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.

A328685 Row sums of A309038.

Original entry on oeis.org

0, 4, 28, 120, 320, 716, 1380, 2464, 3984, 6196, 9124, 13128, 18048, 24476, 32244, 42096, 53440, 67460, 83604, 103192, 124944, 150892, 179908, 214080, 251184, 294356, 341700, 396264, 454624, 521276, 593364, 675088, 761568, 858916, 963124, 1079736, 1202160, 1338380
Offset: 0

Views

Author

Stefano Spezia, Oct 25 2019

Keywords

Comments

All the terms are even.

Links

Crossrefs

Cf. A000583, A309038, A326118, A327479, A327480.

Programs

  • Mathematica
    (* The function T is defined in A309038. *)
Flatten[Table[Sum[T[n, k], {k, 0, n^2}], {n, 0, 37}]]

Formula

Conjectures from Colin Barker, Oct 25 2019: (Start)
G.f.: 4*x*(1 + 5*x + 17*x^2 + 27*x^3 + 46*x^4 + 52*x^5 + 54*x^6 + 28*x^7 + 29*x^8 - 7*x^9+ 5*x^10 - 17*x^11 + 4*x^12 - 6*x^13 + 12*x^14 - 14*x^15 + 8*x^16 - 4*x^17) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3).
a(n) = 2*a(n-1) - a(n-2) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) - 3*a(n-8) + 6*a(n-9) - 3*a(n-10) + a(n-12) - 2*a(n-13) + a(n-14) for n > 18.
(End)
a(n) ~ 5*n^4/8. - Conjectured by Stefano Spezia, Sep 08 2021

A326118 a(n) is the largest number of squares of unit area connected only at corners and without holes that can be inscribed in an n X n square.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 14, 21, 24, 29, 36, 45, 50, 57, 66, 77, 84, 93, 104, 117, 126, 137, 150, 165, 176, 189, 204, 221, 234, 249, 266, 285, 300, 317, 336, 357, 374, 393, 414, 437, 456, 477, 500, 525, 546, 569, 594, 621, 644, 669, 696, 725, 750, 777, 806, 837, 864, 893
Offset: 0

Views

Author

Stefano Spezia, Sep 10 2019

Keywords

Comments

a(n) is equal to h_4(n) as defined in A309038.
a(n) is the maximum size of an induced subtree in the graph of the black squares of an n X n checkerboard, where edges connect diagonally adjacent squares. - Andrew Howroyd, Sep 10 2019

Examples

			Illustrations for n = 1..7:
     __              __              __    __
    |__|            |__|__          |__|__|__|
                       |__|          __|__|__
                                    |__|  |__|
    a(1) = 1        a(2) = 2         a(3) = 5
     __    __                  __    __
    |__|__|__|                |__|__|__|
     __|__|__                  __|__|__    __
    |__|  |__|__              |__|  |__|__|__|
             |__|                    __|__|__
                                    |__|  |__|
        a(4) = 6                  a(5) = 9
     __    __    __      __    __    __    __
    |__|__|__|  |__|__  |__|__|__|  |__|__|__|
     __|__|__    __|__|  __|__|__    __|__|__
    |__|  |__|__|__|    |__|  |__|__|__|  |__|
     __    __|__|__      __    __|__|__    __
    |__|__|__|  |__|__  |__|__|__|  |__|__|__|
       |__|        |__|  __|__|__    __|__|__
                        |__|  |__|  |__|  |__|
       a(6) = 14              a(7) = 21

Links

Crossrefs

Cf. A000290, A309038, A338329 (1st differences).

