Allan Wilks - cp's OEIS frontend

A217943 Triangle read by rows: T(n,k) = 2*C(n-1,k)-C(n,k) for kA216955(n,k).

2, -2, 0, 2, -2, 2, -2, 2, -2, 0, 0, 0, 2, -2, 4, -2, -2, 0, 2, -2, 0, 0, 0, 0, 0, 2, -2, 6, -6, 2, -2, 0, 0, 2, -2, 0, 6, -6, 0, 0, 0, 0, 2, -2, 10, -10, 0, 2, -2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 20, -10, -4, -6, 2, -2, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2

Offset: 2

N. J. A. Sloane, following a suggestion from Allan Wilks, Oct 25 2012

			Triangle begins:
[2, -2]
[0, 2, -2]
[2, -2, 2, -2]
[0, 0, 0, 2, -2]
[4, -2, -2, 0, 2, -2]
[0, 0, 0, 0, 0, 2, -2]
[6, -6, 2, -2, 0, 0, 2, -2]
[0, 6, -6, 0, 0, 0, 0, 2, -2]
[10, -10, 0, 2, -2, 0, 0, 0, 2, -2]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2]
[20, -10, -4, -6, 2, -2, 0, 0, 0, 0, 2, -2]
...

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102, Dec 25 2012.
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
N. J. A. Sloane, Rows 2 through 48 of A217943
Index entries for sequences related to curling numbers

Cf. A216955.

The nonzero entries in the first column form A216958.

A119632 Lengths of successive runs in A160357, where a run here means a string of alternating terms.

Original entry on oeis.org

1, 1, 3, 1, 11, 1, 4, 10, 1, 4, 28, 1, 10, 24, 1, 8, 1, 2, 1, 1, 4, 1, 9, 4, 1, 2, 36, 1, 12, 4, 1, 2, 1, 3, 28, 1, 10, 52, 1, 18, 1, 32, 1, 12, 15, 38, 1, 14, 32, 1, 12, 1, 44, 1, 16, 1, 148, 1, 50, 7, 22, 1, 8, 3, 4, 1, 2, 70, 1, 24, 1, 114, 1, 42, 1, 200, 1, 68, 6, 1, 2, 13

Offset: 1

N. J. A. Sloane and Allan Wilks, Jun 10 2006

nonn

Gives a highly compressed version of A005132.
The encoding of Recamán's sequence a(n) = A005132 using A119632 is easy - A119632 counts runs of alternating i(n)'s, where i(n) = (a(n)-a(n-1))/n = A160357(n).
Note that i(n) is always +1 or -1.  Each run ends when i(n) = i(n+1).
Here is pseudo-code to reconstruct Recamán's sequence from A119632, which we will call I(n):
        a(0) = 0
        n = 1
        i = 1
        for k = 1..oo {
                for j = 1..I(k) {
                        a(n) = a(n-1) + n*i
                        n = n+1
                        i = -i
                }
                i = -i
        }
The gzipped file attached to A119632 represents the first 1470117206801829 terms of A005132. The longest run of alternating i(n)'s (maximal value found so far in A119632) is 232144588914. There are 64094657 runs encoded in the gzipped file.

			A160357 begins 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; ..., where semicolons demark the successive runs.

Allan Wilks, Table of n, a(n) for n = 1..100000
Allan Wilks, The first 64094657 terms (gzipped). (A large file. This encodes the first 1470117206801829 terms of A005132!)
Index entries for sequences related to Recamán's sequence

Cf. A005132, A160357.

Entry expanded by N. J. A. Sloane, Jul 15 2011

A065051 Let R(n) = n-th term of Recamán's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of q.

Original entry on oeis.org

1, 1, 2, 0, 1, 2, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 1, 2, 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, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Allan Wilks, Nov 06 2001

nonn

Index entries for sequences related to Recamán's sequence

Cf. A065052, A005132.

A065052 Let R(n) = n-th term of Recamán's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of r.

Original entry on oeis.org

0, 1, 0, 2, 2, 1, 6, 4, 3, 1, 0, 10, 10, 9, 9, 8, 8, 7, 5, 2, 0, 19, 18, 18, 17, 17, 16, 16, 15, 15, 14, 14, 13, 11, 8, 6, 3, 1, 0, 38, 38, 37, 37, 36, 36, 35, 35, 34, 34, 33, 33, 32, 32, 31, 31, 30, 30, 29, 29, 28, 28, 27, 27, 26, 26, 25, 23, 20, 18, 15, 13, 10, 8, 5, 3

Offset: 1

Allan Wilks, Nov 06 2001

nonn

Index entries for sequences related to Recamán's sequence

Cf. A065051, A005132.

A065053 Lengths of intervals between special points in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 3, 5, 10, 18, 37, 58, 114, 205, 391, 676, 1232, 2293, 4272, 7906, 13998, 25850, 42670, 81952, 146562, 261677, 444906, 753699, 1437381, 2558667, 4427625, 8187574, 13749010, 25908081, 43712354, 71626690, 129494829, 208670472, 392475704, 706150114, 1178963385, 2121607974, 3505821957, 5391635001, 9198342183, 16482140162, 26365869602, 41224268971, 81392278953, 136903510922, 210381061873, 336712115872, 532252635160, 998243799920, 1480053999356, 2400149352850, 3842190007803, 6040948051078, 9569679848801, 15070952302611, 23821706586022, 37852375185843, 57407444706709, 90871005894367, 143419003461175, 205641408919347, 339635878467789, 530661445779104

Offset: 1

Allan Wilks, Nov 06 2001

nonn

Index entries for sequences related to Recamán's sequence

First differences of A064492.

