Philippe A.J.G. Chevalier - cp's OEIS frontend

A270950 Number of distinct cardinalities of orbits of lattice points under the automorphism group of the n-dimensional integer lattice.

0, 1, 2, 5, 9, 12, 20, 29, 40, 53, 76, 99, 132, 172, 216, 270, 341, 424, 532, 660, 810, 983, 1210, 1446, 1750, 2111, 2508, 2975, 3569, 4197, 4948, 5807, 6817, 7963, 9351, 10863, 12604, 14598, 16892, 19439, 22472, 25780, 29588, 33892, 38800, 44206, 50463, 57297, 65086, 73919, 83842, 94510

Offset: 0

Philippe A.J.G. Chevalier, Mar 26 2016

A finite number of orbits partition hypercubic shells of infinity norm s in the n-dimensional integer lattice. The number of orbits is given by C(n+s-1,s). The number of distinct cardinalities of the orbits of lattice points under the automorphism group of the n-dimensional integer lattice is found under the condition that n <= s.

A new connection was discovered using the partition of the dimension 'n'. These partitions create a base set of cardinalities. Each of these cardinalities can be subjected to the process of prime factorization. The prime factorization yields the exponents of the primes that form lattice points in a new integer lattice of dimension 'n'. These lattice points become elements of a set A. The unique summands of a specific partition of 'n' give the multipliers of the base vector (1,0^n) that need to be subtracted from the specific partition representative element of set A. The cardinality of the set A increases until all the specific partitions of 'n' have been processed. This augmented set A* has the correct cardinality. This method is much faster than the brute force technique. - Philippe A.J.G. Chevalier, Jun 24 2022

			For n=0 the a(0)=0.
For n=3 we have the following distinct cardinalities of the orbits 6, 8, 12, 24, 48 and thus a(3)=5.
For n=4 we have the distinct cardinalities of the orbits 8, 16, 24, 32, 48, 64, 96, 192, 384 and thus a(4)=9.
For n=5 we have the distinct cardinalities of the orbits 10, 32, 40, 160, 240, 320, 480, 640, 960, 1920, 3840 and thus a(5)=12.

Philippe A.J.G. Chevalier, Table of n, a(n) for n = 0..54

Cf. A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287.

a(17) corrected and a(18)-a(51) from Philippe A.J.G. Chevalier, Jun 24 2022

A266398 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 13440.

Original entry on oeis.org

0, 0, 12, 37, 76, 130, 200, 287, 392, 516, 660, 825, 1012, 1222, 1456, 1715, 2000, 2312, 2652, 3021, 3420, 3850, 4312, 4807, 5336, 5900, 6500, 7137, 7812, 8526, 9280, 10075, 10912, 11792, 12716, 13685, 14700, 15762, 16872, 18031, 19240, 20500, 21812, 23177

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002112, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(2), Vec(x^3*(12-11*x)/(1-x)^4 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (n^3+30*n^2-97*n+66)/6.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x^3*(12-11*x) / (1-x)^4.
(End)

A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

Original entry on oeis.org

0, 0, 9, 31, 70, 130, 215, 329, 476, 660, 885, 1155, 1474, 1846, 2275, 2765, 3320, 3944, 4641, 5415, 6270, 7210, 8239, 9361, 10580, 11900, 13325, 14859, 16506, 18270, 20155, 22165, 24304, 26576, 28985, 31535, 34230, 37074, 40071, 43225, 46540, 50020, 53669

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(2), Vec(x^3*(9-5*x)/(1-x)^4 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (4*n^3+3*n^2-37*n+30)/6.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x^3*(9-5*x) / (1-x)^4.
(End)

A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

Original entry on oeis.org

0, 0, 0, 10, 41, 105, 215, 385, 630, 966, 1410, 1980, 2695, 3575, 4641, 5915, 7420, 9180, 11220, 13566, 16245, 19285, 22715, 26565, 30866, 35650, 40950, 46800, 53235, 60291, 68005, 76415, 85560, 95480, 106216, 117810, 130305, 143745, 158175, 173641, 190190

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

LinearRecurrence[{5,-10,10,-5,1},{0,0,0,10,41},50] (* Harvey P. Dale, Nov 18 2024 *)

concat(vector(3), Vec(x^4*(10-9*x)/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (n^4+30*n^3-205*n^2+390*n-216)/24.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: x^4*(10-9*x) / (1-x)^5.
(End)

A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

Original entry on oeis.org

0, 0, 0, 0, 15, 75, 225, 525, 1050, 1890, 3150, 4950, 7425, 10725, 15015, 20475, 27300, 35700, 45900, 58140, 72675, 89775, 109725, 132825, 159390, 189750, 224250, 263250, 307125, 356265, 411075, 471975, 539400, 613800, 695640, 785400, 883575, 990675, 1107225

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(4), Vec(15*x^5/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 5*(n-1)*(n-2)*(n-3)*(n-4)/8 = 15*A000332(n-1).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: 15*x^5 / (1-x)^5.
(End)

