A005132 -id:A005132 - cp's OEIS frontend

A057167 Term in Recamán's sequence A005132 where n appears for first time, or -1 if n never appears.

0, 1, 4, 2, 131, 129, 3, 5, 16, 14, 12, 10, 8, 6, 31, 29, 27, 25, 23, 99734, 7, 9, 11, 13, 15, 17, 64, 62, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 40, 38, 111, 22, 20, 18, 28, 30, 32, 222, 220, 218, 216, 214, 212, 210, 208, 206, 204, 202, 200, 198, 196

Offset: 0

N. J. A. Sloane, Sep 14 2000

T. D. Noe, Table of n, a(n) for n=0..1642 (taking a(1355) from A064227)
Nick Hobson, Python program for this sequence
C. L. Mallows, Plot (jpeg) of first 10000 terms of A005132
C. L. Mallows, Plot (postscript) of first 10000 terms of A005132
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057165, A057166.

Cf. also A187943, A187922.

w := array(1..10000); for j from 1 to 100 do l := 0; for k from 1 to nops(a) do if a[k] = j then l := k; exit; fi; od: w[j] := l; od: s := [seq(w[j],j=1..100)]; # where a is an array formed from sequence A005132

A005132 = {0}; Do[If[(r = Last[A005132] - n) <= 0 || MemberQ[ A005132, r], r = r + 2n]; AppendTo[ A005132, r], {n, 1, 10^5}]; a[n_] := If[p = Position[ A005132, n]; p == {}, 0, p[[1, 1]] - 1]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 18 2012 *)

first(n) = my(a=vector(n), r=[0]); while(#Set(a)Iain Fox, Jul 11 2022

from itertools import count, islice
def agen(): # generator of terms
    an, A005132set, inv, y = 0, {0}, {0: 0}, 0
    for n in count(1):
        t = an - n
        an = t if t >= 0 and t not in A005132set else an + n
        A005132set.add(an)
        inv[an] = n
        while y in inv: yield inv[y]; y += 1
print(list(islice(agen(), 61))) # Michael S. Branicky, Jul 12 2022

I conjecture a(n) is never -1 - but see A064227, A064228.

a(0)=0 added and escape clause value changed to -1 by N. J. A. Sloane, May 01 2020

A064289 Height of n-th term in Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7

Offset: 0

N. J. A. Sloane, Sep 25 2001

The height of a term in A005132 = number of addition steps - number of subtraction steps to produce it.

Partial sums of A160357. - Allan C. Wechsler, Sep 08 2019

			A005132 begins 1, 3, 6, 2, 7, 13, 20, 12, ... and these terms have heights 1, 2, 3, 2, 3, 4, 5, 4, ...

Alois P. Heinz, Table of n, a(n) for n = 0..100000
Nick Hobson, Python program for this sequence
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
Index entries for sequences related to Recamán's sequence

Cf. A005132, A064288, A064290, A064292, A064293, A064294, A160357.

g:= proc(n) is(n=0) end:
b:= proc(n) option remember; local t;
      if n=0 then 0 else t:= b(n-1)-n; if t<=0 or g(t)
      then t:= b(n-1)+n fi; g(t):= true; t fi
    end:
a:= proc(n) option remember; `if`(n=0, 0,
       a(n-1)+signum(b(n)-b(n-1)))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 08 2019

g[n_] := n == 0;
b[n_] := b[n] = Module[{t}, If[n == 0, 0, t = b[n - 1] - n; If[t <= 0 || g[t], t = b[n - 1] + n]; g[t] = True; t]];
a[n_] := a[n] = If[n == 0, 0, a[n - 1] + Sign[b[n] - b[n - 1]]];
a /@ Range[0, 100] (* Jean-François Alcover, Apr 11 2020, after Alois P. Heinz *)

a(0)=0 prepended by Allan C. Wechsler, Sep 08 2019

A064227 From Recamán's sequence (A005132): record values in A057167.

Original entry on oeis.org

0, 1, 4, 131, 99734, 181653, 328002, 325374625245, 394178473633984

Offset: 1

N. J. A. Sloane, Sep 22 2001

Let R(n) = n-th term of Recamán's sequence (A005132); consider values of R(n) that take a record number of steps to appear (A064228); sequence gives corresponding values of n.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057167, A064228.

a(8)-a(9) from Allan Wilks, Nov 06 2001
a(10) > 10^25. - Benjamin Chaffin, Jun 13 2006
a(10) > 4.28*10^73. - Benjamin Chaffin, Mar 22 2010
a(10) > 10^230. - Benjamin Chaffin, 2018
Initial term 0 added by N. J. A. Sloane, Feb 26 2023

A064290 First number of height n in Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 3, 6, 13, 20, 43, 62, 113, 224, 367, 494, 833, 1815, 3379, 5551, 9169, 17864, 32978, 58964, 106218, 131313, 155719, 180118, 591890, 881467, 1345004, 3012446, 5728819, 9309579, 17512700, 25641318, 52978675, 61998980, 125130665, 244636214, 280766754, 566273517, 1031389697, 2182394227, 3045423658, 3454917187

Offset: 1

N. J. A. Sloane, Sep 25 2001

nonn

The height of a term in A005132 = number of addition steps - number of subtraction steps to reach it (see A064289).

