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 1-6 of 6 results.

A283027 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to rotations.

Original entry on oeis.org

0, 1, 16456, 10763361, 1073758336, 38147070625, 705277897416, 8308234084801, 70368748374016, 463255057977921, 2500000025005000, 11487432519490081, 46221064831264896, 166354152499741921, 544488334821309256, 1642102089568970625, 4611686019501162496
Offset: 0

Views

Author

David Nacin, Feb 27 2017

Keywords

Comments

Cycle index of cyclic group C4 acting on the 16 entries is (2s(4)^4 + s(2)^8 + s(1)^16)/24.

Examples

			For n=2 we get a(2)=16456 inequivalent 4 X 4 binary matrices up to rotations.

Links

Crossrefs

Row n=4 of A343095.
Cf. A282612, A282614, A283026, A283028, A283029, A283030, A283032, A283033.
Cf. A283031 (5 X 5 version), A282613 (3 X 3 version), A006528 (2 X 2 version).

Programs

  • GAP
    List([0..20], n -> n^4*(n^4+1)*(n^8-n^4+2)/4); # G. C. Greubel, Dec 07 2018
  • Magma
    [n^4*(n^4+1)*(n^8-n^4+2)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018
  • Maple
    [n^4*(n^4+1)*(n^8-n^4+2)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018
  • Mathematica
    Table[n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4,{n,0,30}]
  • PARI
    a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4; \\ Indranil Ghosh, Feb 27 2017
  • Python
    def A283027(n) : return n**4*(n**4 + 1)*(n**8 - n**4 + 2)/4 # Indranil Ghosh, Feb 27 2017
  • Python
    def A283027(n): return n**4*(n**4*(n**8+1)+2)>>2 # Chai Wah Wu, Jan 24 2023
  • Sage
    [n^4*(n^4+1)*(n^8-n^4+2)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
G.f.: -x*(x + 1)*(x^14 + 16438*x^13 + 10467307*x^12 + 882551228*x^11 + 20463257081*x^10 + 175065647018*x^9 + 626805100059*x^8 + 968894657736*x^7 + 626805100059*x^6 + 175065647018*x^5 + 20463257081*x^4 + 882551228*x^3 + 10467307*x^2 + 16438*x + 1)/(x - 1)^17. (End)
E.g.f.: (1/4)*x*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764054*x^7 + 3281882632*x^6 + 2734926824*x^5 + 1096191600*x^4 + 171800604*x^3 + 7142664*x^2 + 32908*x + 4)*exp(x). - Stefano Spezia, Dec 07 2018
a(n) = n^4*(n^12 + n^4 + 2)/4. - Chai Wah Wu, Jan 24 2023

A283033 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations and reflections.

Original entry on oeis.org

0, 1, 4211744, 105918450471, 140738033618944, 37252918396015625, 3553786240466361696, 167633579843887699759, 4722366500530551259136, 89737248564744874067889, 1250000000501250002500000, 13543382431328404683826391, 119245270812803151147085824
Offset: 0

Views

Author

David Nacin, Feb 27 2017

Keywords

Comments

Cycle index of dihedral group D4 acting on the 25 entries is (2*s(4)^6*s(1) + s(2)^{12}*s(1) + 4*s(2)^10*s(1)^5 + s(1)^25)/8.

Examples

			For n=2 we get a(2)=4211744 inequivalent 5 X 5 binary matrices up to rotations and reflections.

Links

Crossrefs

Row n=5 of A343097.
Cf. A282612, A282613, A282614, A283026, A283027, A283028, A283029, A283030, A283031, A283032.
Cf. A217338 (4 X 4 version), A217331 (3 X 3 version), A002817 (2 X 2 version).

