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 1-9 of 9 results.

A378933 Number of minimal edge cuts in the 3 X n grid graph.

Original entry on oeis.org

2, 15, 53, 146, 356, 809, 1759, 3716, 7702, 15763, 31993, 64582, 129912, 260749, 522627, 1046616, 2094858, 4191639, 8385533, 16773690, 33550412, 67104305, 134212583, 268429676, 536864446, 1073734619, 2147475649, 4294958446, 8589924832, 17179858453
Offset: 1

Views

Author

Andrew Howroyd, Dec 11 2024

Keywords

Links

Crossrefs

Row 3 of A378932.
Cf. A166761, A359988, A378934.

Programs

  • Mathematica
    LinearRecurrence[{6, -14, 16, -9, 2}, {2, 15, 53, 146, 356}, 30] (* Paolo Xausa, Mar 02 2025 *)
  • PARI
    a(n) = {16*2^n - (2*n^3 + 9*n^2 + 73*n + 96)/6}

Formula

a(n) = 16*2^n - (2*n^3 + 9*n^2 + 73*n + 96)/6.
G.f.: x*(2 + 3*x - 9*x^2 + 6*x^3)/((1 - 2*x)*(1 - x)^4).
a(n) = A166761(n)/2.

A378934 Number of minimal edge cuts in the 4 X n grid graph.

Original entry on oeis.org

3, 28, 146, 627, 2471, 9292, 33878, 120771, 423251, 1463908, 5011690, 17021179, 57450167, 192966908, 645696454, 2154226075, 7170606795, 23825657596, 79055534746, 262031761435, 867792229799, 2872103661988, 9501035284286, 31417942222787, 103862506390523, 343276150243020
Offset: 1

Views

Author

Andrew Howroyd, Dec 11 2024

Keywords

Links

Crossrefs

Row 4 of A378932.
Cf. A166766, A378933.

Formula

G.f.: x*(3 - 2*x - 23*x^2 + 26*x^3 + 32*x^4 - 2*x^5 - 29*x^6 - 18*x^7 - 3*x^8)/((1 - x)^3*(1 - 2*x - x^2)^2*(1 - 3*x - x^2)).
a(n) = 10*a(n-1) - 37*a(n-2) + 59*a(n-3) - 27*a(n-4) - 25*a(n-5) + 21*a(n-6) + 5*a(n-7) - 4*a(n-8) - a(n-9) for n >= 10.
a(n) = A166766(n)/2.

A166761 Number of n X 3 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

4, 30, 106, 292, 712, 1618, 3518, 7432, 15404, 31526, 63986, 129164, 259824, 521498, 1045254, 2093232, 4189716, 8383278, 16771066, 33547380, 67100824, 134208610, 268425166, 536859352, 1073728892, 2147469238, 4294951298, 8589916892
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=4
...2.2.2...2.2.2...1.1.2...1.1.1...2.1.1...1.1.1...1.2.2...1.2.2...1.1.1
...2.1.1...1.2.2...1.2.2...2.2.1...2.2.2...2.1.1...1.2.2...1.2.2...1.2.1
...2.1.1...1.2.1...1.2.2...2.2.1...2.2.2...2.2.1...1.1.2...1.1.2...1.2.1
...2.2.1...1.1.1...1.1.2...2.2.1...2.2.2...2.2.2...2.2.2...1.1.1...1.1.1
------
...1.2.2...2.2.2...1.1.2...1.1.1...1.1.1...1.1.2...1.2.2...1.2.2...1.1.1
...1.1.2...2.1.2...1.1.2...2.2.1...1.1.2...1.1.2...1.1.2...1.2.1...1.1.1
...1.1.2...1.1.1...1.2.2...2.2.1...1.1.2...1.1.2...1.1.2...1.1.1...2.2.1
...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...2.2.2...1.1.1...2.1.1

Links

Crossrefs

Twice row 3 of A378932.
Cf. A378933.

Formula

Empirical: a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) -9*a(n-4) + 2*a(n-5).
From G. C. Greubel, May 26 2016: (Start)
Empirical a(n) = (3*2^(n + 5) - 2*n^3 - 9*n^2 - 73*n - 96)/3.
Empirical G.f.: (1/3)*( 96/(1 - 2*x) + 6*(-16 + 34*x - 25*x^2 + 5*x^3)/(1 - x)^4 ).
Empirical E.g.f.: (1/3)*(96*exp(x) - (96 + 84*x + 15*x^2 + 2*x^3 ) )*exp(x). (End)
From Andrew Howroyd, Dec 12 2024: (Start)
The above empirical formulas are correct.
a(n) = 2*A378933(n).
(End)

