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 1-10 of 11 results. Next

A224334 Number of idempotent 3 X 3 0..n matrices of rank 2.

Original entry on oeis.org

21, 51, 93, 147, 213, 291, 381, 483, 597, 723, 861, 1011, 1173, 1347, 1533, 1731, 1941, 2163, 2397, 2643, 2901, 3171, 3453, 3747, 4053, 4371, 4701, 5043, 5397, 5763, 6141, 6531, 6933, 7347, 7773, 8211, 8661, 9123, 9597, 10083, 10581, 11091, 11613, 12147
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Row 3 of A224333.

Examples

			Some solutions for n=3:
..1..2..0....1..0..0....1..0..0....1..0..1....1..0..2....0..0..0....0..0..0
..0..0..0....1..0..2....0..1..0....0..1..3....0..1..2....3..1..0....1..1..0
..0..1..1....0..0..1....0..0..0....0..0..0....0..0..0....1..0..1....2..0..1

Links

Crossrefs

Cf. A224333.

Programs

Formula

a(n) = 6*n^2 + 12*n + 3.
From Colin Barker, Feb 23 2018: (Start)
G.f.: 3*x*(7 - 4*x + x^2) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
(End)

A224327 Number of idempotent n X n 0..2 matrices of rank n-1.

Original entry on oeis.org

1, 10, 51, 212, 805, 2910, 10199, 34984, 118089, 393650, 1299067, 4251516, 13817453, 44641030, 143489055, 459165008, 1463588497, 4649045850, 14721978563, 46490458660, 146444944821, 460255540910, 1443528741991, 4518872583672
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Column 2 of A224333.

Examples

			Some solutions for n=3:
..1..0..0....1..0..0....1..0..0....1..1..0....1..0..0....1..0..1....1..0..0
..0..0..2....0..1..0....0..1..2....0..0..0....0..0..1....0..1..2....0..1..0
..0..0..1....0..0..0....0..0..0....0..2..1....0..0..1....0..0..0....1..0..0

Links

Crossrefs

Cf. A224333.

Formula

a(n) = n*(2*3^(n-1)-1).
a(n) = 8*a(n-1) - 22*a(n-2) + 24*a(n-3) - 9*a(n-4).
G.f.: x*(1 + 2*x - 7*x^2) / ((1 - x)^2*(1 - 3*x)^2). - Colin Barker, Aug 29 2018

A224328 Number of idempotent n X n 0..3 matrices of rank n-1.

Original entry on oeis.org

1, 14, 93, 508, 2555, 12282, 57337, 262136, 1179639, 5242870, 23068661, 100663284, 436207603, 1879048178, 8053063665, 34359738352, 146028888047, 618475290606, 2611340115949, 10995116277740, 46179488366571, 193514046488554
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Column 3 of A224333

Examples

			Some solutions for n=3
..0..1..3....1..0..0....0..0..0....0..0..0....0..1..0....1..3..0....1..0..0
..0..1..0....0..0..1....0..1..0....2..1..0....0..1..0....0..0..0....0..1..0
..0..0..1....0..0..1....1..0..1....0..0..1....0..0..1....0..3..1....0..3..0

Links

Formula

a(n) = n*(2*4^(n-1)-1).
a(n) = 10*a(n-1) -33*a(n-2) +40*a(n-3) -16*a(n-4).
G.f.: -x*(-1-4*x+14*x^2) / ( (4*x-1)^2*(x-1)^2 ). - R. J. Mathar, Oct 21 2014

A224329 Number of idempotent n X n 0..4 matrices of rank n-1.

Original entry on oeis.org

1, 18, 147, 996, 6245, 37494, 218743, 1249992, 7031241, 39062490, 214843739, 1171874988, 6347656237, 34179687486, 183105468735, 976562499984, 5187988281233, 27465820312482, 144958496093731, 762939453124980
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Column 4 of A224333.

Examples

			Some solutions for n=3:
..0..0..0....1..0..0....0..4..2....1..0..0....1..0..0....1..0..0....1..2..0
..3..1..0....0..1..2....0..1..0....1..0..3....0..1..3....0..0..0....0..0..0
..3..0..1....0..0..0....0..0..1....0..0..1....0..0..0....0..0..1....0..0..1

Links

Crossrefs

Cf. A224333.

Programs

  • PARI
    Vec(x*(1 + 6*x - 23*x^2) / ((1 - x)^2*(1 - 5*x)^2) + O(x^40)) \\ Colin Barker, Aug 29 2018

Formula

a(n) = n*(2*5^(n-1)-1).
a(n) = 12*a(n-1) - 46*a(n-2) + 60*a(n-3) - 25*a(n-4).
G.f.: x*(1 + 6*x - 23*x^2) / ((1 - x)^2*(1 - 5*x)^2). - Colin Barker, Aug 29 2018

A224330 Number of idempotent n X n 0..5 matrices of rank n-1.