A065038 Values of Recamán's sequence A005132 at start of n-th segment (see A064492).

Original entry on oeis.org

1, 3, 2, 20, 10, 41, 38, 75, 268, 247, 1361, 2533, 3041, 2751, 15135, 18635, 51668, 62443, 57070, 398963, 181693, 1313022, 2359729, 1034838, 5365613, 3225918, 17353757, 10212210, 73599139, 96446382, 58056874, 407076917, 520187758, 908672243, 2046244881, 2712110771, 6440748154, 11156601694, 14732275193, 8416580354, 41424646066, 23006557538, 78977395399, 65854567302, 107078836273, 188471115226, 650749252297, 1071511376043, 872467803893, 2809440878107, 2402964238973, 7766036476659, 18849502773536, 10125357598982, 32332611300121, 102943941995445, 163227751205887, 193885933162482, 307443058720011, 159888464280046, 250759470174413, 394178473635587, 599819882554934, 2818367283068173

Offset: 1

Allan Wilks, Nov 06 2001

nonn

Index entries for sequences related to Recamán's sequence

A064970 a(1)=1; thereafter, values of n for which r(n)-r(n-1) and r(n-1)-r(n-2) have the same sign, where r(n) = A005132(n).

Original entry on oeis.org

1, 2, 3, 6, 7, 18, 19, 23, 33, 34, 38, 66, 67, 77, 101, 102, 110, 111, 113, 114, 115, 119, 120, 129, 133, 134, 136, 172, 173, 185, 189, 190, 192, 193, 196, 224, 225, 235, 287, 288, 306, 307, 339, 340, 352, 367, 405, 406, 420, 452, 453, 465, 466

Offset: 1

Allan Wilks, Oct 30 2001

nonn

			E.g., going from r(4)=2 to r(5)=7 to r(6)=13 we increase twice in a row, so 6 is a member. Going from r(21)=63 to r(22)=41 to r(23)=18 we decrease twice in a row, so 23 is a member.

Index entries for sequences related to Recamán's sequence

Cf. A005132.

Better description from Dean Hickerson, Feb 15 2006

A042946 Frequency of occurrence of numbers appearing in A042944.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 6, 8, 4, 4, 4, 4, 4, 4, 8, 8, 6, 8, 4, 4, 4, 8, 8, 8, 4, 8, 12, 2, 8, 8, 12, 8, 8, 4, 4, 4, 4, 12, 8, 4, 4, 10, 8, 8, 8, 4, 4, 4, 12, 8, 8, 8, 12, 12, 8, 8, 8, 12, 6, 8, 12, 8, 4, 12, 8, 12, 12, 8, 12, 16, 4, 4, 8, 12, 8, 6, 8, 12, 12, 4, 4, 8, 12, 8, 4

Offset: 0

Allan Wilks

nonn

			E.g. 35 occurs 6 times.

A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 24, 25, 28, 33, 36, 40, 49, 52, 57, 60, 64, 72, 73, 76, 81, 84, 88, 96, 97, 100, 105, 108, 112, 121, 124, 129, 136, 144, 145, 148, 153, 156, 160, 168, 169, 172, 177, 180, 184, 192, 193, 196, 201, 204, 216, 217, 220, 225, 228, 232

Offset: 0

Allan Wilks

nonn

Appears to be a superset of {A008784 - 1}. - Ralf Stephan, Jan 26 2005

See A189226 for additional comments, references, links, examples, and crossrefs. - Jonathan Sondow, Aug 24 2012

K. E. Stange, The sensual Apollonian circle packing, arXiv:1208.4836 [math.NT], 2012-2014; see section 14. - _Jonathan Sondow_, Aug 24 2012

Cf. A042944-A042946, A045679, A189226.

A045679 Numbers congruent to 0,1,4,9 mod 12 missing from A045673 (conjectured to be finite).

Original entry on oeis.org

13, 21, 37, 45, 48, 61, 69, 85, 93, 109, 117, 120, 132, 133, 141, 157, 165, 181, 189, 205, 208, 213, 229, 237, 241, 252, 253, 261, 277, 285, 300, 301, 309, 325, 328, 333, 340, 349, 357, 360, 373, 381, 397, 405, 421, 429, 445, 453, 468, 469, 477

Offset: 1

Allan Wilks

nonn

Cf. A042944, A042945, A042946, A045674, A112652.

Offset 1 and name corrected by Michel Marcus, Sep 13 2019

cp's OEIS Frontend

User: Allan Wilks

A217943 Triangle read by rows: T(n,k) = 2*C(n-1,k)-C(n,k) for kA216955(n,k).

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A119632 Lengths of successive runs in A160357, where a run here means a string of alternating terms.

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A065051 Let R(n) = n-th term of Recamán's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of q.

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A065052 Let R(n) = n-th term of Recamán's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of r.

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Author

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Crossrefs

A065053 Lengths of intervals between special points in Recamán's sequence A005132.

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Author

Keywords

Links

Crossrefs

A065038 Values of Recamán's sequence A005132 at start of n-th segment (see A064492).

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A064970 a(1)=1; thereafter, values of n for which r(n)-r(n-1) and r(n-1)-r(n-2) have the same sign, where r(n) = A005132(n).

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Keywords

Examples

Links

Crossrefs

Extensions

A042946 Frequency of occurrence of numbers appearing in A042944.

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Examples

A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.

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Author

Keywords

Comments

Links

Crossrefs

A045679 Numbers congruent to 0,1,4,9 mod 12 missing from A045673 (conjectured to be finite).

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Author

Keywords

Crossrefs

Extensions