Programs

  • Magma
    I:=[0, 1, 2, 5, 6, 9, 14, 21, 24]; [n le 9 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..58]];
  • Mathematica
    Join[{0,1,2},Table[(1/8)*(-29+12*n+2*n^2-3(-1)^n-12*Sin[n*Pi/2]),{n,3,57}]]
  • PARI
    concat([0], Vec(x*(-1-2*x^2+2*x^3-x^4-2*x^5+2*x^7)/((-1+x)^3*(1+x)*(1+x^2))+O(x^58)))

Formula

O.g.f.: x*(1 + 2*x^2 - 2*x^3 + x^4 + 2*x^5 - 2*x^7)/((1 - x)^3*(1 + x)*(1 + x^2)).
E.g.f.: -3*exp(-x)/8 + (2 + x)^2 + exp(x)/8*(-29 + 2*x*(7 + x)) - 3*sin(x)/2.
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 8.
a(n) = (1/8)*(-29 + 12*n + 2*n^2 - 3*(-1)^n - 12*sin(n*Pi/2)) for n > 2, a(0) = 0, a(1) = 1, a(2) = 2.
Limit_{n->oo} a(n)/A000290(n) = 1/4.

A327480 a(n) is the maximum number of squares of unit area that can be removed from an n X n square while still obtaining a connected figure without holes and of the longest perimeter.

Original entry on oeis.org

0, 0, 2, 4, 8, 12, 22, 28, 40, 48, 64, 76, 94, 108, 130, 148, 172, 192, 220, 244, 274, 300, 334, 364, 400, 432, 472, 508, 550, 588, 634, 676, 724, 768, 820, 868, 922, 972, 1030, 1084, 1144, 1200, 1264, 1324, 1390, 1452, 1522, 1588, 1660, 1728, 1804, 1876, 1954
Offset: 0

Views

Author

Stefano Spezia, Sep 16 2019

Keywords

Comments

a(n) is equal to h_1(n) + h_2(n) as defined in A309038.

Examples

			Illustrations for n = 2..7:
      __                      __    __                __    __
     |__|__                  |__|__|__|              |__|__|__|
        |__|                  __|__|__                __|__|__ __
                             |__|  |__|              |__|  |     |
                                                           |__ __|
      a(2) = 2                a(3) = 4                  a(4) = 8
   __    __ __ __     __    __    __         __    __    __    __
  |__|__|__ __ __|   |__|__|__|  |__|__     |__|__|__|  |__|__|__|
   __|__|__    __     __|__|__    __|__|     __|__|__    __|__|__
  |  |  |__|__|__|   |__|  |__|__|__|       |__|  |__|__|__|  |__|
  |  |   __|__|__     __    __|__|__         __    __|__|__    __
  |__|  |__|  |__|   |__|__|__|  |__|__     |__|__|__|  |__|__|__|
                        |__|        |__|     __|__|__    __|__|__
                                            |__|  |__|  |__|  |__|
      a(5) = 12           a(6) = 22               a(7) = 28

Links

Crossrefs

Cf. A000290, A056594, A309038, A326118, A327479.

Programs

  • Magma
    I:=[0, 0, 2, 4, 8, 12, 22, 28, 40, 48, 64]; [n le 11 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..53]];
  • Maple
    gf := (1/24)*exp(-x)*(33+9*exp(2*x)*(2*x^2-2*x+7)-2*exp(x)*(x^4+12*x^2+48)-12*exp(x)*sin(x)); ser := series(gf, x, 53):
seq(factorial(n)*coeff(ser, x, n), n = 0 .. 52)
  • Mathematica
    Join[{0,0,2,4,8},Table[(1/8)*(21-12n+6n^2+11*(-1)^n-4*Sin[n*Pi/2]),{n,5,52}]]
  • PARI
    concat([0, 0], Vec(2*x^2*(1+x^2+2*x^4-2*x^5+2*x^6-2*x^7+x^8)/((1-x)^3*(1+x)*(1+x^2))+O(x^53)))

Formula

O.g.f.: 2*x^2*(1 + x^2 + 2*x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8)/((1 - x)^3*(1 + x)*(1 + x^2)).
E.g.f.: (1/24)*exp(-x)*(33 + 9*exp(2*x)*(7 - 2*x + 2*x^2) - 2*exp(x)*(48 + 12*x^2 + x^4) - 12*exp(x)*sin(x)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 10.
a(n) = (1/8)*(21 - 12*n + 6*n^2 + 11*(-1)^n + 4*A056594(n+1)) for n > 4, a(0) = 0, a(1) = 0, a(2) = 2, a(3) = 4, a(4) = 8. [corrected by Jason Yuen, Dec 17 2024]
Limit_{n->oo} a(n)/A000290(n) = 3/4.

