cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A169752 A variation on Recamán's sequence A005132: see Comments for definition.

Original entry on oeis.org

1, 3, 2, 5, 4, 8, 6, 11, 17, 10, 7, 15, 24, 14, 9, 20, 12, 24, 37, 23, 38, 22, 39, 21, 40, 27, 47, 26, 48, 25, 49, 28, 53, 19, 45, 18, 46, 75, 105, 74, 42, 75, 41, 76, 112, 57, 94, 56, 95, 55, 96, 54, 97, 141, 52, 97, 51, 98, 50, 99, 149, 200, 148, 201, 147, 92, 36, 93, 35, 94
Offset: 1

Views

Author

Rodolfo Kurchan, Apr 08 2010

Keywords

Comments

Similar to A169748, but now B = 1,1,2,3,5,8,... (the Fibonacci numbers).

Extensions

Corrected and extended by D. S. McNeil, May 09 2010

A209386 a(n) = 2*A005132(n).

Original entry on oeis.org

0, 2, 6, 12, 4, 14, 26, 40, 24, 42, 22, 44, 20, 46, 18, 48, 16, 50, 86, 124, 84, 126, 82, 36, 84, 34, 86, 32, 88, 30, 90, 28, 92, 158, 226, 156, 228, 154, 78, 156, 76, 158, 74, 160, 72, 162, 70, 164, 68, 166, 66, 168, 64, 170, 62, 172, 60, 174, 58, 176, 56, 178, 54, 180, 52, 182, 314, 448, 312, 450, 310
Offset: 0

Views

Author

N. J. A. Sloane, Mar 07 2012

Keywords

Comments

This is the even-numbers version of Recamán's sequence A005132: a(0)=0; for n>0, a(n) = a(n-1)-2n if that number is >= 0 and not already in the sequence, otherwise a(n) = a(n-1)+2n.

Links

Crossrefs

Cf. A005132, A209387.

Programs

  • Mathematica
    f[s_List]:= Block[{a = s[[-1]], len = Length@s}, Append[s, If[a > len && ! MemberQ[s, a - len], a - len, a + len]]]; A005132:= Nest[f, {0}, 130]; Table[2*A005132[[n]], {n, 1, 50}] (* G. C. Greubel, Jan 03 2018 *)

A209387 a(n) = 2*A005132(n) + 1.

Original entry on oeis.org

1, 3, 7, 13, 5, 15, 27, 41, 25, 43, 23, 45, 21, 47, 19, 49, 17, 51, 87, 125, 85, 127, 83, 37, 85, 35, 87, 33, 89, 31, 91, 29, 93, 159, 227, 157, 229, 155, 79, 157, 77, 159, 75, 161, 73, 163, 71, 165, 69, 167, 67, 169, 65, 171, 63, 173, 61, 175, 59, 177, 57, 179, 55, 181, 53, 183, 315, 449, 313, 451, 311
Offset: 0

Views

Author

N. J. A. Sloane, Mar 07 2012

Keywords

Comments

This is the odd-numbers version of Recamán's sequence A005132: a(0)=1; for n>0, a(n) = a(n-1)-(2n+1) if that number is >= 0 and not already in the sequence, otherwise a(n) = a(n-1)+(2n+1).

Links

Crossrefs

Cf. A005132, A209386.

Programs

  • Mathematica
    f[s_List]:= Block[{a = s[[-1]], len = Length@s}, Append[s, If[a > len && ! MemberQ[s, a - len], a - len, a + len]]]; A005132:= Nest[f, {0}, 130]; Table[2*A005132[[n]] + 1, {n, 1, 50}] (* G. C. Greubel, Jan 03 2018 *)

A210604 Bisection of Recamán's sequence A005132.

Original entry on oeis.org

0, 3, 2, 13, 12, 11, 10, 9, 8, 43, 42, 41, 42, 43, 44, 45, 46, 113, 114, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 157, 156, 155, 154, 153, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 367, 368, 369, 370, 151, 152
Offset: 1

Views

Author

Omar E. Pol, Mar 23 2012

Keywords

Comments

Also length of the horizontal edges of a rectangular polyspiral path. For the vertical edges see A210605. The mentioned polyspiral path is one of the three views of a rectangular polyhelix which is a three-dimensional graphical representation of Recamán's sequence A005132.

Examples

			1) Written as an irregular array in which each row lists consecutive numbers in ascending order or in descending order:
0;
3,2;
13,12,11,10,9,8;
43,42,41;
42,43,44,45,46;
113,114;
39,38,37,36,35,34,33,32,31,30,29,28,27,26;
...
2) Written as an irregular array in which each row lists consecutive numbers in ascending order, in descending order or in descending-ascending order:
0;
3,2;
13,12,11,10,9,8;
43,42,41,42,43,44,45,46;
113,114;
39,38,37,36,35,34,33,32,31,30,29,28,27,26;
...

Links

Crossrefs

Cf. A005132, A065056, A171175, A171178, A210605-A210607.

A210605 Bisection of Recamán's sequence A005132.

Original entry on oeis.org

1, 6, 7, 20, 21, 22, 23, 24, 25, 62, 63, 18, 17, 16, 15, 14, 79, 78, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 224, 225, 226, 227, 228, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 265, 264, 263, 262, 261, 40, 265, 494, 495, 258, 259
Offset: 1

Views

Author

Omar E. Pol, Mar 23 2012

Keywords

Comments

Also length of the vertical edges of a rectangular polyspiral path. For the horizontal edges see A210604. The mentioned polyspiral path is one of the three views of a rectangular polyhelix which is a three-dimensional graphical representation of Recamán's sequence A005132.