Original entry on oeis.org

1, 22, 213, 1724, 12955, 93306, 653177, 4478968, 30233079, 201553910, 1330255861, 8707129332, 56596340723, 365699432434, 2350924922865, 15045919506416, 95917736853487, 609359740010478, 3859278353399789, 24374389600419820
Offset: 1

Views

Author

R. H. Hardin, formula from M. F. Hasler, William J. Keith and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Column 5 of A224333.

Examples

			Some solutions for n=3:
  0 5 0    1 0 0    1 0 0    0 0 0    0 3 3    0 0 0    1 5 0
  0 1 0    0 1 2    5 0 1    4 1 0    0 1 0    3 1 0    0 0 0
  0 0 1    0 0 0    0 0 1    2 0 1    0 0 1    1 0 1    0 4 1

Links

Programs

  • Mathematica
    Table[n*(2*6^(n-1)-1),{n, 1, 40}] (* or *)
CoefficientList[Series[(1 + 8*x - 34*x^2) / ((1 - x)^2*(1 - 6*x)^2), {x, 0, 40}], x] (* Stefano Spezia, Aug 29 2018 *)
  • PARI
    Vec(x*(1 + 8*x - 34*x^2) / ((1 - x)^2*(1 - 6*x)^2) + O(x^40)) \\ Colin Barker, Aug 29 2018

Formula

a(n) = n*(2*6^(n-1) - 1).
a(n) = 14*a(n-1) - 61*a(n-2) + 84*a(n-3) - 36*a(n-4).
G.f.: x*(1 + 8*x - 34*x^2) / ((1 - x)^2*(1 - 6*x)^2). - Colin Barker, Aug 29 2018

A224331 Number of idempotent n X n 0..6 matrices of rank n-1.

Original entry on oeis.org

1, 26, 291, 2740, 24005, 201678, 1647079, 13176680, 103766409, 807072130, 6214455467, 47455841820, 359873467213, 2712892291382, 20346692185455, 151921968318160, 1129919639366417, 8374698503539434, 61879716720597043
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Column 6 of A224333.

Examples

			Some solutions for n=3:
..1..5..0....0..3..6....1..0..0....0..0..0....1..0..0....1..0..0....0..0..0
..0..0..0....0..1..0....0..1..0....6..1..0....0..1..0....0..1..0....2..1..0
..0..2..1....0..0..1....0..0..0....4..0..1....1..6..0....3..4..0....3..0..1

Links

Crossrefs

Cf. A224333.

Programs

  • Mathematica
    Table[n*(2*7^(n-1)-1), {n, 1, 40}] (* or *)
CoefficientList[Series[(1 + 10*x - 47*x^2) / ((1 - x)^2*(1 - 7*x)^2) , {x, 0, 40}], x] (* Stefano Spezia, Aug 29 2018 *)
  • PARI
    Vec(x*(1 + 10*x - 47*x^2) / ((1 - x)^2*(1 - 7*x)^2) + O(x^40)) \\ Colin Barker, Aug 29 2018

Formula

a(n) = n*(2*7^(n-1)-1).
a(n) = 16*a(n-1) - 78*a(n-2) + 112*a(n-3) - 49*a(n-4).
G.f.: x*(1 + 10*x - 47*x^2) / ((1 - x)^2*(1 - 7*x)^2). - Colin Barker, Aug 29 2018

A224332 Number of idempotent n X n 0..7 matrices of rank n-1.

Original entry on oeis.org

1, 30, 381, 4092, 40955, 393210, 3670009, 33554424, 301989879, 2684354550, 23622320117, 206158430196, 1786706395123, 15393162788850, 131941395333105, 1125899906842608, 9570149208162287, 81064793292668910, 684547143360315373
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Column 7 of A224333.

Examples

			Some solutions for n=3:
..1..0..3....0..0..4....0..3..0....0..0..1....1..0..6....0..7..7....1..0..3
..0..1..3....0..1..0....0..1..0....0..1..0....0..1..4....0..1..0....0..1..6
..0..0..0....0..0..1....0..0..1....0..0..1....0..0..0....0..0..1....0..0..0

Links

Crossrefs

Cf. A224333.

Programs

  • Mathematica
    Table[n*(2*8^(n-1)-1), {n, 1, 40}] (* Stefano Spezia, Aug 29 2018 *)
  • PARI
    Vec(x*(1 + 12*x - 62*x^2) / ((1 - x)^2*(1 - 8*x)^2) + O(x^40)) \\ Colin Barker, Aug 29 2018
  • PARI
    a(n) = n*(2*8^(n-1)-1); \\ Altug Alkan, Aug 31 2018

Formula

a(n) = n*(2*8^(n-1)-1).
a(n) = 18*a(n-1) - 97*a(n-2) + 144*a(n-3) - 64*a(n-4).
G.f.: x*(1 + 12*x - 62*x^2) / ((1 - x)^2*(1 - 8*x)^2). - Colin Barker, Aug 29 2018

A224335 Number of idempotent 4X4 0..n matrices of rank 3.

