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.

Showing 1-4 of 4 results.

A282613 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to rotations.

Original entry on oeis.org

0, 1, 140, 4995, 65824, 489125, 2521476, 10092775, 33562880, 96870249, 250025500, 589527851, 1290008160, 2651218765, 5165397524, 9611031375, 17180133376, 29647326545, 49590297900, 80672546899, 128000804000, 198571037301, 301818598180, 450289780535
Offset: 0

Views

Author

David Nacin, Feb 19 2017

Keywords

Comments

Cycle index of symmetry group (cyclic rotation group of order 4 acting on the 9 cells of the square) is (2s(4)^2*s(1) + s(2)^4*s(1) + s(1)^9)/4.

Examples

			The number of 3 X 3 binary matrices up to rotations is 140.

Links

Crossrefs

Row n=3 of A343095.
Cf. A282612, A282614, A217331, A168555.
Cf. A006528 (2 x 2 version), A283027 (4 X 4 version).

Programs

  • Mathematica
    Table[(2n^3+n^5+n^9)/4, {n, 0, 24}]
  • PARI
    concat(0, Vec(x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

Formula

a(n) = n^3*(n^2+1)*(n^4-n^2+2)/4.
G.f.: x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10. - Colin Barker, Feb 23 2017

A282614 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to vertical and horizontal reflections.

Original entry on oeis.org

0, 1, 168, 5346, 67840, 496875, 2544696, 10151428, 33693696, 97135605, 250525000, 590412966, 1291500288, 2653631071, 5169160920, 9616725000, 17188519936, 29659392873, 49607301096, 80696066410, 128032800000, 198613915731, 301875282808, 450363792396
Offset: 0

Views

Author

David Nacin, Feb 19 2017

Keywords

Comments

Cycle index of symmetry group is (2*s(2)^3*s(1)^3 + s(2)^4*s(1) + s(1)^9)/4.

Examples

			The number of 3 X 3 binary matrices up to vertical and horizontal reflections is 168.

Links

Crossrefs

Cf. A282613, A282614, A217331, A168555. (For 2x2 version see A039623.)

Programs

  • Mathematica
    Table[(2n+1+n^4)n^5/4, {n, 0, 24}]
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,168,5346,67840,496875,2544696,10151428,33693696,97135605},30] (* Harvey P. Dale, Oct 01 2024 *)
  • PARI
    concat(0, Vec(x*(1 + 158*x + 3711*x^2 + 21820*x^3 + 39095*x^4 + 22254*x^5 + 3577*x^6 + 104*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

Formula

a(n) = n^5*(n+1)*(n^3-n^2+n+1)/4.
G.f.: x*(1 + 158*x + 3711*x^2 + 21820*x^3 + 39095*x^4 + 22254*x^5 + 3577*x^6 + 104*x^7) / (1 - x)^10. - Colin Barker, Feb 23 2017

A282612 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to row permutations.

Original entry on oeis.org

0, 1, 120, 3654, 45760, 333375, 1703016, 6784540, 22500864, 64836045, 167167000, 393877506, 861456960, 1769830699, 3447273480, 6412923000, 11461636096, 19776716505, 33076889784, 53804808190, 85365336000, 132422893911, 201268229800, 300266132244, 440396812800
Offset: 0

Views

Author

David Nacin, Feb 19 2017

Keywords

Comments

Cycle index of symmetry group is (3*s(2)^3*s(1)^3 + 2*s(3)^3 + s(1)^9)/6.

Examples

			The number of 3 X 3 binary matrices up to row permutations is 120.

Links

Crossrefs

Cf. A282613, A282614, A217331, A168555. A037270 (2x2 version.)

Programs

  • Mathematica
    Table[(3n^6+2n^3+n^9)/6, {n, 0, 24}]
  • PARI
    concat(0, Vec(x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

Formula

a(n) = n^3*(n^3+2)*(n+1)*(n^2-n+1)/6.
G.f.: x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10. - Colin Barker, Feb 23 2017

A283029 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.

Original entry on oeis.org

0, 1, 16793600, 423651479175, 562950490292224, 149011627197265625, 14215144250057342976, 670534312205763205375, 18889465949070766899200, 358948993948871860432449, 5000000000500000000000000, 54173529719030485105622951, 476981083228048575587942400
Offset: 0

Views

Author

David Nacin, Feb 27 2017

Keywords

Comments

Cycle index of symmetric group S2 acting on the set of 25 entries is (s(2)^10*s(1)^5 + s(1)^25)/2.

Examples

			For n=2 we get a(2)=16793600 inequivalent 5x5 binary matrices up to the action of transposition.

Crossrefs

Cf. A282612,A282613,A282614. A283026, A283027, A283028, A283030, A283031, A283032, A283033. A170798 (4x4 version). A168555 (3x3 version). A019582 (2x2 version)

Programs

  • Mathematica
    Table[n^15 (n^2 + 1) (n^8 - n^6 + n^4 - n^2 + 1)/2, {n, 0, 12}]
  • PARI
    a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2; \\ Indranil Ghosh, Feb 27 2017
  • Python
    def A283029(n): return n**15*(n**2+1)*(n**8-n**6+n**4-n**2+1)/2 # Indranil Ghosh, Feb 27 2017

Formula

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 16793574*x^23 + 423214845900*x^22 + 551941009751074*x^21 + 134512557517054626*x^20 + 10522699609491808746*x^19 + 347912001753554722204*x^18 + 5696453728178627889150*x^17 + 50977946159336791604079*x^16 + 265857130683340877431996*x^15 + 842694350441988138095256*x^14 + 1667306282568523129263444*x^13 + 2089823554970188253479900*x^12 + 1667306282568523129263444*x^11 + 842694350441988138095256*x^10 + 265857130683340877431996*x^9 + 50977946159336791604079*x^8 + 5696453728178627889150*x^7 + 347912001753554722204*x^6 + 10522699609491808746*x^5 + 134512557517054626*x^4 + 551941009751074*x^3 + 423214845900*x^2 + 16793574*x + 1)/(x - 1)^26. (End)
Showing 1-4 of 4 results.

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