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 1-8 of 8 results.

A232295 T(n,k)=Number of nXk 0..2 arrays with every 0 next to a 1 and every 1 next to a 2 horizontally or antidiagonally.

Original entry on oeis.org

1, 3, 1, 7, 15, 1, 15, 97, 73, 1, 33, 587, 1313, 355, 1, 73, 3615, 20563, 17759, 1727, 1, 161, 22387, 336079, 718483, 240241, 8401, 1, 355, 138505, 5546705, 31119789, 25116353, 3249889, 40867, 1, 783, 856719, 91293443, 1370434057, 2885285507
Offset: 1

Views

Author

R. H. Hardin, Nov 22 2013

Keywords

Comments

Table starts
.1......3..........7.............15.................33.....................73
.1.....15.........97............587...............3615..................22387
.1.....73.......1313..........20563.............336079................5546705
.1....355......17759.........718483...........31119789.............1370434057
.1...1727.....240241.......25116353.........2885285507...........339565321435
.1...8401....3249889......877968487.......267483142619.........84130295708483
.1..40867...43963319....30690409685.....24797475083765......20843967523877175
.1.198799..594719777..1072818688305...2298890419506403....5164265865545602229
.1.967065.8045152705.37501616113029.213122398219612007.1279489656786836660869

Examples

			Some solutions for n=3 k=4
..2..2..2..1....2..1..2..2....0..1..0..1....2..2..2..2....1..2..1..0
..2..1..0..0....1..2..2..1....2..2..2..0....1..1..2..0....2..0..2..2
..0..0..1..2....1..0..2..2....2..2..1..2....2..2..1..2....2..2..1..0

Links

Crossrefs

Row 1 is A193641

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 4*a(n-1) +4*a(n-2) +a(n-3)
k=3: a(n) = 12*a(n-1) +20*a(n-2) +9*a(n-3)
k=4: [order 10]
k=5: [order 13] for n>14
k=6: [order 37] for n>38
Empirical for row n:
n=1: a(n) = 2*a(n-1) +a(n-3)
n=2: [order 9]
n=3: [order 31]

A232149 T(n,k)=Number of nXk 0..2 arrays with every 0 next to a 1 and every 1 next to a 2 horizontally, diagonally or antidiagonally.

Original entry on oeis.org

1, 3, 1, 7, 23, 1, 15, 191, 145, 1, 33, 1299, 3669, 887, 1, 73, 9097, 67725, 67311, 5487, 1, 161, 65837, 1345057, 3409361, 1270511, 33957, 1, 355, 474721, 27888353, 191897041, 177194147, 23931701, 210039, 1, 783, 3410799, 572956549, 11346946019
Offset: 1

Views

Author

R. H. Hardin, Nov 19 2013

Keywords

Comments

Table starts
.1.......3..........7.............15................33....................73
.1......23........191...........1299..............9097.................65837
.1.....145.......3669..........67725...........1345057..............27888353
.1.....887......67311........3409361.........191897041...........11346946019
.1....5487....1270511......177194147.......28299613241.........4807617578085
.1...33957...23931701.....9181257593.....4162183572673......2029709037695893
.1..210039..450210003...475203378037...611509764410977....855821105743586179
.1.1299219.8472530835.24603676419865.89867078251266793.360977861939388247605

Examples

			Some solutions for n=3 k=4
..0..1..1..0....1..2..2..1....0..1..0..2....2..2..2..2....1..1..1..2
..1..1..2..2....2..1..0..1....2..1..2..2....2..2..0..1....1..2..2..1
..2..2..2..1....2..2..1..2....0..2..1..1....2..2..2..2....0..1..0..1

Links

Crossrefs

Row 1 is A193641

Formula

Empirical for column k:
k=2: a(n) = 5*a(n-1) +6*a(n-2) +8*a(n-3) +3*a(n-4) -8*a(n-5) -5*a(n-6)
k=3: [order 9]
k=4: [order 32]
k=5: [order 81]
Empirical for row n:
n=1: a(n) = 2*a(n-1) +a(n-3)
n=2: a(n) = 6*a(n-1) +4*a(n-2) +25*a(n-3) +60*a(n-4) -24*a(n-5) -72*a(n-6)
n=3: [order 19]
n=4: [order 58]