Original entry on oeis.org

60, 212, 508, 996, 1724, 2740, 4092, 5828, 7996, 10644, 13820, 17572, 21948, 26996, 32764, 39300, 46652, 54868, 63996, 74084, 85180, 97332, 110588, 124996, 140604, 157460, 175612, 195108, 215996, 238324, 262140, 287492, 314428, 342996, 373244
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Row 4 of A224333.

Examples

			Some solutions for n=3
..1..0..0..0....1..0..0..0....1..0..0..0....1..0..0..3....1..0..0..0
..0..1..0..0....2..0..1..1....0..1..1..0....0..1..0..2....0..1..0..0
..3..2..0..0....0..0..1..0....0..0..0..0....0..0..1..1....1..2..0..1
..0..0..0..1....0..0..0..1....0..0..2..1....0..0..0..0....0..0..0..1

Links

Programs

  • PARI
    Vec(-4*x*(x^3-5*x^2+7*x-15)/(x-1)^4 + O(x^100)) \\ Colin Barker, Sep 20 2014

Formula

a(n) = 8*n^3 + 24*n^2 + 24*n + 4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Colin Barker, Sep 20 2014
G.f.: -4*x*(x^3-5*x^2+7*x-15) / (x-1)^4. - Colin Barker, Sep 20 2014

A224336 Number of idempotent 5 X 5 0..n matrices of rank 4.

Original entry on oeis.org

155, 805, 2555, 6245, 12955, 24005, 40955, 65605, 99995, 146405, 207355, 285605, 384155, 506245, 655355, 835205, 1049755, 1303205, 1599995, 1944805, 2342555, 2798405, 3317755, 3906245, 4569755, 5314405, 6146555, 7072805, 8099995, 9235205, 10485755, 11859205, 13363355
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Examples

			Some solutions for n=3:
..1..0..0..1..0....0..0..0..0..0....0..2..3..2..2....1..0..0..1..0
..0..1..0..1..0....1..1..0..0..0....0..1..0..0..0....0..1..0..2..0
..0..0..1..2..0....3..0..1..0..0....0..0..1..0..0....0..0..1..2..0
..0..0..0..0..0....2..0..0..1..0....0..0..0..1..0....0..0..0..0..0
..0..0..0..2..1....3..0..0..0..1....0..0..0..0..1....0..0..0..1..1

Links

Crossrefs

Row 5 of A224333.

Programs

  • PARI
    Vec(-5*x*(x^4-6*x^3+16*x^2+6*x+31)/(x-1)^5 + O(x^100)) \\ Colin Barker, Sep 20 2014

Formula

a(n) = 10*n^4 + 40*n^3 + 60*n^2 + 40*n + 5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Colin Barker, Sep 20 2014
G.f.: -5*x*(x^4-6*x^3+16*x^2+6*x+31) / (x-1)^5. - Colin Barker, Sep 20 2014
E.g.f.: 5*(exp(x)*(1 + 30*x + 50*x^2 + 20*x^3 + 2*x^4) - 1). - Stefano Spezia, Aug 25 2025

A224337 Number of idempotent 6X6 0..n matrices of rank 5.

Original entry on oeis.org

378, 2910, 12282, 37494, 93306, 201678, 393210, 708582, 1199994, 1932606, 2985978, 4455510, 6453882, 9112494, 12582906, 17038278, 22674810, 29713182, 38399994, 49009206, 61843578, 77236110, 95551482, 117187494, 142576506, 172186878
Offset: 1

Views

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

Comments

Row 6 of A224333.

Examples

			Some solutions for n=3
..1..0..0..0..0..0....1..0..0..0..0..0....1..0..1..0..0..0....1..0..0..0..2..0
..3..0..0..2..2..1....0..1..0..0..2..0....0..1..0..0..0..0....0..1..0..0..1..0
..0..0..1..0..0..0....0..0..1..0..2..0....0..0..0..0..0..0....0..0..1..0..1..0
..0..0..0..1..0..0....0..0..0..1..0..0....0..0..1..1..0..0....0..0..0..1..2..0
..0..0..0..0..1..0....0..0..0..0..0..0....0..0..3..0..1..0....0..0..0..0..0..0
..0..0..0..0..0..1....0..0..0..0..0..1....0..0..0..0..0..1....0..0..0..0..2..1

Links

Programs

  • PARI
    Vec(-6*x*(x^5-7*x^4+18*x^3-82*x^2-107*x-63)/(x-1)^6 + O(x^100)) \\ Colin Barker, Sep 20 2014

Formula

a(n) = 12*n^5 + 60*n^4 + 120*n^3 + 120*n^2 + 60*n + 6.
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). - Colin Barker, Sep 20 2014
G.f.: -6*x*(x^5-7*x^4+18*x^3-82*x^2-107*x-63) / (x-1)^6. - Colin Barker, Sep 20 2014
Showing 1-10 of 11 results. Next

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