A005132 -id:A005132 - cp's OEIS frontend

A324785 Values of A005132 at its low points.

0, 2, 8, 14, 26, 64, 40, 4, 115, 266, 47, 229, 165, 92, 459, 404, 287, 139, 111, 844, 724, 516, 1639, 255, 181, 1424, 1099, 2974, 2930, 803, 644, 475, 380, 205, 150, 2694, 2428, 1942, 1314, 963, 5320, 637, 426, 371, 5094, 4852, 4422, 3700, 3138, 2534, 9953, 1821, 1592, 1363, 1188, 9680, 616

Offset: 1

N. J. A. Sloane, Sep 01 2019

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..230

Cf. A005132, A309226, A324784, A324786.

A324786 First differences of A324784: distances in A005132 from one low point to the next.

Original entry on oeis.org

4, 12, 15, 33, 35, 12, 20, 39, 17, 35, 63, 52, 66, 47, 58, 166, 80, 16, 73, 140, 244, 75, 43, 64, 303, 262, 89, 24, 55, 194, 82, 74, 130, 30, 215, 216, 784, 268, 358, 103, 49, 146, 34, 199, 216, 656, 632, 544, 492, 109, 429, 114, 186, 210, 127, 169, 106, 64, 477, 564, 422, 354, 644, 440, 1162

Offset: 1

N. J. A. Sloane, Sep 01 2019

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..229

Cf. A005132, A309226, A324784, A324785.

A334148 a(n) is the first term to repeat when following the same rules as Recamán's sequence A005132 but starting at n instead of 0.

Original entry on oeis.org

42, 20, 33, 3, 4, 5, 28, 6, 7, 8, 16, 15, 9, 10, 11, 19, 24, 21, 12, 13, 14, 15, 19, 23, 26, 15, 16, 17, 27, 21, 42, 44, 49, 18, 19, 20, 30, 36, 27, 48, 34, 59, 21, 22, 23, 21, 25, 29, 33, 36, 40, 45, 24, 25, 26, 23, 27, 31, 55, 79, 42, 46, 49, 27, 28, 29, 25, 29

Offset: 0

Scott R. Shannon, Apr 16 2020

nonn

The terms of this sequence grow slowly as n increases and are confined to bands of certain values, see the link image. Between n = 998000 and n = 1000000 the smallest term is 2829 and the largest is 19331.
The number of terms in each sequence starting from n required to reach a(n) is given in A334219.
The values where a(n) = n are given in A334225.
The number of terms in each sequence starting from n required to reach a value greater than n given in A334149.

			a(0) = 42 as a(0) corresponds to the standard Recamán's sequence A005132 in which 42 is the first term to repeat, appearing at A005132(20) and then again at A005132(24).
a(3) = 3 as starting from 3 the sequence of visited numbers is 3,2,0,3 and 3 is the first term to repeat.
a(6) = 28 as starting from 6 the sequence of visited numbers is 6,5,3,0,4,9,15,8,16,7,17,28,40,27,13,28 and 28 is the first number to repeat.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Plot of the terms for n = 0 to 1000000.
Index entries for sequences related to Recamán's sequence

Cf. A005132, A334149, A334219, A334225.

A334149 a(n) is the number of terms required beyond the starting value n before a value larger than n first appears when following the same rules as Recamán's sequence A005132 but starting at n instead of 0.

Original entry on oeis.org

1, 2, 2, 4, 5, 5, 5, 7, 9, 10, 6, 8, 10, 12, 14, 8, 9, 11, 13, 15, 17, 9, 11, 13, 14, 16, 18, 20, 20, 12, 14, 16, 17, 19, 21, 23, 23, 14, 15, 17, 19, 21, 22, 24, 26, 26, 30, 34, 18, 20, 22, 24, 26, 28, 29, 29, 33, 37, 37, 21, 23, 25, 27, 29, 31, 33, 33, 36, 19, 40, 44, 27, 26, 28

Offset: 0

Scott R. Shannon, Apr 16 2020

nonn

For 100 <= n <= 100000 the largest number of terms to surpass the starting value n is for n = 97646 which takes 26867 terms to surpass 97646, see the link image. The longest in terms of ratio of terms required compared to starting value is for n = 133 which takes 80 terms, see the link image. The shortest ratio is for n = 82148 which only takes 8587, see the link image.
The first repeated number in each sequence starting from n is given in A334148.
The number of terms in each sequence starting from n required to reach the first repeated number is given in A334219.