A193648 T(n,k)=Number of arrays of -k..k integers x(1..n) with every x(i) being in a substring of length 1 or 2 with sum zero. Array listed by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 5, 7, 1, 7, 13, 15, 1, 9, 19, 37, 33, 1, 11, 25, 67, 105, 73, 1, 13, 31, 105, 217, 297, 161, 1, 15, 37, 151, 369, 721, 841, 355, 1, 17, 43, 205, 561, 1393, 2377, 2381, 783, 1, 19, 49, 267, 793, 2361, 5105, 7855, 6741, 1727, 1, 21, 55, 337, 1065, 3673, 9361
Offset: 1

Views

Author

R. H. Hardin, Aug 02 2011

Keywords

Examples

			Table starts
....1.....1.....1......1......1.......1.......1.......1.......1........1
....3.....5.....7......9.....11......13......15......17......19.......21
....7....13....19.....25.....31......37......43......49......55.......61
...15....37....67....105....151.....205.....267.....337.....415......501
...33...105...217....369....561.....793....1065....1377....1729.....2121
...73...297...721...1393...2361....3673....5377....7521...10153....13321
..161...841..2377...5105...9361...15481...23801...34657...48385....65321
..355..2381..7855..18937..38171...68485..113191..175985..260947...372541
..783..6741.25939..69897.153591..295453..517371..844689.1306207..1934181
.1727.19085.85675.258521.621911.1291237.2416835.4187825.6835951.10639421
Some solutions for n=7 k=6
.-6....4....1....2...-5...-2....1....5...-5...-1....4...-3...-4...-6....0....0
..6...-4...-1...-2....5....2...-1...-5....5....1...-4....3....4....6....1....5
.-6....4....1...-4...-3....0....3....5....6...-3...-1....0...-5....0...-1...-5
.-4....2...-3....4....3...-4...-3...-5...-6....3....1....2....5....3...-1....5
..4...-2....3...-1...-3....4....4....4....1...-3....1...-2...-5...-3....1....0
.-4....3...-4....1...-6...-3...-4...-4...-1...-2...-1....3...-5....0...-1...-6
..4...-3....4...-1....6....3....0....4....1....2....1...-3....5....0....0....6

Links

Crossrefs

Cf. A193641 (column 1) to A193647 (column 7).

Programs

  • Maple
    F:= normal @ gfun:-rectoproc({t(n) = 2*t(n-1)+2*(k-1)*t(n-2)+t(n-3),t(1)=1,t(2)=2*k+1,t(3)=6*k+1},t(n),remember):
seq(seq(eval(F(j),k=m-j),j=1..m-1),m=2..20); # Robert Israel, May 26 2016
  • Mathematica
    nmax = 12;
col[k_] := col[k] = CoefficientList[(x + (2 k - 1) x^2 + x^3)/
   (1 - 2 x + 2 (1 - k) x^2 - x^3) + O[x]^(nmax + 1), x] // Rest;
T[n_, k_] := col[k][[n]];
Table[T[n - k + 1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 24 2022 *)

Formula

Empirical for column k: T(n,k)=2*T(n-1,k)+2*(k-1)*T(n-2,k)+T(n-3,k); with T(1,k)=1, T(2,k)=2*k+1, T(3,k)=6*k+1.
From Robert Israel, May 26 2016: (Start)
G.f. for column k: (x+(2k-1)x^2+x^3)/(1-2x+2(1-k)x^2-x^3).
The recursion for column k can be obtained from this.
G.f. for array: A(x,y) = y/(y-1) - (1-x+x^2)*y*LerchPhi(y,1,(-1+2*x+x^3)/(2*x^2))/(2*x^2). (End)

A232309 T(n,k)=Number of nXk 0..2 arrays with every 0 next to a 1 and every 1 next to a 2 horizontally, vertically or antidiagonally.

