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 1-7 of 7 results.

A318268 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that all but 3 such pairs are joined by an edge.

Original entry on oeis.org

0, 0, 0, 2, 34, 250, 1234, 4830, 16174, 48444, 133416, 344220, 843020, 1978804, 4484228, 9865742, 21166390, 44439910, 91570126, 185614242, 370846914, 731502296, 1426514540, 2753525208, 5266164280, 9987859912, 18799814312, 35141997050, 65274659562, 120540177522
Offset: 0

Views

Author

Donovan Young, Aug 22 2018

Keywords

Comments

This is also the number of "(n-3)-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

Examples

			See example in A318267.

Links

Crossrefs

Cf. A046741, A318243, A318244, A318267, A318269, A318270.

Programs

  • Mathematica
    CoefficientList[Normal[Series[x^2(2*x + 20*x^2 + 46*x^3 + 40*x^4 + 30*x^5 + 4*x^6 + 4*x^7)/(1 - x)^3/(1 - x - x^2)^4, {x, 0, 30}]], x]
LinearRecurrence[{7,-17,11,19,-29,-3,21,-3,-7,1,1},{0,0,0,2,34,250,1234,4830,16174,48444,133416},30] (* Harvey P. Dale, Aug 05 2019 *)

Formula

G.f.: x^2*(2*x + 20*x^2 + 46*x^3 + 40*x^4 + 30*x^5 + 4*x^6 + 4*x^7)/(1 - x)^3/(1 - x - x^2)^4 (conjectured).
The above conjecture is true. The PARI program given in the links can be used to establish an upper limit on the order of the linear recurrence and sufficient number of terms to prove correctness. - Andrew Howroyd, Sep 03 2018

Extensions

Terms a(14) and beyond from Andrew Howroyd, Sep 03 2018

A318243 Triangle read by rows giving the sum of the number of k-matchings of the graphs obtained by deleting one edge and its two vertices from the ladder graph L_n = P_2 X P_n in all possible ways.

Original entry on oeis.org

1, 4, 4, 7, 22, 9, 10, 58, 78, 20, 13, 112, 282, 224, 40, 16, 184, 702, 1052, 570, 78, 19, 274, 1419, 3260, 3335, 1338, 147, 22, 382, 2514, 7928, 12520, 9462, 2968, 272, 25, 508, 4068, 16460, 35955, 42108, 24766, 6312, 495, 28, 652, 6162, 30584, 86330, 140586, 128352, 60976, 12996, 890, 31, 814, 8877, 52352
Offset: 1

Views

Author

Donovan Young, Aug 22 2018

Keywords

Comments

T(n,k) is useful for computing the number of configurations of n indistinguishable pairs placed on the vertices of P_2 X P_n such that only one such pair is joined by an edge. The g.f. given below can be proven using the calculus of the rook polynomial associated with A046741.

Examples

			The first few rows of T(n,k) are
   1;
   4,   4;
   7,  22,   9;
  10,  58,  78,  20;
  13, 112, 282, 224,  40;
For n = 2, the four ways of deleting an edge and its vertices from P_2 X P_2 all yield a graph with two vertices joined by an edge. This graph has 1 0-matching and 1 1-matching, thus T(2,k) = 4, 4.

Links

Crossrefs

Cf. A046741, A318244, A318267, A318268, A318269, A318270.

Programs

  • Mathematica
    CoefficientList[Normal[Series[(1 + 2*t*z + 2*z/(1-t*z)^2)*(1-t*z)^2/(1 - z - 2*t*z - t*z^2 + t^3*z^3)^2,{z,0,10}]],{z,t}]//MatrixForm

Formula

G.f.: (1 + 2*t*z + 2*z/(1-t*z)^2)*(1-t*z)^2/(1 - z - 2*t*z - t*z^2 + t^3*z^3)^2.

A318244 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that only one such pair is joined by an edge.

