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.

Previous Showing 21-30 of 43 results. Next

A354511 Number of SAWs crossing a square domain of the hexagonal lattice.

Original entry on oeis.org

2, 14, 264, 21512, 5663596, 6478476233, 23432328776346, 365121393771314359, 18039965927005597824652, 3847346539490622663060402802, 2604549807872636495439504536518768, 7613280873970130888072912524910312775000, 70659728324509466176595292882340210105184200002
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 16 2022

Keywords

Links

Crossrefs

Cf. A001006, A002026, A007764, A116485.

A356610 Number of SAWs crossing a rhomboidal domain of the hexagonal lattice.

Original entry on oeis.org

2, 14, 316, 25092, 7374480, 8029311942, 32223151155864, 476605408516689238, 26016526700583361056456, 5246595079903462547245876694, 3911053741699230141571030313824664, 10780907768757190963361134040036893772360, 109919900687141309301630828947780890728732496678
Offset: 1

Views

Author

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

Keywords

Links

Crossrefs

Cf. A001006, A002026, A007764, A116485.

A356616 Number of SAPs crossing a triangular domain of the hexagonal lattice and including top vertex.

Original entry on oeis.org

1, 1, 4, 36, 666, 24696, 1808820, 259300148, 72369408510, 39205936157880, 41152969216872016, 83592236529606631688, 328284931491454739745904, 2490876950205850778116435156, 36494758452603010620499864088198, 1032033208911845667821292289616451218
Offset: 1

Views

Author

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

Keywords

Links

Crossrefs

Cf. A001006, A002026, A007764, A116485.

A145403 Number of nonintersecting rook paths joining opposite corners of 6 X n board.

Original entry on oeis.org

1, 32, 414, 5382, 79384, 1262816, 20562673, 336067810, 5493330332, 89803472792, 1468381290905, 24012936982592, 392716580997352, 6422777815120738, 105043595925333255, 1717976646746942760
Offset: 1

Views

Author

N. J. A. Sloane, Feb 03 2009

Keywords

Links

Crossrefs

Row 6 of A064298. Cf. A007764.

