Howard Givner - cp's OEIS frontend

A345470 Number of self-complementary score sequences that are possible in an n-team round-robin tournament.

1, 1, 1, 2, 2, 5, 6, 15, 19, 48, 64, 161, 223, 557, 796, 1971, 2887, 7090, 10596, 25826, 39256, 95016, 146533, 352411, 550328, 1315827, 2077418, 4940587, 7876036, 18639221, 29971423, 70608885, 114422037, 268436473, 438068242

Offset: 0

Howard Givner, Jun 20 2021

nonn

See A000571 for the definition of a score sequence.

A self-complementary score sequence W is a score sequence of win counts such that W = {s(1), s(2), ..., s(n)} and its complement, L={n-1-s(n), n-1-s(n-1), ..., n-1-s(1)}, a score sequence of loss counts, are identical.

			For n = 4 there are 4 score sequences of which only 2, those marked with an asterisk, are self-complementary.  These are the sequences for n=4.
    {0,1,2,3} *
    {0,2,2,2}
    {1,1,1,3}
    {1,1,2,2} *
For n = 5, there are 9 score sequences of which only 5, those marked with an asterisk, are self-complementary.  These are the sequences for n=5.
    {0,1,2,3,4} *
    {0,1,3,3,3}
    {0,2,2,2,4} *
    {0,2,2,3,3}
    {1,1,2,3,3} *
    {1,1,1,3,4}
    {1,1,2,2,4}
    {1,2,2,2,3} *
    {2,2,2,2,2} *

Paul K. Stockmeyer, Table of n, a(n) for n = 0..500
Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, J. Integer Seq. 26 (2023), Article 23.5.2.

Cf. A000571.

a(30) corrected by Howard Givner, Jun 28 2021

a(0)=1 prepended and a(1) changed from 0 to 1 by Howard Givner, Feb 22 2022

A345388 a(n) = 0, 1, or 2 according to whether A065091(n), the n-th odd prime, is in A001122, A139035, or A268923, respectively.

Original entry on oeis.org

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

Offset: 1

Howard Givner, Jun 17 2021

nonn

The three OEIS sequences A001122, A139035, and A268923 are implicitly described in a Zoom lecture that was given May 14, 2021, by James Tanton. Here is a link to the video, followed by a description of how the sequences can be obtained by carrying out the procedure that the speaker described in his talk.
Description of the method:
James Tanton defined GOOD, HALF-GOOD, and BAD odd prime integers and a procedure for determining which of the three categories an odd prime integer belongs to.
Procedure for categorizing an odd prime integer P:
Step 1. Begin with an initial partition (1,P-1) of P.
Step 2. Generate a successor partition, derived from an existing partition.
When (x,y) is an existing partition and x is even, the successor partition is (s,t), where s=x/2 and t=P-s.
When (x,y) is an existing partition and x is odd, the successor partition is (s,t), where t=y/2 and s=P-t.
Step 3. Repeat step 2 until you return to (1,P-1).
He then classified P as either GOOD, HALF-GOOD, or BAD as follows:
P is GOOD when every integer from 1 to P-1 appears among the left parts of the set of generated partitions.
P is HALF-GOOD when P does not meet the requirements for GOOD, but every integer from 1 to P-1 appears somewhere in the set of generated partitions.
P is BAD when P does not meet the requirements for GOOD or HALF-GOOD.
The sequence of GOOD odd prime integers is identical to A001122.
The sequence of HALF-GOOD odd prime integers is identical to A139035.
The sequence of BAD odd prime integers is identical to A268923.

			For P=5, the generated partition set is:
  (1,4), (3,2), (4,1), (2,3), (1,4), and thus 5 is GOOD, so a(2)=0.
For P=7, the generated partition set is:
  (1,6), (4,3), (2,5), (1,6), and thus 7 is HALF-GOOD, so a(3)=1.
For P=17, the generated partition set is:
  (1,16), (9,8), (13,4), (15,2), (16,1), (8,9), (4,13), (2,15), (1,16),
  but 3, 5, 6, 7, 10, 11, 12, and 14 do not appear, and thus 17 is BAD, so a(6)=2.