Original entry on oeis.org

1, 0, 8, 34, 347, 3666, 47484, 707480, 11971341, 226599568, 4744010444, 108834109034, 2714992695407, 73169624071138, 2118530753728184, 65582753432993648, 2161565971116312537, 75572040870327124064, 2793429487732659591888, 108847840347732886117874, 4459207771645802095292995
Offset: 1

Views

Author

Donovan Young, Aug 22 2018

Keywords

Comments

This is a companion entry to A318243 and uses an inclusion-exclusion method on the matching numbers given there.
This is also the number of "1-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

Examples

			For the case n = 2, if one pair is joined by an edge, then the remaining pair is forced to be joined by the remaining edge. Thus a(2) = 0.

Links

Crossrefs

Cf. A046741, A318243, A318267, A318268, A318269, A318270. When no pair is joined by an edge, the number of configurations is given by A265167.

Formula

a(n) = Sum_{k=0..n-1} (-1)^k*(2*n-2*k-3)!! * A318243(n,k) where and 0!! = (-1)!! = 1; proved by inclusion-exclusion.

A318269 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that all but 4 such pairs are joined by an edge.

Original entry on oeis.org

0, 0, 0, 0, 21, 347, 2919, 17050, 78815, 309075, 1072617, 3386970, 9921030, 27338000, 71614370, 179788174, 435311905, 1021684125, 2333955085, 5207067714, 11377225161, 24403026561, 51484962205, 107024887620, 219528748908, 444886466640, 891735024852, 1769575953980
Offset: 0

Views

Author

Donovan Young, Aug 23 2018

Keywords

Comments

This is also the number of "(n-4)-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

Examples

			See example in A318267.

Links

Crossrefs

Cf. A046741, A318243, A318244, A318267, A318268, A318270.

Programs

  • Mathematica
    CoefficientList[Normal[Series[x^2(5*x^10 + 10*x^9 + 93*x^8 + 230*x^7 + 502*x^6 + 612*x^5 + 447*x^4 + 158*x^3 + 21*x^2)/(1 - x)^4/(1 - x - x^2)^5, {x, 0, 30}]], x]

Formula

G.f.: x^2*(5*x^10 + 10*x^9 + 93*x^8 + 230*x^7 + 502*x^6 + 612*x^5 + 447*x^4 + 158*x^3 + 21*x^2)/(1 - x)^4/(1 - x - x^2)^5 (conjectured).
The above conjecture is true. See A318268. - Andrew Howroyd, Sep 03 2018

Extensions

Terms a(14) and beyond from Andrew Howroyd, Sep 03 2018

A318270 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that all but 5 such pairs are joined by an edge.

Original entry on oeis.org

0, 0, 0, 0, 0, 186, 3666, 36714, 253386, 1369260, 6209700, 24668742, 88338174, 290968686, 894709790, 2597386330, 7181246394, 19040425628, 48684375292, 120592523460, 290476059204, 682548818802, 1568744083242, 3534725236308, 7823387477220, 17037467831748
Offset: 0

Views

Author

Donovan Young, Aug 23 2018

Keywords

Comments

This is also the number of "(n-5)-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

Examples

			See example in A318267.

Links

Crossrefs

Cf. A046741, A318243, A318244, A318267, A318268, A318269.

Programs

  • Mathematica
    CoefficientList[Normal[Series[x^2(6*x^13+20*x^12+228*x^11+888*x^10+3012*x^9+6612*x^8+10020*x^7+9636*x^6+5502*x^5+1620*x^4+186*x^3)/(1-x)^5/(1-x-x^2)^6,{x,0,30}]],x]

Formula

G.f.: x^2*(6*x^13 + 20*x^12 + 228*x^11 + 888*x^10 + 3012*x^9 + 6612*x^8 + 10020*x^7 + 9636*x^6 + 5502*x^5 + 1620*x^4 + 186*x^3)/(1 - x)^5/(1 - x - x^2)^6 (conjectured).
The above conjecture is true. See A318268. - Andrew Howroyd, Sep 03 2018