Formula

Recurrence:
a(1) = 1,
a(2) = 32,
a(3) = 414,
a(4) = 5382,
a(5) = 79384,
a(6) = 1262816,
a(7) = 20562673,
a(8) = 336067810,
a(9) = 5493330332,
a(10) = 89803472792,
a(11) = 1468381290905,
a(12) = 24012936982592,
a(13) = 392716580997352,
a(14) = 6422777815120738,
a(15) = 105043595925333255,
a(16) = 1717976646746942760,
a(17) = 28097347987645295129,
a(18) = 459529700981496318610,
a(19) = 7515570007661530339293,
a(20) = 122916531487036730334780,
a(21) = 2010289859051351461718841,
a(22) = 32878127252299185360551934,
a(23) = 537719101299048122399217869,
a(24) = 8794352250919537166665750722,
a(25) = 143830917261013287829855929053,
a(26) = 2352342978307852368872254574110,
a(27) = 38472378495706095194731534070125,
a(28) = 629212627935457125950913558054726,
a(29) = 10290721464101586255448326254366900,
a(30) = 168303914369885958800758915526318474,
a(31) = 2752596860300114955964065429361536989,
a(32) = 45018498254837163421818726088041699166,
a(33) = 736273885345044284085688553892457204990,
a(34) = 12041699640279371326340375422350041719446,
a(35) = 196941020336151050199143987475335247318191,
a(36) = 3220954404252653214796052011262240269847376,
a(37) = 52678447875240888447093955411712504021593807,
a(38) = 861551739720563513304275975426292082337631174,
a(39) = 14090608781288751611582325118090142798190478571,
a(40) = 230450763051941815978795941071686604125891198442,
a(41) = 3769003526784804976816338101329440702079133017666,
a(42) = 61641746795668086369885223391335280193549793452454,
a(43) = 1008145766120479656207584228935637479155797947389803,
a(44) = 16488142185777157345793212901099082094584264689337958,
a(45) = 269662227303264323330785234671779693565559562284410182,
a(46) = 4410303842290033896172439105038616399715156984924650402,
a(47) = 72130161409086529608951854829851816002712963801157839787,
a(48) = 1179682935903881340573479585181430128337758733576064749582,
a(49) = 19293618675966098340238272567572020098236154654850930308513,
a(50) = 315545567613362204775242670274937424600545170340425654393866,
a(51) = 5160722149260222882522006042304141173305206572051170726255899,
a(52) = 84403191917113277982043589954202883741227100622483260510931370,
a(53) = 1380407353807693358300087458031214954879276213089038340492802025,
a(54) = 22576450240384821778027453624243941724086228917427154372144432134,
a(55) = 369236011421291236034279467148078460540271871269615690271797866884,
a(56) = 6038825000328308509532140773346231121610228947574581160028281180694,
a(57) = 98764492781235197642079658639806209320318476451903082795917783012815,
a(58) = 1615285263905535856420093270568679674123758786480792514826877881411406,
a(59) = 26417859397806804999115463757296013189790610675906972739777532953432044,
a(60) = 432061946429732109168468779744829065082966074684439846926537350314283068,
a(61) = 7066338068562305854471591381542565889032938460560686542553098493028726504,
a(62) = 115569385621266871108822160868123881005301075723863358645704767187297221382,
a(63) = 1890127922452274019805513045202943498801049603564334398540115110078021072823,
a(64) = 30912888772650264652219507061031956074793682526018605864614278139682619190156,
a(65) = 505577787047572692090462300937222384232557420150184666960671714745016065033080,
a(66) = 8268683675466366377840360356400869587932159727058836866913105126545228412490614,
a(67) = 135233650442183190541354312834185782890515070868821995834750746327337159470828189,
a(68) = 2211735377685386523121420331929400511514963984542634134765620183963171569729235278,
a(69) = 36172752601960652644405183597210303325660884461711588396278289372424954431031849588,
a(70) = 591602433095763079343906237098879371053254029141187068993235175242965360620853115872,
a(71) = 9675609779993804523757669609814376179537455425273511736449480229116222799072745849896,
a(72) = 158243812698379899306192927052283225599988748265808627411791715806385192535377775606282,
a(73) = 2588064713926068829323899654190495449456281961482820545222829156707671713277546826822289,
a(74) = 42327588354029840959980586262134563828846542737714855516560279055906134220418117167021544,
a(75) = 692264272306516416237168808269386146006151827583985698688727056187756308291243240771646474, and
a(n) = 76a(n-1) - 2640a(n-2) + 55984a(n-3) - 812934a(n-4) + 8556872a(n-5)
- 67099242a(n-6) + 393958772a(n-7) - 1692942183a(n-8) + 4884527404a(n-9) - 6187506869a(n-10)
- 19086405626a(n-11) + 128174201130a(n-12) - 327127420664a(n-13) + 297315119122a(n-14) + 743733332720a(n-15)
- 3157843190533a(n-16) + 5268656094548a(n-17) - 3941342671128a(n-18) - 3509217289604a(n-19) + 25691997627302a(n-20)
- 79177609422932a(n-21) + 124810724415142a(n-22) + 32165552119276a(n-23) - 559590816744166a(n-24) + 954577325227640a(n-25)
+ 45695215480520a(n-26) - 2489003696662264a(n-27) + 3079811130140804a(n-28) + 1436343394106164a(n-29) - 6800600057977368a(n-30)
+ 3717237179493356a(n-31) + 6652945245605814a(n-32) - 9432540370407444a(n-33) - 2036411447626966a(n-34) + 12103828254803672a(n-35)
- 3892070556133820a(n-36) - 11936409494863372a(n-37) + 8331936811395842a(n-38) + 10790544774261660a(n-39) - 9791814381222907a(n-40)
- 9774483491028244a(n-41) + 8082925131170466a(n-42) + 8591527532922680a(n-43) - 4558074323604317a(n-44) - 6507699416893516a(n-45)
+ 1335741921421883a(n-46) + 3811541403121978a(n-47) + 265590026556815a(n-48) - 1596050169969560a(n-49) - 489317457105434a(n-50)
+ 441751378351184a(n-51) + 251839358248300a(n-52) - 69448285619300a(n-53) - 76332173161850a(n-54) + 1539583576296a(n-55)
+ 15557344027403a(n-56) + 2097787252080a(n-57) - 2266145094960a(n-58) - 598133889956a(n-59) + 240729252424a(n-60)
+ 98573852340a(n-61) - 17808243041a(n-62) - 11420445450a(n-63) + 718791367a(n-64) + 980442116a(n-65)
+ 34587845a(n-66) - 51217686a(n-67) - 4961985a(n-68) + 1519440a(n-69) + 196028a(n-70)
- 26928a(n-71) - 3486a(n-72) + 308a(n-73) + 25a(n-74) - 2a(n-75).