James Tanton, How to fold things into thirds, sevenths, and thirty-sevenths!, video, May 14, 2021.

Cf. A001122, A065091, A139035, A268923.

Name edited by Felix Fröhlich, Jun 28 2021

A054245 Beethoven's Fifth Symphony; 1 stands for the first note in the minor scale, etc.

Original entry on oeis.org

5, 5, 5, 3, 4, 4, 4, 2, 5, 5, 5, 3, 6, 6, 6, 5, 10, 10, 10, 8, 5, 5, 5, 2, 6, 6, 6, 5, 11, 11, 11, 9, 12, 12, 11, 10, 9, 12, 12, 11, 10, 9, 12, 12, 11, 10, 8, 12, 13, 13, 13, 11

Offset: 0

Howard Givner, circa 1996; Rebecca Bellovin, Apr 27 2000

I'm including this only because it was independently submitted by two people. It violates almost all my rules. - N. J. A. Sloane, May 08 2000

The explanation "1 stands for the first note in the minor scale, etc." is not satisfying. It appears that the current data refers to the full tones in that scale, but this does not allow us to encode correctly half-tones occurring later in the score but which are not part of the C minor scale. It would be more satisfying to record the half-tones, using, e.g., 0=c, 1=c#, 2=d, 3=d#, ..., 12=c', -12=C, ... - M. F. Hasler, Jun 12 2012

Richard Friedberg, An Adventurer's Guide to Number Theory, 1968, McGraw-Hill; reprinted by Dover Publications.

Classical Archives, Ludwig van Beethoven, Symphony No.5 in C-, Op.67 (MP3 files). [From Ami Fischman (ami(AT)fischman.org), May 27 2000, link updated by _Brian Galebach_, Sep 13 2003, and by _M. F. Hasler_, Jun 12 2012]
Project Gutenberg, Symphony No. 5 in C minor Opus 67 by Ludwig van Beethoven (MIDI files). [From _M. F. Hasler_, Jun 12 2012]
Tower Records, Beethoven's 5th symphony [From _Brian Galebach_, Mar 20 2001, link updated by _M. F. Hasler_, Jun 12 2012]
Wikipedia, Symphony No. 5 (Beethoven). [From _M. F. Hasler_, Jun 12 2012]
OEIS index entry for music.

More terms from Brian Galebach, Mar 20 2001

A001049 Numbered stops in Manhattan on the Lexington Avenue subway.

Original entry on oeis.org

8, 14, 23, 28, 33, 42, 51, 59, 68, 77, 86, 96, 103, 110, 116, 125

Offset: 1

Howard Givner

These are the numbered stops for the #6 train.

An Adventurer's Guide to Number Theory by R. Friedberg, 1968, McGraw-Hill, p. 30 (corrected).
NYC Transit, NYC Subway Map, 2005.

R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
Map, schedule, etc.
NYC Transit

Cf. A000053 A000054 A007826.

Additional comments from Nyck Byrd (nyc10024(AT)lycos.com), Dec 25 2005

A001165 Position of first even digit after decimal point in sqrt(n).

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 7, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2

Offset: 1

Howard Givner

T. D. Noe, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, April 17 2019)

Flatten[Table[First[Position[Rest[RealDigits[Sqrt[n],10,20][[1]]], ?(EvenQ[#]&)]],{n,100}]] (* _Harvey P. Dale, Oct 27 2011 *)

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User: Howard Givner

A345470 Number of self-complementary score sequences that are possible in an n-team round-robin tournament.

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A345388 a(n) = 0, 1, or 2 according to whether A065091(n), the n-th odd prime, is in A001122, A139035, or A268923, respectively.

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Examples

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Crossrefs

Extensions

A054245 Beethoven's Fifth Symphony; 1 stands for the first note in the minor scale, etc.

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Author

Keywords

Comments

References

Links

Extensions

A001049 Numbered stops in Manhattan on the Lexington Avenue subway.

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Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A001165 Position of first even digit after decimal point in sqrt(n).

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Author

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Mathematica