3, ...) move a distance of k in the direction of zero. If the result has already occurred before, move a distance of k away from zero instead. See A228474 and A248939. - David Nacin, Mar 15 2019

It is currently unproved that all orbits are finite, and therefore unclear whether all a(n) are well defined. In particular, the orbit of n = 11281 is of unknown length > 32*10^9 steps. - M. F. Hasler, Mar 18 2019

			a(0) = max {0} = 0;
a(1) = max {1,0} = 1;
a(2) = max {2,1,-1,-4,0} = 2;
a(3) = max {3,2,0} = 3;
a(4) = max {4,3,1,-2,2,-3,-9,-16,-8,-17,-7,-18,-6,7,21,6,-10,-27,...} = 21;
a(5) = max {5,4,2,-1,3,-2,-8,-15,-7,-16,-6,-17,-5,8,22,7,-9,-26,...} = 26;
a(6) = max {6,5,3,0} = 6;
a(7) = max {7,6,4,1,-3,2,-4,3,-5,-14,-24,-13,-1,12,-2,13,29,46,...} = 6843;
a(8) = max {8,7,5,2,-2,3,-3,4,-4,-13,-23,-12,0} = 8;
a(9) = max {9,8,6,3,-1,4,-2,5,-3,-12,-22,-11,1,14,0} = 14.

M. F. Hasler, Table of n, a(n) for n = 0..5000 (terms up to n = 1000 from Reinhard Zumkeller), Mar 18 2019
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A248939 (entire orbit of n), A248952 (minimum of the orbit), A000217, A228474 (length of orbits - 1: main entry for wrecker ball sequences).

a248953 n = a248953_list !! n
-- See A248952 for definition of a248953_list.

#This and sequences A324660-A324692 generated by manipulating this trip function
#spots - positions in order with possible repetition
#flee - positions from which we move away from zero with possible repetition
#stuck - positions from which we move to a spot already visited with possible repetition
def trip(n):
    stucklist = list()
    spotsvisited = [n]
    leavingspots = list()
    turn = 0
    forbidden = {n}
    while n != 0:
        turn += 1
        sign = n // abs(n)
        st = sign * turn
        if n - st not in forbidden:
            n = n - st
        else:
            leavingspots.append(n)
            if n + st in forbidden:
                stucklist.append(n)
            n = n + st
        spotsvisited.append(n)
        forbidden.add(n)
    return {'stuck':stucklist, 'spots':spotsvisited,
            'turns':turn, 'flee':leavingspots}
#Actual sequence
def a(n):
    d = trip(n)
    return max(d['spots'])
# David Nacin, Mar 15 2019
(C++) #include
long A248953(long n) { long c=0, s, m=n; for(std::map seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { if(n>m) m=n; seen[n]=true; ++c; } return m; } // M. F. Hasler, Mar 18 2019

a(n) = largest term in row n of triangle A248939.

a(A000217(n)) = A000217(n).

Edited by M. F. Hasler, Mar 18 2019

A248940 Wrecker ball sequence starting with 7.

Original entry on oeis.org

7, 6, 4, 1, -3, 2, -4, 3, -5, -14, -24, -13, -1, 12, -2, 13, 29, 46, 28, 9, -11, 10, -12, 11, 35, 60, 34, 61, 33, 62, 32, 63, 31, 64, 30, 65, 101, 138, 100, 139, 99, 58, 16, -27, 17, -28, 18, -29, 19, -30, 20, -31, 21, -32, 22, -33, 23, -34, 24, -35, 25, -36

Offset: 0

Reinhard Zumkeller, Oct 18 2014

Finite sequence with A228474(7) + 1 = 1726 terms;
row 7 of table A248939: a(n) = A248939(7,n) for n = 0..1725;
A248952(7) = -6362 <= a(n) <= 6843 = A248953(7).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1725
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A248939, A228474, A248941 (start with 17), A248942 (start with 20).

a248940 n = a248940_list !! n
a248940_list = a248939_row 7

A248941 Wrecker ball sequence starting with 17.

Original entry on oeis.org

17, 16, 14, 11, 7, 2, -4, 3, -5, 4, -6, 5, -7, 6, -8, -23, -39, -22, -40, -21, -1, 20, -2, 21, -3, 22, 48, 75, 47, 18, -12, 19, -13, -46, -80, -45, -9, 28, -10, 29, -11, 30, 72, 115, 71, 26, -20, 27, 75, 124, 74, 23, -29, 24, -30, 25, -31, -88, -146, -87

Offset: 0

Reinhard Zumkeller, Oct 18 2014

This is a finite sequence, containing exactly A228474(17) + 1 = 781803 terms.
This is row 17 of table A248939: a(n) = A248939(17,n) for n = 0..781802;
A248952(17) = -4843985 <= a(n) <= 4674389 = A248953(17).