Needs a b-file. - N. J. A. Sloane, May 01 2020

			A005132 begins 0, 1, 3, 6, 2, 7, 13, 20, 12, ... and these terms have heights 0, 1, 2, 3, 2, 3, 4, 5, 4, ...

Cf. A005132, A064288, A064289, A064292, A064293, A064294.

a(0) = 0 added by N. J. A. Sloane, May 01 2020

A064389 Variation (4) on Recamán's sequence (A005132): to get a(n), we first try to subtract n from a(n-1): a(n) = a(n-1) - n if positive and not already in the sequence; if not then we try to add n: a(n) = a(n-1) + n if not already in the sequence; if this fails we try to subtract n+1 from a(n-1), or to add n+1 to a(n-1), or to subtract n+2, or to add n+2, etc., until one of these produces a positive number not already in the sequence - this is a(n).

Original entry on oeis.org

1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 44, 19, 45, 72, 100, 71, 101, 70, 38, 5, 39, 4, 40, 77, 115, 76, 36, 78, 120, 163, 119, 74, 28, 75, 27, 79, 29, 80, 132, 185, 131, 186, 130, 73, 15, 81, 141, 202

Offset: 1

N. J. A. Sloane, Sep 28 2001

This is the nicest of these variations. Is this a permutation of the natural numbers?
The number of steps before n appears is the inverse series, A078758. The height of n is in A126712.
See A078758 for the inverse permutation (in case this is a permutation of the positive integers). - M. F. Hasler, Nov 03 2014
After 10^12 terms, the smallest number which has not appeared is 5191516. - Benjamin Chaffin, Oct 09 2016

Suggested by J. C. Lagarias.

Cf. A005132, A046901, A064387, A064388. Agrees with A064387 for first 187 terms, then diverges.

h := array(1..100000); maxt := 100000; a := array(1..1000); a[1] := 1; h[1] := 1; for nx from 2 to 1000 do for i from 0 to 100 do t1 := a[nx-1]-nx-i; if t1>0 and h[t1] <> 1 then a[nx] := t1; if t1 < maxt then h[t1] := 1; fi; break; fi; t1 := a[nx-1]+nx+i; if h[t1] <> 1 then a[nx] := t1; if t1 < maxt then h[t1] := 1; fi; break; fi; od; od; evalm(a);

h[1] = 1; h[] = 0; maxt = 100000; a[1] = 1; a[] = 0; For[nx = 2, nx <= 1000, nx++, For[i = 0, i <= 100, i++, t1 = a[nx - 1] - nx - i; If[t1 > 0 && h[t1] != 1, a[nx] = t1; If[t1 < maxt, h[t1] = 1]; Break[]]; t1 = a[nx - 1] + nx + i; If[h[t1] != 1, a[nx] = t1; If[t1 < maxt, h[t1] = 1]; Break[]]]]; Table[a[n], {n, 1, 100}](* Jean-François Alcover, May 09 2012, after Maple *)

A064389(n=1000,show=0)={ my(k,s,t); for(n=1,n, k=n; while( !(t>k && !bittest(s, t-k) && t-=k) && !(!bittest(s, t+k)  && t+=k), k++); s=bitor(s,1<M. F. Hasler, Nov 03 2014

A160357 Sign of first differences of Recamán's sequence A005132.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1

Offset: 0

M. F. Hasler, Jun 03 2009

sign

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A005132, A119632.