Original entry on oeis.org

1, 3, 3, 7, 37, 7, 15, 257, 257, 15, 33, 1915, 5149, 1915, 33, 73, 14223, 107047, 107047, 14223, 73, 161, 105411, 2245709, 6363511, 2245709, 105411, 161, 355, 781873, 46996227, 382995685, 382995685, 46996227, 781873, 355, 783, 5798587, 983668985
Offset: 1

Views

Author

R. H. Hardin, Nov 22 2013

Keywords

Comments

Table starts
....1.........3.............7.................15......................33
....3........37...........257...............1915...................14223
....7.......257..........5149.............107047.................2245709
...15......1915........107047............6363511...............382995685
...33.....14223.......2245709..........382995685.............66287652717
...73....105411......46996227........22952606015..........11412010021707
..161....781873.....983668985......1376103637041........1965722343353835
..355...5798587...20589521605.....82507009197797......338617181188008457
..783..43004317..430963205159...4946765617435461....58328939899371775421
.1727.318935945.9020580137409.296587635447682217.10047560921392694812519

Examples

			Some solutions for n=3 k=4
..2..1..1..1....2..1..1..2....1..2..0..1....2..0..2..2....1..2..2..1
..1..2..0..2....1..2..1..1....1..1..2..1....2..1..2..1....0..1..2..1
..0..2..1..1....1..0..2..0....1..2..1..2....0..0..1..0....1..2..0..2

Links

Crossrefs

Column 1 is A193641

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-3)
k=2: a(n) = 6*a(n-1) +9*a(n-2) +11*a(n-3) -a(n-4) +16*a(n-5)
k=3: [order 15]
k=4: [order 33]
k=5: [order 91]

A232459 T(n,k)=Number of nXk 0..2 arrays with every 0 next to a 1 and every 1 next to a 2 horizontally, vertically, diagonally or antidiagonally.

Original entry on oeis.org

1, 3, 3, 7, 51, 7, 15, 381, 381, 15, 33, 3015, 8227, 3015, 33, 73, 24651, 187989, 187989, 24651, 73, 161, 198945, 4423815, 12521603, 4423815, 198945, 161, 355, 1607067, 103104905, 865692227, 865692227, 103104905, 1607067, 355, 783, 12991131
Offset: 1

Views

Author

R. H. Hardin, Nov 24 2013

Keywords

Comments

Table starts
...1.......3..........7............15...............33...................73
...3......51........381..........3015............24651...............198945
...7.....381.......8227........187989..........4423815............103104905
..15....3015.....187989......12521603........865692227..........59192688269
..33...24651....4423815.....865692227.....176788379233.......35630507175345
..73..198945..103104905...59192688269...35630507175345....21136983701111465
.161.1607067.2403848151.4047993038079.7183387830867849.12543745301001506913

Examples

			Some solutions for n=3 k=4
..1..0..1..1....0..1..1..2....1..2..2..1....0..2..0..2....1..1..0..2
..1..2..2..1....0..2..0..2....1..1..1..2....2..1..0..1....2..1..1..2
..2..2..2..2....2..2..1..1....2..1..2..2....2..0..2..1....1..0..1..2

Links

Crossrefs

Column 1 is A193641

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-3)
k=2: a(n) = 6*a(n-1) +12*a(n-2) +39*a(n-3)
k=3: [order 9]
k=4: [order 15]
k=5: [order 46]

A232368 T(n,k)=Number of nXk 0..2 arrays with every 0 next to a 1 and every 1 next to a 2 horizontally or vertically.