A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 42, 147, 392, 882, 1764, 3234, 5544, 9009, 14014, 21021, 30576, 43316, 59976, 81396, 108528, 142443, 184338, 235543, 297528, 371910, 460460, 565110, 687960, 831285, 997542, 1189377, 1409632, 1661352, 1947792, 2272424, 2638944, 3051279

Offset: 1

Philippe A.J.G. Chevalier, Dec 28 2015

The sequence was discovered by enumerating all orbits of Aut(Z^7) and sorting the orbits as function of the infinity norm of the representative integer lattice points. This sequence is one of the 30 sequences that are obtained by classifying the orbits in a table with the rows being the infinity norm and the columns being the 30 cardinalities (1, 14, 84, 128, 168, 280, 448, 560, 672, 840, 896, 1680, 2240, 2688, 3360, 4480, 5376, 6720, 8960, 13440, 17920, 20160, 26880, 40320, 53760, 80640, 107520, 161280, 322560, 645120) generated by signed permutations of integer lattice points of Z^7.

The continued fraction expansion of this sequence is finite and simplifies to the g.f. 7*x^6/(1-x)^6 (see Mathematica). - Benedict W. J. Irwin, Feb 09 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Other sequences that give the number of orbits of Aut(Z^7) as function of the infinity norm for different cardinalities of these orbits: A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

Join[{0, 0, 0, 0, 0},Table[Abs[SeriesCoefficient[Series[7/(x+6/(x - 5/2/(x + ContinuedFractionK[If[Mod[k, 2] ==0, (7 + k/2)/(6 + 2 k), ((k + 1)/2 - 5)/(2 (k - 1) +6)], x, {k, 0, 8}]))), {x, Infinity, 101}],2 n + 1]], {n, 0, 50}]] - (* Benedict W. J. Irwin, Feb 09 2016 *)

concat(vector(5), Vec(7*x^6/(1-x)^6 + O(x^50))) \\ Colin Barker, May 04 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/120.
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) for n>6.
G.f.: 7*x^6 / (1-x)^6.
(End)

A247557 Number of rectangles formed by the absolute leader classes of the seven-dimensional integer lattice as a function of the infinity norm n and having a unique perimeter, where the rectangles have one common lattice point being the origin of the seven-dimensional integer lattice.

Original entry on oeis.org

1, 7, 26, 79, 182, 333, 693, 1180, 1999, 3247

Offset: 1

Philippe A.J.G. Chevalier, Sep 19 2014

An absolute leader class is a term used in coding theory to label special integer lattice points. In the seven-dimensional integer lattice Z^7 we have for the infinity norm n=1 the following absolute leader classes using the Conway abbreviation: (1,0^6),(1^2,0^5),(1^3,0^4),(1^4,0^3),(1^5,0^2),(1^6,0^1),(1^7). These lattice points are the representatives of sets of lattice points formed by the signed permutation of the representative lattice point. The number of absolute leader classes as function of the infinity norm in a d-dimensional integer lattice is given by C(d+n-1,n). This sequence has been found by creating a histogram of the perimeters of the rectangles found in sequence A240934 and counting the ones with frequency 1.

			For n=1 the a(1)=1 unique perimeter is found in the absolute leader class (1^2,0^5). The perimeters of rectangles that are found in the absolute leader classes (1,0^6), (1^3,0^4), (1^4,0^3), (1^5,0^2), (1^6,0^1), (1^7) generate perimeters with multiplicity higher than 1.

P. A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014.
P. A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
Philippe A. J. G. Chevalier, Dimensional exploration techniques for photonics, Slides of a talk, 2016.
Patrick Rault and Christine M. Guillemot, Lattice vector quantization with reduced or without look-up table, Proc. SPIE 3309, Visual Communications and Image Processing '98,(1998), 851.
Adriana Vasilache and Ioan Tabus, Image coding using multiple scale leader lattice vector quantization, Proc. SPIE 5014, Image Processing: Algorithms and Systems II, 9(2003).

Cf. A240934.

A240934 Number of rectangles formed by the absolute leader classes of the seven dimensional integer lattice as function of the infinity norm n, where the rectangles have one common lattice point being the origin of the seven dimensional integer lattice.

Original entry on oeis.org

120, 7196, 162554, 1341957, 9255603, 40532530, 168302117, 523421602, 1637895896, 4129547423

Offset: 1

Philippe A.J.G. Chevalier, Aug 03 2014

Philippe A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
Philippe A. J. G. Chevalier, Dimensional exploration techniques for photonics, Slides of a talk, 2016.

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User: Philippe A.J.G. Chevalier

A270950 Number of distinct cardinalities of orbits of lattice points under the automorphism group of the n-dimensional integer lattice.

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A266398 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 13440.

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A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

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A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

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A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

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A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

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A247557 Number of rectangles formed by the absolute leader classes of the seven-dimensional integer lattice as a function of the infinity norm n and having a unique perimeter, where the rectangles have one common lattice point being the origin of the seven-dimensional integer lattice.

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A240934 Number of rectangles formed by the absolute leader classes of the seven dimensional integer lattice as function of the infinity norm n, where the rectangles have one common lattice point being the origin of the seven dimensional integer lattice.

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