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.

A152132 Maximal length of rook tour on an n X n+1 board.

Original entry on oeis.org

2, 8, 24, 54, 104, 174, 270, 396, 558, 756, 996, 1282, 1620, 2010, 2458, 2968, 3546, 4192, 4912, 5710, 6592, 7558, 8614, 9764, 11014, 12364, 13820, 15386, 17068, 18866, 20786, 22832, 25010, 27320, 29768, 32358, 35096, 37982, 41022, 44220, 47582
Offset: 1

Views

Author

R. J. Mathar, Mar 22 2009

Keywords

References

  • M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 76.

Links

Crossrefs

Cf. A006071, A087960, A152133, A152134, A152135.

Programs

  • Magma
    I:=[2, 8, 24, 54, 104, 174, 270]; [n le 7 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3) + Self(n-4) - 3*Self(n-5) + 3*Self(n-6)- Self(n-7): n in [1..50]]; // Vincenzo Librandi, Dec 14 2012
  • Maple
    # Figure 43 of the Gardner book:
C := proc(n,m)
if type(m,even) and type(n,even) then
2 ;
elif type(m,odd) and type(n,odd) then
1 ;
elif type(m,even) and type(n,odd) and type(floor(n/2),even) then
3/2 ;
elif type(m,even) and type(n,odd) and type(floor(n/2),odd) then
1/2 ;
elif type(m,odd) and type(n,even) and type(floor(n/2),even) then
0 ;
elif type(m,odd) and type(n,even) and type(floor(n/2),odd) then
1 ;
fi;
end:
# formula for n X m boards, from the Gardner book:
T := proc(n,m)
n*(3*m^2+n^2-10)/6+C(n,m) ;
end:
for n from 1 to 24 do
m := n+3 ; # third diagonal here, for example
printf("%d,",T(n,m)) ;
od:
  • Mathematica
    CoefficientList[Series[-2 * (-1 - x - 2*x^3 - 2*x^4 - 3*x^2 + x^5)/(1 + x)/(x^2 + 1)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 14 2012 *)

Formula

G.f.: -2*x*(-1-x-2*x^3-2*x^4-3*x^2+x^5)/(1+x)/(x^2+1)/(x-1)^4.
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7).
a(n) = 2*n^3/3+n^2-7*n/6+3/4-(-1)^n/4-A087960(n)/2.

Extensions

More terms from R. J. Mathar, Sep 22 2009

A152133 Maximal length of rook tour on an n X n+2 board.

Original entry on oeis.org

4, 16, 38, 78, 136, 220, 330, 474, 652, 872, 1134, 1446, 1808, 2228, 2706, 3250, 3860, 4544, 5302, 6142, 7064, 8076, 9178, 10378, 11676, 13080, 14590, 16214, 17952, 19812, 21794, 23906, 26148, 28528, 31046, 33710, 36520, 39484, 42602, 45882
Offset: 1

Views

Author

R. J. Mathar, Mar 22 2009

Keywords

References

  • M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 76.

Links

Crossrefs

Cf. A006071, A152132, A152134, A152135.

Programs

  • Magma
    I:=[4,16,38,78,136]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 11 2012
  • Mathematica
    LinearRecurrence[{3,-2,-2,3,-1},{4,16,38,78,136},40] (* Harvey P. Dale, Dec 16 2011 *)

Formula

G.f.: -2*x*(-2-2*x+x^2-2*x^3+x^4)/(1+x)/(x-1)^4.
a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5).
a(n) = 2*n^3/3+2*n^2+n/3+3/2+(-1)^n/2. [R. J. Mathar, Oct 20 2009]

A152135 Maximal length of rook tour on an n X n+4 board.

Original entry on oeis.org

12, 36, 74, 134, 216, 328, 470, 650, 868, 1132, 1442, 1806, 2224, 2704, 3246, 3858, 4540, 5300, 6138, 7062, 8072, 9176, 10374, 11674, 13076, 14588, 16210, 17950, 19808, 21792, 23902, 26146, 28524, 31044, 33706, 36518, 39480, 42600, 45878
Offset: 1

Views

Author

R. J. Mathar, Mar 22 2009

Keywords

References

  • M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 76.

Links

Crossrefs

Cf. A006071, A152132, A152133, A152134.

