Revision History for A053253
(Underlined text is an addition;
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Showing entries 1-10
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#55 by Joerg Arndt at Fri Apr 26 01:42:53 EDT 2024
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| EXTENSIONS
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A053253 coincides with the number of configurations of unit 3D spheres stacking with 180 degree of rotational and reflexive symmetry in octahedrons
with side length L, N(L)= 1,2...5347,6072... Similar number of stackings with a different symmetry coincides with another mock theta function A053254,
while number series for other symmetries are not found in oeis.org. We obtained these number series in 2012 by computer without a formula or recurrence relation.
Similar questions have been studied in alternating sign matrices with certain symmetries (book "Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture"). Phys. Rev. E 81, 041118 (2010) shows that six-vertex model or alternating-sign matrices have one-to-one mapping to stacking of spheres in tetrahedron; thus sphere stacking in other containers (e.g. octahedron here) deserve further study. Further, square stacking (i.e. integer partition, one square sitting on top of two) and cube stacking (i.e. plane partition, one cube sitting on top of three) have been intensively studied, but surprisingly high-D cube stacking or 3D sphere stacking (one sphere sitting on top of four spheres) has not been explored (except in tetrahedrons which is essentially the alternating-sign matrices).
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| STATUS
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editing
approved
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#54 by Michel Marcus at Mon Jan 22 11:27:27 EST 2024
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Discussion
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| Mon Jan 29
| 13:05
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| Mon Feb 05
| 17:26
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| Mon Feb 12
| 20:18
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| Mon Feb 19
| 21:37
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| Mon Feb 26
| 23:39
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| Tue Mar 05
| 02:48
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| Tue Mar 12
| 07:40
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| Tue Mar 19
| 10:15
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| Tue Mar 26
| 14:59
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| Tue Apr 02
| 17:22
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| Tue Apr 09
| 19:27
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| Tue Apr 16
| 21:58
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| Wed Apr 24
| 00:17
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#53 by Yilong Han at Mon Jan 22 10:32:33 EST 2024
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Discussion
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| Mon Jan 22
| 11:27
| Michel Marcus: this should go to comments section with a better layout (see the unwanted linebreaks) and be signed
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#52 by Yilong Han at Mon Jan 15 02:50:18 EST 2024
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| EXTENSIONS
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A053253 coincides with the number of configurations of unit 3D spheres stacking with 180 degree of rotational and reflexive symmetry in octahedrons
with side length L, N(L)= 1,2...5347,6072... Similar number of stackings with a different symmetry coincides with another mock theta function A053254,
while number series for other symmetries are not found in oeis.org. We obtained these number series in 2012 by computer without a formula or recurrence relation.
Similar questions have been studied in alternating sign matrices with certain symmetries (book "Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture"). Phys. Rev. E 81, 041118 (2010) shows that six-vertex model or alternating-sign matrices have one-to-one mapping to stacking of spheres in tetrahedron; thus sphere stacking in other containers (e.g. octahedron here) deserve further study. Further, square stacking (i.e. integer partition, one square sitting on top of two) and cube stacking (i.e. plane partition, one cube sitting on top of three) have been intensively studied, but surprisingly high-D cube stacking or 3D sphere stacking (one sphere sitting on top of four spheres) has not been explored (except in tetrahedrons which is essentially the alternating-sign matrices).
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| STATUS
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approved
editing
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Discussion
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| Mon Jan 22
| 08:02
| OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review. If it is ready for review, please
visit https://oeis.org/draft/A053253 and click the button that reads
"These changes are ready for review by an OEIS Editor."
Thanks.
- The OEIS Server
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#51 by Michael De Vlieger at Sat Aug 12 23:00:08 EDT 2023
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#50 by Jon E. Schoenfield at Sat Aug 12 21:35:15 EDT 2023
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#49 by Jon E. Schoenfield at Sat Aug 12 21:35:11 EDT 2023
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| NAME
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Coefficients of the '3rd -order' mock theta function omega(q)).
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| FORMULA
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G.f.: (1 - G(0) )/())/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
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| CROSSREFS
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Other '3rd -order' mock theta functions are at A000025, A053250, A053251, A053252, A053254, A053255, A261401.
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| STATUS
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approved
editing
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#48 by Michael De Vlieger at Mon Jun 27 08:49:55 EDT 2022
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#47 by Gus Wiseman at Mon Jun 27 03:45:23 EDT 2022
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#46 by Gus Wiseman at Mon Jun 27 03:37:13 EDT 2022
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| MATHEMATICA
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(* second program *)
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[2*n+1], And@@OddQ/@#&&And@@OddQ/@conj[#]&]], {n, 0, 15}] (* Gus Wiseman, Jun 26 2022 *)
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| CROSSREFS
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AppearsConjectured to count the partitions ranked by A352143.
A035444A069911 = strict partitions w/ even partsall andodd multiplicitiesparts, ranked by A352141A258116.
A035457A078408 = partitions w/ all odd parts and even mults, ranked by A351979A066208.
A055922A117958 = partitions w/ all odd parts and multiplicities, ranked by A268335A352142.
A055922 aerated = partitions w/ even parts and odd mults, ranked by A352140.
A069911 = strict partitions with all odd parts, ranked by A258116.
A078408 = partitions with all odd parts, ranked by A066208.
A117958 = partitions with all odd parts and mults, ranked by A352142.
A162642 counts odd prime exponents, even A162641.
A257991 counts odd prime indices, even A257992.
Cf. A000009, A000290, `, A000701, A035363 (complement A086543), A035444, A035457, A045931, `A046682, `A066207, A055922, A258117, `A346635.
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