A215527 Number of nonintersecting (or self-avoiding) rook paths joining opposite poles of a sphere with n horizontal sectors and n vertical sectors (demarcated by longitudes and latitudes).

Original entry on oeis.org

1, 8, 441, 23436, 3274015, 1279624470, 1429940707685, 4632832974994840, 44016723796115276451, 1236712122885961369684270, 103348977536357696768748889161, 25793194766828189243602379528079372, 19283754194866506189223991782133012219131
Offset: 1

Views

Author

Alex Ratushnyak, Aug 15 2012

Keywords

Comments

Overall there are n*n sectors.
The length of the step is 1. The length of the path varies.
Equivalently, the number of directed paths in the graph C_n X P_n that start at any one of the n vertices on one side of the cylinder and terminate at any of the n vertices on the opposite side. - Andrew Howroyd, Apr 09 2016

Examples

			With n=2 there are four sectors: North-Western, North-Eastern, South-Western, South Eastern. Eight nonintersecting (self-avoiding) rook paths joining opposite poles exist:
NorthPole NW SW SouthPole
NorthPole NW SW SE SouthPole
NorthPole NW NE SE SouthPole
NorthPole NW NE SE SW SouthPole
NorthPole NE SE SouthPole
NorthPole NE SE SW SouthPole
NorthPole NE NW SW SouthPole
NorthPole NE NW SW SE SouthPole
So a(2)=8.

Crossrefs

Cf. A007764, A268894.

Programs

  • C
    #include   // GCC -O3  // a(7) in ~1.5 hours
char grid[8][8];
long long SIZE;
long long calc_ways(long long x, long long y) {
  if (grid[x][y]) return 0;
  grid[x][y] = 1;
  long long n = calc_ways( x==0? SIZE-1 : x-1, y);  // try West
  if (SIZE>2)
            n+= calc_ways( x==SIZE-1? 0 : x+1, y);  // East
  if (y>0)  n+= calc_ways(x,y-1);  // North
  if (y==SIZE-1)  n++;
  else      n+= calc_ways(x,y+1);  // South
  grid[x][y] = 0;
  return n;
}
int main(int argc, char **argv)
{
  for (SIZE=1; SIZE<7; ++SIZE) {
    memset(grid, 0, sizeof(grid));
    printf("%llu, ",calc_ways(0,0)*SIZE);
  }
  printf("\n      ");
  for (SIZE=3; SIZE<9; ++SIZE) {
    unsigned long long r;
    memset(grid, 0, sizeof(grid));
    grid[0][0]=1;
    grid[0][1]=1;
    r =  calc_ways(0,2)*SIZE;    if (SIZE>6) printf(".");
    r += calc_ways(1,1)*SIZE*2;  if (SIZE>6) printf(".");
    grid[0][1]=0;
    grid[1][0]=1;
    r += calc_ways(1,1)*SIZE*2;  if (SIZE>6) printf(".");
    r += calc_ways(2,0)*SIZE*2;  printf("%llu, ", r);
  }
}

Extensions

a(8)-a(13) from Andrew Howroyd, Apr 09 2016

A236753 Number of simple (non-intersecting) directed paths on the grid graph P_n X P_n.

Original entry on oeis.org

1, 28, 653, 28512, 3060417, 873239772, 687430009069, 1532025110398168, 9829526954625359697, 183563561823425961932572, 10056737067604248527218979485, 1626248896102138091401810358337184
Offset: 1

Views

Author

Jaimal Ichharam, Jan 30 2014

Keywords

Comments

This is the number of directed paths on P_n X P_n of any length and also includes one zero length path per vertex. - Andrew Howroyd, May 27 2017

Examples

			For n=2 there are 4 zero length paths (one for each vertex), 8 paths with 1 one edge, 8 paths with 2 edges and 8 paths with 3 edges, so a(2)=28. - _Andrew Howroyd_, May 27 2017

Crossrefs

Cf. A236690 (includes diagonal edges).
Cf. A288032, A007764, A121785, A120443.

Formula

a(n) = 2*A288032(n) + n^2. - Andrew Howroyd, Jun 10 2017

Extensions

a(6) corrected and a(8) added from Jaimal Ichharam, Feb 13 2014
a(6)-a(8) corrected and a(9)-a(12) from Andrew Howroyd, May 27 2017

A333520 Triangle read by rows: T(n,k) is the number of self-avoiding paths of length 2*(n-1+k) connecting opposite corners in the n X n grid graph (0 <= k <= floor((n-1)^2/2), n >= 1).