A327479 a(n) is the minimum number of squares of unit area that must be removed from an n X n square to obtain a connected figure without holes and of the longest perimeter.

Original entry on oeis.org

0, 0, 0, 4, 6, 12, 16, 28, 32, 44, 52, 68, 76, 92, 104, 124, 136, 156, 172, 196, 212, 236, 256, 284, 304, 332, 356, 388, 412, 444, 472, 508, 536, 572, 604, 644, 676, 716, 752, 796, 832, 876, 916, 964, 1004, 1052, 1096, 1148, 1192, 1244, 1292, 1348, 1396, 1452, 1504
Offset: 0

Views

Author

Stefano Spezia, Sep 16 2019

Keywords

Comments

a(n) is equal to h_1(n) as defined in A309038.
All the terms are even numbers (A005843).

Examples

			Illustrations for n = 3..8:
      __    __               __    __.__             __    __.__.__
     |__|__|__|             |__|__|__.__|           |__|__|__.__.__|
      __|__|__               __|__|__.__             __|__|__    __
     |__|  |__|             |  |  |     |           |  |  |__|__|__|
                            |__|  |__.__|           |  |   __|__|__
                                                    |__|  |__|  |__|
      a(3) = 4                a(4) = 6                  a(5) = 12
   __    __    __.__     __    __    __    __     __    __    __    __.__
  |__|__|__|  |__   |   |__|__|__|  |__|__|__|   |__|__|__|  |__|__|__   |
   __|__|__    __|  |    __|__|__    __|__|__     __|__|__    __|  |  |__|
  |__|  |__|__|__.__|   |__|  |__|__|__|  |__|   |__|  |__|__|__.__|   __
   __    __|__|__.__     __    __|__|__    __     __    __|__|__    __|  |
  |  |__|  |  |     |   |__|__|__|  |__|__|__|   |__|__|  |  |__|__|__.__|
  |__.__.__|  |__.__|    __|__|__    __|__|__     __|__.__|   __|__|__.__
                        |__|  |__|  |__|  |__|   |  |__    __|  |  |     |
                                                 |__.__|  |__.__|  |__.__|
     a(6) = 16                a(7) = 28                 a(8) = 32

Links

Crossrefs

Cf. A000290, A005843, A309038, A326118, A327480.

Programs

  • Magma
    I:=[0, 0, 0, 4, 6, 12, 16, 28, 32, 44, 52]; [n le 11 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..55]];
  • Maple
    gf := 8+4*x+2*x^2+(1/12)*x^4+1/4*(-7*exp(-x)+exp(x)*(2*x^2+6*x-25)-4*sin(x)):
ser := series(gf, x, 55): seq(factorial(n)*coeff(ser, x, n), n = 0..54);
  • Mathematica
    Join[{0,0,0,4,6},Table[(1/4)*(-25+2n*(2+n)-7*Cos[n*Pi]-4*Sin[n*Pi/2]),{n,5,54}]]
  • PARI
    concat([0, 0, 0], Vec(2*x^3*(-2+x-2*x^2+x^3-2*x^4+3*x^5-2*x^6+x^7)/((-1+x)^3*(1+x+x^2+x^3))+O(x^55)))

Formula

O.g.f.: 2*x^3*(-2 + x - 2*x^2 + x^3 - 2*x^4 + 3*x^5 - 2*x^6 + x^7)/((-1 + x)^3*(1 + x + x^2 + x^3)).
E.g.f.: 8 + 4*x + 2*x^2 + x^4/12 + (1/4)*(-7*exp(-x) + exp(x)*(-25 + 6*x + 2*x^2) - 4*sin(x)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 10.
a(n) = (1/4)*(- 25 + 2*n*(2 + n) - 7*cos(n*Pi) - 4*sin(n*Pi/2)) for n > 4, a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 4, a(4) = 6.
Lim_{n->inf} a(n)/A000290(n) = 1/2.
Showing 1-4 of 4 results.

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