Examples

			1) Written as an irregular array in which each row lists consecutive numbers in ascending order or in descending order:
1;
6,7;
20,21,22,23,24,25;
62,63;
18,17,16,15,14;
79,78,77;
78,79,80,81,82,83,84,85,86,87,88,89,90,91;
...
2) Written as an irregular array in which each row lists consecutive numbers in ascending order, in descending order or in descending-ascending order:
1;
6,7;
20,21,22,23,24,25;
62,63;
18,17,16,15,14;
79,78,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91;
...

Links

Crossrefs

Cf. A005132, A065056, A171175, A171178, A210604-A210607.

A210762 Triangle read by rows in which row n lists the positive integers smaller than the currently largest number in Recamán's sequence A005132 after the n-th stage, but not yet present in Recamán's sequence.

Original entry on oeis.org

2, 2, 4, 5, 4, 5, 4, 5, 4, 5, 8, 9, 10, 11, 12, 4, 5, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 19, 4, 5, 8, 9, 10, 14, 15, 16, 17
Offset: 2

Views

Author

Omar E. Pol, Mar 25 2012

Keywords

Comments

The number of positive integers in row n is A210612(n).

Examples

			Written as an irregular triangle:
2;
2, 4, 5;
4, 5;
4, 5;
4, 5, 8, 9, 10, 11, 12;
4, 5, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19;
4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19;
4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19;
4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 19;

Links

Crossrefs

Cf. A005132, A210606-A210613.

Programs

  • Maple
    A210762 := proc(n)
        local L,maxa ;
        rec := [seq(A005132(j),j=0..n)] ;
        maxa := max(op(rec)) ;
        L := [] ;
        for i from 0 to maxa do
                if not member(i,rec) then
                        L := [op(L),i] ;
                end if;
        end do;
        if nops(L) = 0 then
                return [0] ;
        end if;
        L ;
end proc:
seq(op(A210762(n)),n=1..11) ; # R. J. Mathar, Apr 01 2012

A227838 2^a(n) is the highest power of 2 dividing A005132(n).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 3, 3, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 4, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 4, 2, 0, 0, 1, 1, 0, 0, 2, 5, 0, 0, 1, 1, 0, 0, 3, 2, 0, 0, 1, 1, 0, 0, 5, 2, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 1, 1, 0, 0, 3, 2, 0, 0, 1, 1, 0, 0, 2, 5, 0, 0, 1, 1, 0, 0, 6, 2, 0, 0, 3, 4, 0, 0, 1, 1, 0
Offset: 1

Views

Author

N. J. A. Sloane, Aug 04 2013

Keywords

Comments

The 2-adic valuation of Recamán's sequence A005132.

Links

Crossrefs

Cf. A005132, A227839, A007814.

Programs

A227839 3^a(n) is the highest power of 3 dividing A005132(n).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2
Offset: 1

Views

Author

N. J. A. Sloane, Aug 04 2013

Keywords

Comments

The 3-adic valuation of Recamán's sequence A005132.

Links

Crossrefs

Cf. A005132, A227838, A007949.

Programs

A277558 A variation 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 a(n) = a(n-1)+n-i, where i >= 0 is the smallest number such that a(n-1)+n-i has not already appeared.

Original entry on oeis.org

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 40, 15, 39, 66, 38, 67, 37, 68, 36, 69, 35, 70, 34, 71, 33, 72, 32, 73, 31, 74, 30, 75, 29, 76, 28, 77, 27, 78, 26, 79, 133, 188, 132, 189, 131, 190, 130, 191, 129, 192
Offset: 0

Views

Author

Benjamin Chaffin, Oct 19 2016

Keywords

Comments

Is it ever impossible to extend the sequence -- meaning there is no number less than a(n-1)+n which has not appeared?
After 10^11 terms, the smallest number which has not appeared is 609790506.

Examples

			a(23) = 18. To get a(24) we try 18-24, but that is negative; so we try 18+24 = 42, but 42 has already appeared; so we try 18+24-1, but 41 has also already appeared; so we try 18+24-2. 40 is positive and has not yet appeared, and so a(24) = 40.

Links

Crossrefs

Cf. A005132, A064387 (chooses a(n-1)+n+i instead of a(n-1)+n-i).

A330788 Values of k at which the ratio k/A005132(k) sets a new record.

Original entry on oeis.org

1, 4, 31, 62, 64, 111, 129, 131, 30764, 30766, 30768, 56127, 56167, 56199, 56513, 56515, 56827, 99734, 58055311, 904012796, 904032692, 1610175617, 1610178287, 1610186343, 2789149734, 13808214835, 325374625053, 325374625245
Offset: 1

Views

Author

Jud McCranie, Dec 31 2019

Keywords

Comments

No other terms < 4.61 * 10^11.

Examples

			A005132(131) = 4, 131/4 = 32.75, a higher ratio than any smaller term in A005132, so 131 is in the sequence.

Links

Crossrefs

Cf. A005132, A064622, A330789.

Programs

  • Mathematica
    Block[{s = MapIndexed[First[#2]/#1 &, Rest@ Nest[Append[#, If[And[#3 > 0, FreeQ[#1, #3]], #3, #1[[-1]] + #2]] & @@ {#, Length@ #, #[[-1]] - Length@ #} &, {0}, 10^5]], t}, t = Union@ FoldList[Max, Rest@ s]; MapAt[# - 1 &, #, 1] &@ Map[FirstPosition[s, #][[1]] &, t]] (* Michael De Vlieger, Jan 01 2020 *)
Previous Showing 51-60 of 230 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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