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.

A254150 Number of independent sets in the generalized Aztec diamond E(L_5,L_{2n-1}).

Original entry on oeis.org

1, 8, 73, 689, 6556, 62501, 596113, 5686112, 54239137, 517383521, 4935293524, 47077513469, 449070034657, 4283656560248, 40861585458553, 389776618229969, 3718059650555596, 35466384896440661, 338312070235103473, 3227141903559443792, 30783545081553045457
Offset: 0

Views

Author

Steve Butler, Jan 26 2015

Keywords

Comments

E(L_5,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=5, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Links

Crossrefs

Row n=3 of A331406.
Cf. A254124, A254125, A254126, A254151, A254152.

Programs

  • PARI
    Vec((1 - 4*x + x^2)/(1 - 12*x + 24*x^2 - 5*x^3) + O(x^25)) \\ Andrew Howroyd, Jan 16 2020

Formula

Empirical g.f.: -(x^2-4*x+1) / (5*x^3-24*x^2+12*x-1). - Colin Barker, Jan 26 2015
The above g.f. is correct. See A331406 for bounds on the order of the recurrence. - Andrew Howroyd, Jan 16 2020

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 15 2020

A254151 Number of independent sets in the generalized Aztec diamond E(L_7,L_{2n-1}).

Original entry on oeis.org

1, 16, 314, 6556, 139344, 2976416, 63663808, 1362242592, 29151501760, 623849225024, 13350628082560, 285709494797952, 6114316283697408, 130849237522680064, 2800235203724240384, 59926350645878761984, 1282452098548524184576, 27445078313878468469760
Offset: 0

Views

Author

Steve Butler, Jan 26 2015

Keywords

Comments

E(L_7,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=7, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Links

Crossrefs

Row n=4 of A331406.
Cf. A254124, A254125, A254126, A254150, A254152.

Programs

  • Mathematica
    LinearRecurrence[{30,-202,396,-248,32},{1,16,314,6556,139344},20] (* Harvey P. Dale, May 31 2024 *)
  • PARI
    Vec((1 - 14*x + 36*x^2 - 28*x^3 + 4*x^4)/(1 - 30*x + 202*x^2 - 396*x^3 + 248*x^4 - 32*x^5) + O(x^20)) \\ Andrew Howroyd, Jan 16 2020

Formula

Empirical g.f.: -(4*x^4-28*x^3+36*x^2-14*x+1) / (32*x^5-248*x^4+396*x^3-202*x^2+30*x-1). - Colin Barker, Jan 26 2015
The above g.f. is correct. See A331406 for bounds on the order of the recurrence. - Andrew Howroyd, Jan 16 2020

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 15 2020

A254152 Number of independent sets in the generalized Aztec diamond E(L_9,L_{2n-1}).

Original entry on oeis.org

1, 32, 1351, 62501, 2976416, 142999897, 6888568813, 332097693792, 16014193762579, 772279980131297, 37243762479698928, 1796118644459454733, 86619824190256627593, 4177339899819872607008, 201457018240598757372431, 9715496740529686006497709, 468541027322402116068858304
Offset: 0

Views

Author

Steve Butler, Jan 26 2015

Keywords

Comments

E(L_9,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=9, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Links

Crossrefs

Row n=5 of A331406.
Cf. A254124, A254125, A254126, A254150, A254151.

Programs

  • PARI
    Vec((1 - 42*x + 433*x^2 - 1745*x^3 + 3002*x^4 - 2275*x^5 + 700*x^6 - 72*x^7)/(1 - 74*x + 1450*x^2 - 10672*x^3 + 34214*x^4 - 50814*x^5 + 34671*x^6 - 9772*x^7 + 936*x^8) + O(x^20)) \\ Andrew Howroyd, Jan 16 2020

Formula

G.f.: (1 - 42*x + 433*x^2 - 1745*x^3 + 3002*x^4 - 2275*x^5 + 700*x^6 - 72*x^7)/(1 - 74*x + 1450*x^2 - 10672*x^3 + 34214*x^4 - 50814*x^5 + 34671*x^6 - 9772*x^7 + 936*x^8). - Andrew Howroyd, Jan 16 2020

Extensions

a(10)-a(11) corrected and a(12) and beyond from Andrew Howroyd, Jan 15 2020

A054867 Number of non-attacking configurations on a diamond of size n, where a prince attacks the four adjacent non-diagonal squares.

Original entry on oeis.org

1, 2, 17, 689, 139344, 142999897, 748437606081, 19999400591072512, 2728539172202554958697, 1900346273206544901717879089, 6755797872872106084596492075448192, 122584407857548123729431742141838309441329, 11352604691637658946858196503018301306800588837281
Offset: 0

Views

Author

Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

Keywords

Comments

A diamond of size n X n contains (n^2 + (n-1)^2) = A001844(n-1) squares.
For n > 0, a(n) is the number of ways to place non-adjacent counters on the black squares of a 2n-1 X 2n-1 checker board. The checker board is such that the black squares are in the corners. - Andrew Howroyd, Jan 16 2020

Examples

			From _Andrew Howroyd_, Jan 16 2020: (Start)
Case n=2: The grid consists of 5 squares as shown below.
        __
     __|__|__
    |__|__|__|
       |__|
If a prince is placed on the central square then a prince cannot be placed on the other 4 squares, otherwise princes can be placed in any combination. The total number of non-attacking configurations is then 1 + 2^4 = 17, so a(2) = 17.
.
Case n=3: The grid consists of 13 squares as shown below:
           __
        __|__|__
     __|__|__|__|__
    |__|__|__|__|__|
       |__|__|__|
          |__|
The total number of non-attacking configurations of princes is 689 so a(3) = 689.
(End)

Links

Crossrefs

Main diagonal of A331406.
Cf. A006506, A001844.

Extensions

a(0)=1 prepended and terms a(5) and beyond from Andrew Howroyd, Jan 15 2020
Showing 1-4 of 4 results.

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