A166766 Number of n X 4 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

6, 56, 292, 1254, 4942, 18584, 67756, 241542, 846502, 2927816, 10023380, 34042358, 114900334, 385933816, 1291392908, 4308452150, 14341213590, 47651315192, 158111069492, 524063522870, 1735584459598, 5744207323976, 19002070568572, 62835884445574, 207725012781046, 686552300486040
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=4
...1.1.1.1...2.2.1.1...2.2.2.2...1.1.2.2...2.2.2.2...2.1.1.1...2.2.2.2
...1.2.1.1...2.1.1.1...1.2.1.2...2.1.1.2...1.1.2.1...2.2.1.1...2.2.2.2
...1.2.1.2...2.1.1.1...1.2.1.1...2.1.2.2...1.1.1.1...2.2.1.1...1.2.2.1
...2.2.2.2...1.1.1.1...1.1.1.1...2.2.2.2...1.1.1.1...2.1.1.1...1.1.1.1
------
...2.2.2.2...2.2.2.2...2.2.1.1...1.1.2.1...2.1.1.1...1.1.1.1...1.1.2.2
...2.1.1.1...1.1.1.2...1.2.2.1...1.1.2.1...2.1.2.2...1.2.1.2...1.1.1.1
...1.1.1.1...1.1.2.2...1.2.1.1...1.1.1.1...2.1.2.2...1.2.2.2...1.1.1.1
...1.1.1.1...2.2.2.2...1.1.1.1...1.1.1.1...2.2.2.2...1.1.2.2...1.1.1.1

Links

Crossrefs

Twice row 4 of A378932.
Cf. A378934.

Formula

a(n) = 2*A378934(n). - Andrew Howroyd, Dec 12 2024

Extensions

a(16) onwards from Andrew Howroyd, Dec 12 2024

A166771 Number of n X 6 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

10, 132, 1618, 18584, 206392, 2247486, 24057962, 253872596, 2646985492, 27314646774, 279336752042, 2834216781868, 28557453089020, 285979131364722, 2848221941695166, 28228788045963448, 278553921030487780, 2737889369953658314, 26815069186724304206, 261785062066218670340
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=3
...1.1.1.1.1.2...2.2.2.2.2.1...2.2.2.2.2.2...2.2.2.2.1.1...2.2.2.2.2.2
...1.1.2.1.1.2...2.2.2.1.2.1...2.1.1.2.2.2...1.1.1.2.2.1...2.1.1.2.1.2
...2.2.2.2.2.2...2.2.2.1.1.1...1.1.1.1.1.1...1.1.1.1.1.1...2.2.1.1.1.2
------
...2.2.2.2.1.1...1.1.1.1.1.1...2.2.1.1.2.2...2.2.1.1.1.2...1.1.1.1.1.1
...2.2.2.2.1.1...1.1.1.1.1.1...2.2.2.2.2.2...2.1.1.1.1.2...1.1.1.2.2.2
...2.2.1.1.1.1...1.2.2.2.1.1...2.2.2.2.2.2...2.2.2.2.2.2...1.1.1.1.1.1

Links

Crossrefs

Twice row 6 of A378932.

Extensions

a(7) onwards from Andrew Howroyd, Dec 12 2024

A378935 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the rook graph K_m X K_n.

Original entry on oeis.org

0, 1, 1, 3, 6, 3, 7, 22, 22, 7, 15, 84, 150, 84, 15, 31, 346, 1276, 1276, 346, 31, 63, 1476, 11538, 23214, 11538, 1476, 63, 127, 6322, 102772, 418912, 418912, 102772, 6322, 127, 255, 26844, 890130, 7290534, 14673870, 7290534, 890130, 26844, 255, 511, 112666, 7525876, 123174016, 496484776, 496484776, 123174016, 7525876, 112666, 511
Offset: 1

Views

Author

Andrew Howroyd, Dec 12 2024

Keywords

Examples

			Array begins:
======================================================
m\n |  1    2      3       4         5           6 ...
----+-------------------------------------------------
  1 |  0    1      3       7        15          31 ...
  2 |  1    6     22      84       346        1476 ...
  3 |  3   22    150    1276     11538      102772 ...
  4 |  7   84   1276   23214    418912     7290534 ...
  5 | 15  346  11538  418912  14673870   496484776 ...
  6 | 31 1476 102772 7290534 496484776 32893769886 ...
  ...

Links

Crossrefs

Main diagonal is A378936.
Rows 1..2 are A000225(n-1), A378937.
Cf. A143088, A360873, A378932.

Programs

  • PARI
    \\ Needs G from A360873.
T(M,N=M) = {G(M,N) + matrix(M,N,m,n, (2^(m-1) - 1)*(2^(n-1) - 1) - 2^(m*n-1))}
{ my(A=T(7)); for(n=1, #A~, print(A[n,])) }

Formula

T(m,n) = A360873(m,n) + (2^(m-1) - 1)*(2^(n-1) - 1) - 2^(m*n-1).
T(m,n) = T(n,m).