A160357 := proc(n)
        sign( A005132(n)-A005132(n-1)) ;
end proc: # R. J. Mathar, Apr 01 2012

f[s_List] := Module[{a = s[[-1]], n = Length[s]}, Append[s, If[a > n && FreeQ[s, a - n], a - n, a + n]]];
Nest[f, {0}, 100] // Differences // Sign (* Jean-François Alcover, Apr 19 2020, using Robert G. Wilson v's code for A005132 *)

a(n) = sign(A005132(n)-A005132(n-1)) = sign(A160356(n)) = (-1)^A160351(n) = -(-1)^A076213(n-1).
a(n) = (A005132(n)-A005132(n-1))/n. - N. J. A. Sloane, Jul 15 2011
A160357 = chi_A057165 - chi_A057166, where chi_A denotes the characteristic function of (the set of values of) A.

a(0)=1 added by N. J. A. Sloane, May 01 2020

A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 5, 3, 4, 4, 5, 11, 13, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17

Offset: 1

Omar E. Pol, Mar 24 2012

Consider a toothpick structure formed by L-toothpicks connected by their endpoints. The endpoints of the L-toothpicks are placed on the main diagonal of the first quadrant. At stage 1 we place an L-toothpick with one of its endpoints on the origin. At stage n we place an L-toothpick of size n. The L-toothpicks are placed alternately, on one or another sector of the first quadrant, trying to make the structure have an exposed endpoint closest to the origin. The total length of all L-toothpicks after the n-th stage is A002378(n). The value of x and y of the endpoint of the structure after the n-th stage is equal to the n-th term of Recamán's sequence A005132(n). Note that we can get other illustrations of initial terms of Recamán's sequence by replacing each L-toothpick by a Q-toothpick or by a semicircumference. This structure is also one of the three views of the three-dimensional model for Recamán's sequence. For more information about L-toothpicks and Q-toothpicks, see A172310 and A187210.

			The summands are the size of the L-toothpicks:
a(1) = 1.
a(2) = 1 + 2 = 3.
a(3) = 2 + 3 = 5.
a(4) = 3.
a(5) = 4.
a(6) = 4.
a(7) = 5.
a(8) = 5 + 6 = 11.
a(9) = 6 + 7 = 13.
a(10) = 7.

First differences of A210607.

Cf. A005132, A064289, A119632, A139250, A160356, A160357, A171175, A171178, A172310, A187210, A210604, A210605, A210608-A210613.

A064228 From Recamán's sequence (A005132): values of n achieving records in A057167.

Original entry on oeis.org

0, 1, 2, 4, 19, 61, 879, 1355, 2406

Offset: 1

N. J. A. Sloane, Sep 22 2001

If R(n) is the n-th term of Recamán's sequence (A005132), sequence gives values of R(n) that take a record number of steps to appear. A064227 gives corresponding values of n.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057167, A064227.

a(8) and a(9) from Allan Wilks, Nov 06 2001. After 10^15 terms of A005132, the smallest missing number was 852655.
After 10^25 terms of A005132 the smallest missing number is still 852655. - Benjamin Chaffin, Jun 13 2006
Even after 4.28*10^73 terms, the smallest missing number is still 852655. - Benjamin Chaffin, Mar 22 2010
Even after 10^230 terms, the smallest missing number is still 852655. - Benjamin Chaffin, 2018
Initial term 0 added by N. J. A. Sloane, Feb 26 2023

A064288 Height of n when it appears for first time in Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 2, 2, 7, 7, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 18, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 19, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 19, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

N. J. A. Sloane, Sep 25 2001

nonn

The height of a term in A005132 = number of addition steps - number of subtraction steps to produce it (see A064289).

			A005132 begins 0, 1, 3, 6, 2, 7, 13, 20, 12, ... and these terms have heights 0, 1, 2, 3, 2, 3, 4, 5, 4, ...

Rémy Sigrist, Table of n, a(n) for n = 0..1354
Rémy Sigrist, C program for A064288
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
Index entries for sequences related to Recamán's sequence

Cf. A005132, A064289, A064290, A064291, A064292, A064293, A064294.

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a(0)=0 added by N. J. A. Sloane, May 01 2020

A064293 Last number of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 7, 8, 14, 26, 4, 47, 92, 111, 181, 150, 371, 361, 781, 828, 366, 19, 61, 879, 4802, 3378, 5329, 9462, 32102, 18954, 18107, 16114, 4202, 25231, 60240, 92404, 92188, 14804

Offset: 1

N. J. A. Sloane, Sep 26 2001

nonn

The height of a term in A005132 = number of addition steps - number of subtraction steps to produce it.

Cf. A005132, A064290, A064292, A064294.

cp's OEIS Frontend

A057167 Term in Recamán's sequence A005132 where n appears for first time, or -1 if n never appears.

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A064289 Height of n-th term in Recamán's sequence A005132.

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A064227 From Recamán's sequence (A005132): record values in A057167.

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A064290 First number of height n in Recamán's sequence A005132.

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A160357 Sign of first differences of Recamán's sequence A005132.

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A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

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A064228 From Recamán's sequence (A005132): values of n achieving records in A057167.

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