Original entry on oeis.org

1, 2, 6, 4, 2, 20, 36, 48, 48, 32, 70, 224, 510, 956, 1586, 2224, 2106, 732, 104, 252, 1200, 3904, 10560, 25828, 58712, 121868, 217436, 300380, 280776, 170384, 61336, 10180, 924, 5940, 25186, 88084, 277706, 821480, 2309402, 6140040, 15130410, 33339900, 62692432, 96096244, 116826664, 110195700, 78154858, 39287872, 12396758, 1879252, 111712
Offset: 1

Views

Author

Seiichi Manyama, Mar 29 2020

Keywords

Examples

			T(3,1) = 4;
   S--*      S--*--*   S  *--*   S
      |            |   |  |  |   |
   *--*         *--*   *--*  *   *  *--*
   |            |            |   |  |  |
   *--*--E      *--E         E   *--*  E
Triangle starts:
=======================================================
n\k|   0     1     2      3      4 ...      8 ...   12
---|---------------------------------------------------
1  |   1;
2  |   2;
3  |   6,    4,    2;
4  |  20,   36,   48,    48,    32;
5  |  70,  224,  510,   956,  1586, ... , 104;
6  | 252, 1200, 3904, 10560, ................. , 10180;

Links

Crossrefs

Row sums give A007764.
T(n,0) gives A000984(n-1).
T(n,1) gives A257888(n).
T(n,floor((n-1)^2/2)) gives A121788(n-1).
T(2*n-1,2*(n-1)^2) gives A001184(n-1).
Cf. A074148, A302337, A329633.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333520(n):
    if n == 1: return [1]
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n * n
    paths = GraphSet.paths(start, goal)
    return [paths.len(2 * (n - 1 + k)).len() for k in range((n - 1) ** 2 // 2 + 1)]
print([i for n in range(1, 8) for i in A333520(n)])

A343307 a(n) is the number of self-avoiding paths connecting consecutive corners of an n X n triangular grid.

Original entry on oeis.org

1, 2, 10, 108, 2726, 168724, 25637074, 9454069104, 8461610420420, 18438745892175008, 97929194419509169380, 1267379450261470833222676, 39964658780097197018058705552, 3071011528804416058638501563820092, 575150143830631835000028468717331605240
Offset: 1

Views

Author

Rémy Sigrist, Apr 11 2021

Keywords

Comments

We use unit moves parallel to the triangle edges.

Examples

			For n = 3:
- we have the following paths:
.                         .
.
.                       .   .
.
.                     o---o---o
.
.
.          .              .              .
.
.        o   .          o   o          .   o
.       / \            / \ / \            / \
.      o   o---o      o   o   o      o---o   o
.
.
.          .              .              .
.
.        o---o          o---o          o---o
.       /   /          /     \          \   \
.      o   o---o      o   .   o      o---o   o
.
.
.          o              o              o
.         / \            / \            / \
.        o   o          o   o          o   o
.       /   /          /     \          \   \
.      o   o---o      o   .   o      o---o   o
- so a(3) = 10.

Links

Crossrefs

Cf. A007764, A112675, A288148, A308144, A343310.

Programs

  • Python
    # See Links section.

Extensions

a(14)-a(15) from Andrew Howroyd, Feb 04 2022

A351109 Number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an n X n grid.

Original entry on oeis.org

1, 0, 2, 8, 40, 1380, 211164, 205331148
Offset: 1

Views

Author

Pontus von Brömssen, Feb 01 2022

Keywords

Examples

			For n = 4 the following paths, together with their reflections in the diagonal, exist. The numbers give the positions of the car after successive steps. In total, there are a(4) = 2*4 = 8 possible paths.
  ...3  ...4  ...4  ...5
  ....  ...3  ..3.  ...4
  ..2.  ..2.  ..2.  ...3
  01..  01..  01..  012.

Links

Crossrefs

Main diagonal of A351108.
Cf. A007764, A351042, A351107.

A121786 "Cow patches" on the square lattice (see Jensen web site for further information).

Original entry on oeis.org

1, 7, 160, 11408, 2522191, 1718769373, 3598611604598, 23098353998190640, 453839082673896579243, 27266319759961440667165921, 5005013940387988257218110301496, 2805250606288167736619664411164848668
Offset: 1

Views

Author

N. J. A. Sloane, Aug 30 2006

Keywords

Links

Crossrefs

Cf. A007764, A121785.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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