Programs

  • GAP
    List([0..20], n -> n^7*(n^18+4*n^8+n^6+2)/8); # G. C. Greubel, Dec 07 2018
  • Magma
    [n^7*(n^18+4*n^8+n^6+2)/8: n in [0..20]]; // G. C. Greubel, Dec 07 2018
  • Maple
    [n^7*(n^18+4*n^8+n^6+2)/8$n=0..16]; # Muniru A Asiru, Dec 07 2018
  • Mathematica
    Table[n^7 (n^18 + 4 n^8 + n^6 + 2)/8, {n, 0, 16}]
  • PARI
    a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8; \\ Indranil Ghosh, Feb 27 2017
  • Python
    def A283033(n): return n**7*(n**18 + 4*n**8 + n**6 + 2)/8 # Indranil Ghosh, Feb 27 2017
  • Sage
    [n^7*(n^18+4*n^8+n^6+2)/8 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8.
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 + 4211718*x^23 + 105808945452*x^22 + 137985522720898*x^21 + 33628142067806706*x^20 + 2630674898090394666*x^19 + 86978000386844370748*x^18 + 1424113432167998385342*x^17 + 12744486540004851097263*x^16 + 66464282669989885009756*x^15 + 210673587611186802329496*x^14 + 416826570643036689533748*x^13 + 522455888740564118388412*x^12 + 416826570643036689533748*x^11 + 210673587611186802329496*x^10 + 66464282669989885009756*x^9 + 12744486540004851097263*x^8 + 1424113432167998385342*x^7 + 86978000386844370748*x^6 + 2630674898090394666*x^5 + 33628142067806706*x^4 + 137985522720898*x^3 + 105808945452*x^2 + 4211718*x + 1)/(x - 1)^26. (End)

A283028 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

Original entry on oeis.org

0, 1, 16576, 10766601, 1073790976, 38147265625, 705278736576, 8308236966001, 70368756760576, 463255079498001, 2500000075000000, 11487432626662201, 46221065046245376, 166354152907593001, 544488335559184576, 1642102090850390625, 4611686021648613376
Offset: 0

Views

Author

David Nacin, Feb 27 2017

Keywords

Comments

Cycle index of dihedral group D2 acting on the 16 entries is (3s(2)^8 + s(1)^16)/4.

Examples

			For n=2 we get a(2)=16576 inequivalent 4 X 4 binary matrices up to vertical and horizontal reflections.

Links

Crossrefs

Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.
Cf. A283032 (5 X 5 version), A282614 (3 X 3 version), A039623 (2 X 2 version).

Programs

  • GAP
    List([0..20], n -> n^8*(n^8+3)/4); # G. C. Greubel, Dec 07 2018
  • Magma
    [n^8*(n^8+3)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018
  • Maple
    [n^8*(n^8+3)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018
  • Mathematica
    Table[n^8(n^8 + 3)/4,{n,0,30}]
  • PARI
    a(n) = n^8 * (n^8 + 3)/4; \\ Altug Alkan, Feb 27 2017
  • Python
    def A283028(n): return n**8*(n**8 + 3)/4 # Indranil Ghosh, Feb 27 2017
  • Sage
    [n^8*(n^8+3)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = n^8 * (n^8 + 3)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
G.f.: -x*(x + 1)*(x^14 + 16558*x^13 + 10468387*x^12 + 882544028*x^11 + 20463263441*x^10 + 175065686258*x^9 + 626804969739*x^8 + 968894839176*x^7 + 626804969739*x^6 + 175065686258*x^5 + 20463263441*x^4 + 882544028*x^3 + 10468387*x^2 + 16558*x + 1)/(x - 1)^17. (End)
E.g.f.: (1/4)*x*exp(x)*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764056*x^7 + 3281882688*x^6 + 2734927356*x^5 + 1096193700*x^4 + 171804004*x^3 + 7144584*x^2 + 33148*x + 4). - Stefano Spezia, Dec 07 2018

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)

A283030 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to row permutations.

Original entry on oeis.org

0, 1, 376992, 7355513529, 9474438804480, 2491483056641250, 237223883948569056, 11182222570880983622, 314920519245916176384, 5983496429606726016735, 83341666958337500020000, 902948225666983587054711, 7950004204832195461143552, 58805000552467321853765064
Offset: 0

Views

Author

David Nacin, Feb 27 2017

Keywords

Comments

Cycle index of symmetric group S4 on the set of 25 entries is (10*s(2)^5*s(1)^15 + 20*s(3)^5*s(1)^10 + 15*s(2)^10*s(1)^5 + 30*s(4)^5*s(1)^5 + 20*s(2)^5*s(3)^5 + 24*s(5)^5+s(1)^25)/120.

Examples

			For n=2 we get a(2)=376992 inequivalent 5 X 5 binary matrices up to row permutations.

Links

Crossrefs

Cf. A282613, A282614, A283027, A283028, A283029, A283031, A283032, A283033.
Cf. A283026 (4 X 4 version), A282612 (3 X 3 version), A037270 (2 X 2 version).