			a(0) = 1 as a(0) corresponds to the standard Recamán's sequence A005132 in which the first term is 0 and it only takes one more term to reach 1 and surpass the start value.
a(4) = 5 as starting from 4 the sequence of visited numbers is 4,3,1,4,0,5 and it takes five more terms to reach 5 and surpass the start value 4.
a(12) = 10 as starting from 12 the sequence of visited numbers is 12,11,9,6,2,7,1,8,0,9,19 and it takes ten more terms to reach 19 and surpass the start value 12.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Line plot of the corresponding Recamán type sequence starting with n = 97646.
Scott R. Shannon, Line plot of the corresponding Recamán type sequence starting with n = 133.
Scott R. Shannon, Line plot of the corresponding Recamán type sequence starting with n = 82148.
Scott R. Shannon, Plot of this sequence's terms for n = 0 to 100000.
Index entries for sequences related to Recamán's sequence

Cf. A005132, A334148, A334219, A334225.

A064492 Start of n-th segment of Recamán's sequence R(m) (A005132): values of n where the change in the fractional parts of successive values of R(n)/n is positive.

Original entry on oeis.org

1, 2, 4, 7, 12, 22, 40, 77, 135, 249, 454, 845, 1521, 2753, 5046, 9318, 17224, 31222, 57072, 99742, 181694, 328256, 589933, 1034839, 1788538, 3225919, 5784586, 10212211, 18399785, 32148795, 58056876, 101769230, 173395920, 302890749, 511561221, 904036925, 1610187039, 2789150424, 4910758398, 8416580355, 13808215356, 23006557539, 39488697701, 65854567303, 107078836274, 188471115227, 325374626149, 535755688022, 872467803894, 1404720439054, 2402964238974, 3883018238330, 6283167591180, 10125357598983, 16166305650061, 25735985498862, 40806937801473, 64628644387495, 102481019573338, 159888464280047, 250759470174414, 394178473635589, 599819882554936, 939455761022725

Offset: 1

John W. Layman, Oct 04 2001

nonn

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064288, A064289, A064292, A064293. This sequence and A064294 keep pretty close together.

See A065038 for the corresponding values of R(n). See also A065053 for first differences.

More terms from John W. Layman and N. J. A. Sloane, Oct 12 2001

Extended to 10^15 by Allan Wilks, Nov 06 2001

A064621 Values of A005132(n) at which the ratio A005132(n)/n sets a new record.

Original entry on oeis.org

1, 3, 6, 13, 20, 62, 113, 224, 367, 494, 833, 9169, 131313, 155719, 180118, 280766754, 3454917187, 31685027464, 152431947617, 629621648241

Offset: 1

Jud McCranie, Sep 26 2001

See A064622 for associated values of n.

No other terms < 4 * 10^12. - Jud McCranie, Dec 29 2019

			A005132(7)=20, 20/7 = 2.857..., larger than the ratio for any smaller value of n.

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064622.

Block[{s = Nest[Append[#1, If[And[#3 > 0, FreeQ[#1, #3]], #3, #1[[-1]] + #2]] & @@ {#1, #2, #1[[-1]] - #2} & @@ {#, Length@ #} &, {0}, 200000], t}, t = MapIndexed[#1/First@ #2 &, s]; Rest@ Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]] ] (* Michael De Vlieger, Dec 29 2019 *)

Offset changed, a(19)-a(20) from Jud McCranie, Dec 28 2019

A064622 Values of n at which the ratio A005132(n)/n sets a new record.

Original entry on oeis.org

1, 2, 3, 6, 7, 19, 34, 67, 102, 115, 190, 2066, 24381, 24398, 24399, 36130540, 409493529, 3744514071, 17951662018, 71184216405

Offset: 1

Jud McCranie, Sep 26 2001

See A064621 for associated values of A005132.

No other terms < 4.61 * 10^11. - Jud McCranie, Dec 29 2019

			A005132(7)=20, 20/7 = 2.857..., larger than the ratio for any smaller value of n. So 7 is in the sequence.

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064621.