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A248939, A228474, A248940 (start with 7), A248942 (start with 20).

a248941 n = a248941_list !! n
a248941_list = a248939_row 17

A248942 Wrecker ball sequence starting with 20.

Original entry on oeis.org

20, 19, 17, 14, 10, 5, -1, 6, -2, 7, -3, 8, -4, 9, -5, -20, -36, -19, -37, -18, 2, 23, 1, -22, -46, -21, -47, -74, -102, -73, -43, -12, -44, -11, -45, -10, 26, 63, 25, -14, -54, -13, 29, 72, 28, -17, -63, -16, 32, 81, 31, 82, 30, -23, -77, -132, -76, -133

Offset: 0

Reinhard Zumkeller, Oct 18 2014

Finite sequence with A228474(20) + 1 = 132897 terms;
row 20 of table A248939: a(n) = A248939(20,n) for n = 0..132896;
A248952(20) = -657367 <= a(n) <= 891114 = A248953(20).

Reinhard Zumkeller, Table of n, a(n) for n = 0..132896
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A248939, A228474, A248940 (start with 7), A248941 (start with 17).

a248942 n = a248942_list !! n
a248942_list = a248939_row 20

A248961 Sums of wrecker ball sequences starting with n.

Original entry on oeis.org

0, 1, -2, 5, -292, -241, 14, -437861, -28, -1, 30, 313, -4472, -4223, -2, 55, 3252, -214246256269, -70, -27, 5260887648, 91, -538, -193, -132, -864538549823, -22, 27, 140, 40053, 53088613819206, 86166834699, 86167898716, 86168962733, 86170026754, 49, 204

Offset: 0

Reinhard Zumkeller, Oct 18 2014

sign

a(n) = A248973(n, A228474(n)) = sum of row n in triangle A248939;

a(A000217(n)) = A000330(n).

			a(1) = 1+0 = 1;
a(2) = 2+1-1-4+0 = -2;
a(3) = 3+2+0 = 5;
a(4) = 4+3+1-2+2-3-9-16-8-17-7-18-6+7+21+6-10-27-45-26-46-25-47-24+0 = -292;
a(5) = 5+4+2-1+3-2-8-15-7-16-6-17-5+8+22+7-9-26-44-25-45-24-46-... = -241;
a(6) = 6+5+3+0 = 14;
a(7) = 7+6+4+1-3+2-4+3-5-14-24-13-1+12-2+13+29+46+28+9-11+10-... = -437861;
a(8) = 8+7+5+2-2+3-3+4-4-13-23-12+0 = -28;
a(9) = 9+8+6+3-1+4-2+5-3-12-22-11+1+14+0 = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A248939, A248973, A228474, A000217, A000330.

import Data.IntSet (singleton, member, insert)
a248961 n = addup 1 n 0 $ singleton n where
   addup  0 sum  = sum
   addup k x sum s = addup (k + 1) y (sum + x) (insert y s) where
                     y = x + (if (x - j) `member` s then j else -j)
                     j = k * signum x
(C++) #include
long A248961(long n) { long c=0, d, S=n; for(std::set A; n; A.insert(n), S += n += A.count(n - (d = n>0 ? c : -c)) ? d : -d) ++c; return S; } // M. F. Hasler, Mar 19 2019

A248961(n,A=[n],c,S=n)={while( n+=sign(n)*if(setsearch(A,n-sign(n)*c+=1), c, -c), A=setunion(A,[n]); S+=n); S} \\ M. F. Hasler, Mar 19 2019

def A248961(n):
  A = {n}; c = 0; S = 0
  while n != 0:
    ++c; s = c if n>0 else -c; n += s if n-s in A else -s; A.add(n); S += n
  return S # M. F. Hasler, Mar 19 2019

cp's OEIS Frontend

A248973 Table read by rows: row n contains the partial sums of the wrecker ball sequences starting with n, cf. A248939.

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A228474 Number of steps required to reach zero in the wrecker ball sequence starting with n: On the k-th step (k = 1, 2, 3, ...) move a distance of k in the direction of zero. If the result has occurred before, move a distance of k away from zero instead. Set a(n) = -1 if 0 is never reached.

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A248952 Smallest term in wrecker ball sequence starting with n.

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A248953 Largest term in wrecker ball sequence starting with n.

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A248940 Wrecker ball sequence starting with 7.

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A248941 Wrecker ball sequence starting with 17.

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A248942 Wrecker ball sequence starting with 20.

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