A166769 Number of n X 5 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

8, 90, 712, 4942, 32426, 206392, 1284728, 7855758, 47333834, 281703020, 1659086104, 9684388606, 56098465446, 322820680276, 1847076802248, 10515831592642, 59608838934806, 336604310563828, 1894399026364292, 10630153480987994, 59494181482941446, 332206086480550664
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=4
...1.1.1.2.2...1.1.1.2.2...1.1.1.1.1...2.2.2.2.1...1.1.1.2.2...1.1.1.2.2
...1.1.1.1.2...1.2.2.2.1...1.2.1.1.1...2.2.1.1.1...1.2.1.1.2...1.1.1.1.2
...1.2.2.2.2...1.1.1.2.1...1.2.1.2.1...2.1.1.1.1...1.2.1.1.2...1.1.1.1.2
...1.1.1.1.2...1.1.1.1.1...1.2.2.2.2...2.2.2.2.2...1.2.2.2.2...1.1.1.1.1
------
...1.1.1.1.2...2.2.2.2.2...1.1.1.1.1...1.1.2.2.2...1.1.1.2.2...1.1.1.1.1
...2.2.2.1.2...1.1.1.1.2...2.2.2.2.1...1.2.2.2.2...1.1.1.1.2...1.2.1.1.1
...2.2.2.2.2...2.2.2.2.2...1.1.1.2.1...1.2.2.1.1...1.1.1.2.2...2.2.1.1.2
...2.2.2.2.2...2.2.2.2.2...1.1.1.1.1...1.1.1.1.1...1.1.1.2.2...2.2.2.2.2

Links

Crossrefs

Twice row 5 of A378932.

Formula

G.f.: 2*x*(4 - 51*x + 256*x^2 - 660*x^3 + 1082*x^4 - 1431*x^5 + 698*x^6 + 4667*x^7 - 15736*x^8 + 23396*x^9 - 16421*x^10 + 1174*x^11 + 4664*x^12 - 1000*x^13 - 500*x^14 + 100*x^15 - 221*x^16 - 99*x^17 - 10*x^18)/((1 - x)^2*(1 - 7*x + 12*x^2 - 7*x^3 + 3*x^4 + 2*x^5)^2*(1 - 8*x + 15*x^2 - 5*x^3 - 9*x^4 + 2*x^5 + x^6)). - Andrew Howroyd, Dec 12 2024

Extensions

a(10) onwards from Andrew Howroyd, Dec 12 2024

A166773 Number of n X 7 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

12, 182, 3518, 67756, 1284728, 24057962, 443968782, 8093169792, 146041169658, 2612337653032, 46368803939634, 817402784921660, 14321271894199646, 249540634207399516, 4326701163983868212, 74685900757658242314, 1284014777976657543342, 21994406111684553359456
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=3
...2.2.2.2.2.2.1...2.2.2.1.1.1.1...2.2.2.2.2.2.2...2.2.2.2.2.2.2
...2.2.2.1.1.1.1...2.2.2.1.1.1.1...2.2.2.2.1.2.2...1.1.2.2.1.2.1
...2.2.1.1.1.1.1...2.2.1.1.1.1.1...2.2.1.1.1.2.2...1.1.1.1.1.1.1
------
...2.2.2.2.2.2.2...2.2.2.2.2.2.2...2.2.2.2.2.2.2...2.2.2.2.2.1.1
...2.2.1.2.1.1.2...2.1.1.2.2.2.2...2.1.1.1.1.2.2...1.2.1.1.1.1.1
...2.1.1.1.1.1.1...1.1.1.1.1.1.2...1.1.1.1.1.1.2...1.1.1.1.1.1.1

Links

Crossrefs

Twice row 7 of A378932.

Extensions

a(6) onwards from Andrew Howroyd, Dec 12 2024

A166774 Number of n X 8 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

14, 240, 7432, 241542, 7855758, 253872596, 8093169792, 255122769986, 7970906327712, 247157616043778, 7612389643759656, 233047562889531042, 7095832481400967212, 214993871402051336458, 6485057217704886418618, 194824321266936718091888, 5831359188043905108147376, 173952785425039482256265510
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=3
...2.2.2.2.2.2.2.2...2.1.1.1.1.1.1.1...2.1.1.1.1.1.1.2...2.2.2.2.2.2.2.1
...1.1.1.2.2.2.2.1...2.1.2.1.1.1.1.1...2.2.1.1.1.1.1.2...2.1.1.1.1.1.1.1
...1.1.1.1.1.1.1.1...2.2.2.2.2.2.2.2...2.2.2.2.2.2.2.2...2.2.2.2.1.1.1.1
------
...1.1.1.1.1.1.1.1...2.2.2.2.1.1.1.1...1.1.1.2.2.2.2.2...2.2.1.1.1.1.1.1
...1.1.1.1.2.1.1.1...2.2.1.1.1.1.1.1...1.2.2.2.2.2.2.2...2.2.2.2.2.2.1.1
...1.1.1.1.2.2.2.1...1.1.1.1.1.1.1.1...1.1.1.1.1.1.2.2...2.2.1.1.1.1.1.1

Links

Crossrefs

Twice row 8 of A378932.

Extensions

a(6) onwards from Andrew Howroyd, Dec 12 2024
Showing 1-9 of 9 results.

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