Programs

  • Magma
    I:=[12, 36, 74, 134, 216]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 11 2012
  • Mathematica
    LinearRecurrence[{3, -2, -2, 3, -1}, {12, 36, 74, 134, 216}, 40] (* Vincenzo Librandi, Dec 11 2012 *)

Formula

G.f.: -2*x*(-6+5*x^2-4*x^3+x^4)/(1+x)/(x-1)^4.
From R. J. Mathar, May 13 2010: (Start)
a(n) = +3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5).
a(n) = 19*n/3+3/2+2*n^3/3+4*n^2+(-1)^n/2. (End)

A152134 Maximal length of rook tour on an n X n+3 board.

Original entry on oeis.org

8, 24, 54, 102, 174, 270, 396, 556, 756, 996, 1282, 1618, 2010, 2458, 2968, 3544, 4192, 4912, 5710, 6590, 7558, 8614, 9764, 11012, 12364, 13820, 15386, 17066, 18866, 20786, 22832, 25008, 27320, 29768, 32358, 35094, 37982, 41022, 44220, 47580
Offset: 1

Views

Author

R. J. Mathar, Mar 22 2009

Keywords

References

  • M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 76.

Links

Crossrefs

Cf. A006071, A152132, A152133, A152135.

Programs

  • Magma
    I:=[8, 24, 54, 102, 174, 270, 396]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-4)-3*Self(n-5)+3*Self(n-6)-Self(n-7): n in [1..40]];// Vincenzo Librandi, Dec 19 2012
  • Mathematica
    CoefficientList[Series[-2*(- 4 - 3*x^2 - 2*x^3 + x^4)/(1+x)/(x^2+1)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 18 2012 *)

Formula

G.f.: -2*x*(-4-3*x^2-2*x^3+x^4)/(1+x)/(x^2+1)/(x-1)^4.
a(n) = 17*n/6+3/4+2*n^3/3+3*n^2+A132429(n+3)/4. - R. J. Mathar, Sep 27 2009
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). - Vincenzo Librandi, Dec 19 2012

Extensions

More terms from R. J. Mathar, Sep 27 2009

A152100 Expansion of g.f. 1 - 2*x*(-7 - 10*x + x^2)/(x - 1)^4.

Original entry on oeis.org

1, 14, 76, 218, 472, 870, 1444, 2226, 3248, 4542, 6140, 8074, 10376, 13078, 16212, 19810, 23904, 28526, 33708, 39482, 45880, 52934, 60676, 69138, 78352, 88350, 99164, 110826, 123368, 136822, 151220, 166594, 182976, 200398, 218892, 238490
Offset: 0

Views

Author

N. J. A. Sloane, Sep 21 2009, based on email from R. J. Mathar, Mar 22 2009

Keywords

Links

Crossrefs

Cf. A006071.

Programs

  • Mathematica
    Join[{1},LinearRecurrence[{4,-6,4,-1},{14,76,218,472},40]] (* Harvey P. Dale, Aug 07 2012 *)

Formula

From R. J. Mathar, Sep 22 2009: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n > 4.
a(n) = 2*n*(8*n^2 + 12*n + 1)/3, n > 0. (End)
E.g.f.: 1 + 2*exp(x)*x*(21 + 36*x + 8*x^2)/3. - Stefano Spezia, Aug 24 2025

Extensions

More terms from R. J. Mathar, Sep 22 2009

A152110 G.f.: -2*(-2 - 11*x - 4*x^2 + x^3)/(x - 1)^4.

Original entry on oeis.org

4, 38, 136, 330, 652, 1134, 1808, 2706, 3860, 5302, 7064, 9178, 11676, 14590, 17952, 21794, 26148, 31046, 36520, 42602, 49324, 56718, 64816, 73650, 83252, 93654, 104888, 116986, 129980, 143902, 158784, 174658, 191556, 209510, 228552, 248714
Offset: 0

Views

Author

N. J. A. Sloane, Sep 21 2009, based on email from R. J. Mathar, Mar 22 2009

Keywords

Links

Crossrefs

Cf. A006071.

Programs

  • Magma
    I:=[4,38,136,330]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 11 2012
  • Mathematica
    CoefficientList[Series[-2*(-2 - 11*x - 4*x^2 + x^3)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2012 *)

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). a(n) = 2*(n+1)*(8*n^2 + 8*n + 3)/3 = A152133(2*n-1). - R. J. Mathar, Sep 22 2009

Extensions

More terms from R. J. Mathar, Sep 22 2009
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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