Programs

  • GAP
    List([0..20], n -> n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120); # G. C. Greubel, Dec 07 2018
  • Magma
    [n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120: n in [0..20]]; // G. C. Greubel, Dec 07 2018
  • Maple
    [(10*n^20+35*n^15+50*n^10+24*n^5+n^25)/120$n=0..16]; # Muniru A Asiru, Dec 07 2018
  • Mathematica
    Table[(10n^20+ 35n^15 + 50n^10 + 24n^5 + n^25)/120, {n, 0, 16}]
  • PARI
    a(n) = (10*n^20 + 35*n^15 + 50*n^10 + 24*n^5 + n^25)/120; \\ Indranil Ghosh, Feb 27 2017
  • Python
    def A283030(n): return (10*n**20 + 35*n**15 + 50*n**10 + 24*n**5 + n**25)/120 # Indranil Ghosh, Feb 27 2017
  • Sage
    [n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = (n^25 + 10*n^20 + 35*n^15 + 50*n^10 + 24*n^5)/120.
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*(201376*x^23 + 6769097812*x^22 + 9115118766616*x^21 + 2236218775591321*x^20 + 175251248958400030*x^19 + 5797456665826176046*x^18 + 94937993285056078902*x^17 + 849635569433212953261*x^16 + 4430970723887327210136*x^15 + 14044903652456409705760*x^14 + 27788396155245137222056*x^13 + 34830392581327241688322*x^12 + 27788479931754180338596*x^11 + 14044908029988217540516*x^10 + 4430933630938187561140*x^9 + 849629302807069561746*x^8 + 94943797840269544152*x^7 + 5799609980863901436*x^6 + 175505398388141776*x^5 + 2247537209457445*x^4 + 9283317972526*x^3 + 7345712062*x^2 + 376966*x + 1)/(x - 1)^26. (End)

A283032 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

Original entry on oeis.org

0, 1, 8407040, 211829725395, 281475530358784, 74505821533203125, 7107572245840091136, 335267157313994232775, 9444732983468915425280, 179474497026544179696969, 2500000000502500000000000, 27086764860568185273201371, 238490541617902791488962560
Offset: 0

Views

Author

David Nacin, Feb 27 2017

Keywords

Comments

Cycle index of dihedral group D2 acting on the 25 entries is (2*s(2)^10*s(1)^5 + s(2)^12*s(1) + s(1)^25)/4.

Examples

			For n=2 we get a(2)=8407040 inequivalent 5 X 5 binary matrices up to vertical and horizontal reflections.

Links

Crossrefs

Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.
Cf. A283028 (4 X 4 version), A282614 (3 X 3 version), A039623 (2 X 2 version).

Programs

  • GAP
    List([0..30], n -> n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4); # G. C. Greubel, Dec 07 2018
  • Magma
    [n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018
  • Maple
    [n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4$n=0..16]; # Muniru A Asiru, Dec 07 2018
  • Mathematica
    Table[n^13*(n^2 + 1)*(n^10- n^8 + n^6 - n^4 + n^2 + 1)/4, {n, 0, 16}]
  • PARI
    a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4; \\ Indranil Ghosh, Feb 27 2017
  • Python
    def A283032(n): return n**13*(n**2 + 1)*(n**10 - n**8 + n**6 - n**4 + n**2 + 1)/4 # Indranil Ghosh, Feb 27 2017
  • Sage
    [n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4.
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 + 8407014*x^23 + 211611142680*x^22 + 275970689783914*x^21 + 67256280546339066*x^20 + 5261349801742569906*x^19 + 173956000842889834744*x^18 + 2848226864171846366430*x^17 + 25488973079778181586319*x^16 + 132928565341114340101276*x^15 + 421347175221451531355376*x^14 + 833653141284854063434884*x^13 + 1044911777483791594156780*x^12 + 833653141284854063434884*x^11 + 421347175221451531355376*x^10 + 132928565341114340101276*x^9 + 25488973079778181586319*x^8 + 2848226864171846366430*x^7 + 173956000842889834744*x^6 + 5261349801742569906*x^5 + 67256280546339066*x^4 + 275970689783914*x^3 + 211611142680*x^2 + 8407014*x + 1)/(x - 1)^26. (End)
Showing 1-6 of 6 results.

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