Extensions

Terms a(14) and beyond from Andrew Howroyd, Sep 03 2018

A325753 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by an edge.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 2, 8, 2, 3, 21, 34, 39, 6, 5, 186, 347, 250, 138, 16, 8, 2113, 3666, 2919, 1234, 414, 36, 13, 27856, 47484, 36714, 17050, 4830, 1104, 76, 21, 422481, 707480, 545788, 253386, 78815, 16174, 2715, 152, 34, 7241480, 11971341, 9195198, 4317996, 1369260, 309075, 48444, 6282, 294, 55
Offset: 0

Views

Author

Donovan Young, May 18 2019

Keywords

Comments

This is the number of "k-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. First column is A265167, second column is A318244. Diagonals are given by A000045, A178523, A318267, A318268, A318269, A318270.

Examples

			The first few rows of T(n,k) are:
   1;
   0,  1;
   1,  0,  2;
   2,  8,  2,  3;
  21, 34, 39,  6, 5;
  ...
For n = 2 there is only one way to place the two pairs such that neither is joined by an edge, hence T(2,0)=1. If one pair is joined by an edge, the other is forced to be, hence T(2,1) = 0, and since the pairs can be joined horizontally or vertically T(2,2) = 2.

Links

Crossrefs

Cf. A046741, A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270.

Programs

  • Mathematica
    CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k*(1-(1-z)*y)^k/(1+(1-z)*y)^k/(1+(1-z)*y-(1-z)^2*y^2)^(k+1),{k,0,20}],{y,0,20}]],{y,z}];

Formula

G.f.: Sum_{j>=0} (2*j-1)!! * y^j * (1-(1-z)*y)^j / (1+(1-z)*y)^j / (1+(1-z)*y-(1-z)^2*y^2)^(j+1).

A325754 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by a horizontal edge.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 7, 4, 4, 0, 43, 38, 21, 2, 1, 372, 360, 168, 36, 9, 0, 4027, 3972, 1818, 478, 93, 6, 1, 51871, 51444, 23760, 6640, 1260, 144, 16, 0, 773186, 768732, 358723, 103154, 20205, 2734, 278, 12, 1, 13083385, 13027060, 6129670, 1796740, 363595, 52900, 5650, 400, 25, 0
Offset: 0

Views

Author

Donovan Young, May 19 2019

Keywords

Comments

This is the number of "k-horizontal-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young].

Examples

			The first few rows of T(n,k) are:
  1;
  1,  0;
  2,  0,  1;
  7,  4,  4,  0;
  43, 38, 21, 2, 1;
  ...
For n=2, let the vertex set of P_2 X P_2 be {A,B,C,D} and the edge set be {AB, AC, BD, CD}, where AB and CD are horizontal edges. For k=0, we may place the pairs on A, C and B, D or on A, D and B, C, hence T(2,0) = 2. If we place a pair on one of the horizontal edges we are forced to place the other pair on the remaining horizontal edge, hence T(2,1)=0 and T(2,2)=1.

Links

Crossrefs

Cf. A046741, A055140, A079267 A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270, A325753.

Programs

  • Mathematica
    CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k/(1-(1-z)*y)/(1+(1-z)*y)^(2*k+1), {k, 0, 20}], {y, 0, 20}]], {y, z}];

Formula

G.f.: Sum_{j>=0} (2*j-1)!! y^j/(1-(1-z)*y)/(1+(1-z)*y)^(2*j+1).
E.g.f.: exp((sqrt(1 - 2 y)-1) (1 - z))/sqrt(1 - 2 y) - exp((y - 2) (1 - z)) sqrt(Pi/2) sqrt(1 - z) (-erfi(sqrt(2) sqrt(1 - z)) + erfi(((1 + sqrt(1 - 2 y)) sqrt(1 - z))/sqrt(2))).
Showing 1-7 of 7 results.

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