Original entry on oeis.org

1, 3, 3, 7, 23, 7, 15, 145, 145, 15, 33, 887, 2411, 887, 33, 73, 5487, 39321, 39321, 5487, 73, 161, 33957, 647287, 1704619, 647287, 33957, 161, 355, 210039, 10652849, 74665511, 74665511, 10652849, 210039, 355, 783, 1299219, 175284259, 3271362341
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2013

Keywords

Comments

Table starts
...1.......3..........7............15................33...................73
...3......23........145...........887..............5487................33957
...7.....145.......2411.........39321............647287.............10652849
..15.....887......39321.......1704619..........74665511...........3271362341
..33....5487.....647287......74665511........8716796353........1017627171097
..73...33957...10652849....3271362341.....1017627171097......316516682114845
.161..210039..175284259..143280358783...118759183620445....98411994699167317
.355.1299219.2884208601.6275473692531.13859593192739275.30598926533212575781

Examples

			Some solutions for n=3 k=4
..0..1..2..2....1..0..2..1....2..2..1..0....1..2..1..1....2..1..2..1
..2..1..1..2....2..1..0..1....0..1..1..1....0..1..2..2....1..1..2..2
..1..2..2..1....2..1..1..2....2..2..2..2....2..2..2..1....2..0..2..2

Links

Crossrefs

Column 1 is A193641
Column 2 is A232145

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-3)
k=2: a(n) = 5*a(n-1) +6*a(n-2) +8*a(n-3) +3*a(n-4) -8*a(n-5) -5*a(n-6)
k=3: [order 21]
k=4: [order 60]

A285184 a(n) = 2*a(n-1) + a(n-3) with initial terms 1,3,5.

Original entry on oeis.org

1, 3, 5, 11, 25, 55, 121, 267, 589, 1299, 2865, 6319, 13937, 30739, 67797, 149531, 329801, 727399, 1604329, 3538459, 7804317, 17212963, 37964385, 83733087, 184679137, 407322659, 898378405, 1981435947, 4370194553, 9638767511, 21258970969, 46888136491, 103415040493, 228089051955
Offset: 0

Views

Author

N. J. A. Sloane, Apr 23 2017

Keywords

Comments

The sequences in Prop. 5.1 and 5.2 should also be added to the OEIS.

Links

Crossrefs

Cf. A193641.

Programs

  • Maple
    a:=proc(n) option remember;
if n=0 then 1
elif n=1 then 3
elif n=2 then 5
else 2*a(n-1)+a(n-3); fi;
end;
[seq(a(n),n=0..40)];
  • Mathematica
    LinearRecurrence[{2,0,1},{1,3,5},40] (* Harvey P. Dale, May 26 2023 *)
  • PARI
    Vec((1 + x - x^2) / (1 - 2*x - x^3) + O(x^40)) \\ Colin Barker, Apr 23 2017

Formula

G.f.: (1 + x - x^2) / (1 - 2*x - x^3). - Colin Barker, Apr 23 2017

A100691 Number of self-avoiding paths with n steps on a triangular lattice in the strip Z x {0,1}.

Original entry on oeis.org

1, 4, 12, 30, 70, 158, 352, 780, 1724, 3806, 8398, 18526, 40864, 90132, 198796, 438462, 967062, 2132926, 4704320, 10375708, 22884348, 50473022, 111321758, 245527870, 541528768, 1194379300, 2634286476, 5810101726, 12814582758
Offset: 0

Views

Author

Emeric Deutsch, Dec 07 2004

Keywords

References

  • J. Labelle, Paths in the Cartesian, triangular and hexagonal lattices, Bulletin of the ICA, 17, 1996, 47-61.

Links

Programs

  • Maple
    g:=series((1+z^2)*(1+z+z^2)/(1-z)/(1-2*z-z^3),z=0,35): 1,seq(coeff(g,z^n), n=1..34);

Formula

G.f.: (1+z^2)(1+z+z^2)/[(1-z)(1-2z-z^3)]= 1+2*(2+z^2)/((z-1)*(z^2+2*z-1)).
a(n) = 2*a(n-1) + a(n-3) + 6 for n >= 4.
a(n) = A008998(n+2) - A052980(n+1) - 3. - Ralf Stephan, May 15 2007
Conjecture: a(n) = A193641(n+2)-3, n>0 - R. J. Mathar, Jul 22 2022
Showing 1-8 of 8 results.

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

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