Block[{s = Nest[Append[#1, If[And[#3 > 0, FreeQ[#1, #3]], #3, #1[[-1]] + #2]] & @@ {#1, #2, #1[[-1]] - #2} & @@ {#, Length@ #} &, {0}, 10^5], t}, t = MapIndexed[#1/First@ #2 &, s]; -1 + Rest@ Map[FirstPosition[t, #][[1]] &, Union@ FoldList[Max, t]] ] (* Michael De Vlieger, Dec 29 2019 *)

Two more terms from John W. Layman and N. J. A. Sloane, Oct 12 2001

Offset changed, a(19)-a(20) from Jud McCranie, Dec 28 2019

A076213 2*a(n)-1 = sign(A005132(n+1)-A005132(n)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0

Offset: 0

Benoit Cloitre, Nov 03 2002

nonn

Characteristic function of A057165 - 1. - M. F. Hasler, Jun 03 2009

Index entries for sequences related to Recamán's sequence

Conjecture: let s(n)=sum(k=1, n, a(k)), then lim n ->infinity s(n)/n = 1/2; for any n, 2*s(n) > n; let v(n)=2*s(n)-n, then v(n)/log(n) is bounded and sum(k=1, n, v(k)) is asymptotic to c*n*log(n) with 1 < c < 3/2.

a(n) = 1-A160351(n+1) = (A160357(n)+1)/2. - M. F. Hasler, Jun 03 2009

Added initial value a(0)=1. - M. F. Hasler, Jun 03 2009

A187943 A057167(A005132(A187922(n) - 1) - A187922(n)).

Original entry on oeis.org

1, 3, 2, 5, 15, 9, 6, 30, 20, 17, 63, 57, 51, 45, 35, 21, 98, 92, 66, 72, 38, 110, 103, 109, 25, 31, 10, 16, 2, 126, 36, 169, 163, 157, 151, 145, 139, 45, 35, 101, 182, 21, 32, 133, 187, 105, 123, 193, 23, 29, 8, 14, 3, 2, 75, 284, 278, 272, 266, 260, 254, 248, 226, 232, 100, 260, 254, 248, 226, 232, 306, 138, 144, 150, 156, 162, 168, 34, 65, 117, 228, 234

Offset: 1

Reinhard Zumkeller, Mar 17 2011

Let x = A187922(n) and r = A005132(x-1) - x, by definition of A187922: r > 0 and r occurs earlier in Recamán's sequence: a(n) = A057167(r) or r = A005132(a(n)), hence A005132(x) <> r but A005132(x) = A005132(x-1) - x.

			n = 5: x = A187922(5) = 19, r = A005132(19-1) - 19 = 43 - 19 = 24 = A005132(15): a(5) = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Recamán's sequence

Cf. A057165, A057167.

A210612 Number of nonnegative integers smaller than the largest number of Recamán's A005132 after the n-th stage that are not yet in Recamán's sequence at that stage.

Original entry on oeis.org

0, 1, 3, 2, 2, 7, 13, 12, 12, 11, 11, 10, 10, 9, 9, 8, 8, 25, 43, 42, 42, 41, 40, 40, 39, 39, 38, 37, 36, 35, 34, 33, 48, 81, 80, 80, 79, 78, 78, 77, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 95, 161, 160, 160

Offset: 1

Omar E. Pol, Mar 25 2012

nonn

Row length of row n of A210762.

Number of grid points that are not covered after n-th stage on the axis of the structure mentioned in A210606 which is a model for the visualization of Recamán's sequence.

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064291, A210606-A210611, A210613, A210762.

A210612 := proc(n)
        nops(A210762(n)) ;
end proc: # R. J. Mathar, Apr 01 2012

a(20) and beyond by R. J. Mathar, Apr 01 2012

cp's OEIS Frontend

A324785 Values of A005132 at its low points.

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A324786 First differences of A324784: distances in A005132 from one low point to the next.

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A334148 a(n) is the first term to repeat when following the same rules as Recamán's sequence A005132 but starting at n instead of 0.

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A334149 a(n) is the number of terms required beyond the starting value n before a value larger than n first appears when following the same rules as Recamán's sequence A005132 but starting at n instead of 0.

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A064492 Start of n-th segment of Recamán's sequence R(m) (A005132): values of n where the change in the fractional parts of successive values of R(n)/n is positive.

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A064621 Values of A005132(n) at which the ratio A005132(n)/n sets a new record.

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A064622 Values of n at which the ratio A005132(n)/n sets a new record.

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A076213 2*a(n)-1 = sign(A005132(n+1)-A005132(n)).

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A187943 A057167(A005132(A187922(n) - 1) - A187922(n)).

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A210612 Number of nonnegative integers smaller than the largest number of Recamán's A005132 after the n-th stage that are not yet in Recamán's sequence at that stage.

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