A223352 -id:A223352 - cp's OEIS frontend

A223345 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of n X n 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

6, 52, 2280, 405988, 329268616, 1079538816324, 16086247351158920, 968803030037550870052, 265167552942550197086665096, 293336730456466043837816011720516

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Diagonal of A223352

			Some solutions for n=3
..4..1..3....1..4..2....4..1..3....1..0..2....0..2..0....1..4..1....5..2..5
..1..3..1....0..2..5....2..4..1....0..2..5....2..5..2....0..1..4....2..5..2
..3..1..4....1..0..2....5..2..0....1..0..2....0..2..5....2..0..1....4..2..4

R. H. Hardin, Table of n, a(n) for n = 1..15

A223346 3 X 3 X 3 triangular graph without horizontal edges coloring a rectangular array: number of n X 1 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

6, 12, 28, 60, 140, 300, 700, 1500, 3500, 7500, 17500, 37500, 87500, 187500, 437500, 937500, 2187500, 4687500, 10937500, 23437500, 54687500, 117187500, 273437500, 585937500, 1367187500, 2929687500, 6835937500, 14648437500, 34179687500

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Column 1 of A223352.

			Some solutions for n=3:
  3  1  5  1  4  3  0  0  1  2  3  4  0  2  2  2
  1  4  2  4  2  1  2  2  0  0  1  2  1  4  4  0
  0  2  0  1  0  4  5  0  2  1  3  5  4  1  2  2

R. H. Hardin, Table of n, a(n) for n = 1..210

Table[2*5^(1/2*(n - 3))*(15 + 7*Sqrt[5] + (-1)^n*(-15 + 7*Sqrt[5])), {n,1,20}] (* Pierre-Louis Giscard, May 17 2013 *)

From Pierre-Louis Giscard, May 17 2013: (Start)
a(n) = 2*5^((1/2)*(n-3))*(15 + 7*sqrt(5) + (-1)^n*(-15 + 7*sqrt(5))) for n > 0, a(0)=6.
G.f: 2*(x^2-6*x-3)/(5*x^2-1).
E.g.f.: (2/5)*(1 + 14*cosh(sqrt(5)*x) + 6*sqrt(5)*sinh(sqrt(5)*x)).  (End)

A223347 3 X 3 X 3 triangular graph without horizontal edges coloring a rectangular array: number of n X 3 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

28, 236, 2280, 20836, 202264, 1851020, 17970056, 164457412, 1596586328, 14611562156, 141852049992, 1298194798372, 12603142057880, 115340831992268, 1119752515193608, 10247697449986948, 99486754138704856, 910478112673392620

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Column 3 of A223352.

			Some solutions for n=3:
..1..4..1....1..4..2....2..0..2....0..2..4....5..2..0....4..1..0....4..2..5
..0..2..4....0..2..0....4..1..0....1..4..1....2..5..2....1..4..1....2..5..2
..1..0..1....2..4..1....2..4..1....0..1..3....0..2..4....4..1..0....4..2..5

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223352.

Empirical: a(n) = 91*a(n-2) - 192*a(n-4) + 64*a(n-6).

Empirical g.f.: 4*x*(7 + 59*x - 67*x^2 - 160*x^3 + 40*x^4 + 64*x^5) / (1 - 91*x^2 + 192*x^4 - 64*x^6). - Colin Barker, Aug 19 2018

A223348 3 X 3 X 3 triangular graph without horizontal edges coloring a rectangular array: number of n X 4 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

60, 1076, 20836, 405988, 7918948, 154482340, 3013692516, 58792282660, 1146943179236, 22375024222628, 436500886445412, 8515433203445028, 166122463695945956, 3240783209292085412, 63222490059635217508

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Column 4 of A223352.

			Some solutions for n=3:
..1..4..1..3....2..0..1..0....1..4..2..4....5..2..0..1....4..2..4..2
..4..2..0..1....4..2..0..1....0..2..4..2....2..0..1..3....1..4..2..4
..1..0..1..0....1..4..1..3....2..0..2..4....0..1..0..1....4..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223352.

Empirical: a(n) = 23*a(n-1) - 66*a(n-2) - 52*a(n-3) + 208*a(n-4) + 32*a(n-5) - 128*a(n-6).

Empirical g.f.: 4*x*(1 - 4*x)*(15 - 16*x - 52*x^2 + 16*x^3 + 32*x^4) / (1 - 23*x + 66*x^2 + 52*x^3 - 208*x^4 - 32*x^5 + 128*x^6). - Colin Barker, Aug 19 2018

A223349 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nX5 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

140, 4908, 202264, 7918948, 329268616, 12912752876, 537001573656, 21060038730884, 875825392204488, 34348003384801484, 1428433095896220376, 56020099749013006372, 2329712263359378360328, 91366346681513601154476

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 5 of A223352

			Some solutions for n=3
..1..0..2..4..1....0..1..0..2..5....4..1..4..1..3....0..2..0..2..4
..3..1..0..2..0....1..4..2..5..2....1..3..1..0..1....2..0..2..4..2
..1..0..2..0..2....0..1..0..2..0....0..1..4..1..0....0..2..4..2..5

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 1711*a(n-2) -132500*a(n-4) +3202240*a(n-6) -28936448*a(n-8) +102379520*a(n-10) -106561536*a(n-12) for n>13

A223350 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nX6 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

300, 22388, 1851020, 154482340, 12912752876, 1079538816324, 90254934876620, 7545802995190884, 630870570544801836, 52744249735952687492, 4409709467224003774348, 368675973859970441084836

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 6 of A223352

			Some solutions for n=3
..3..1..0..2..4..2....0..1..0..2..4..1....0..1..0..2..0..1....3..1..0..1..4..1
..1..0..1..4..2..4....2..0..1..0..2..0....1..4..1..4..1..4....1..4..1..3..1..3
..3..1..3..1..0..2....0..2..4..2..5..2....0..1..4..2..4..1....0..1..0..1..3..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 105*a(n-1) -1690*a(n-2) -12608*a(n-3) +372616*a(n-4) -412320*a(n-5) -25492864*a(n-6) +95198976*a(n-7) +669344256*a(n-8) -3814535168*a(n-9) -4806082560*a(n-10) +56384290816*a(n-11) -49291067392*a(n-12) -244915896320*a(n-13) +525445627904*a(n-14) -268301238272*a(n-15) -40399536128*a(n-16) +53687091200*a(n-17) -8589934592*a(n-18)

A223351 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nX7 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

700, 102124, 17970056, 3013692516, 537001573656, 90254934876620, 16086247351158920, 2703767114101083332, 481898364416857970648, 80997279330454571309932, 14436325294811073087488072

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 7 of A223352

			Some solutions for n=3
..0..1..0..2..0..1..3....0..1..0..2..0..2..4....0..1..0..1..0..1..0
..1..3..1..0..2..4..1....1..0..1..0..1..0..2....1..0..1..4..2..0..2
..0..1..3..1..0..2..0....0..2..4..1..4..2..5....0..2..0..1..4..1..4

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 32411*a(n-2) -75899152*a(n-4) +72830232672*a(n-6) -37138383320832*a(n-8) +11267693571807232*a(n-10) -2156486785001701376*a(n-12) +268134896419496787968*a(n-14) -21861109815107756490752*a(n-16) +1158919426002765454770176*a(n-18) -38882391728037610366435328*a(n-20) +785674063552141870019641344*a(n-22) -8851677936996895344820224000*a(n-24) +50604844225802475480926912512*a(n-26) -142492256277654023889441783808*a(n-28) +199942375039742209972462682112*a(n-30) -136858193061759279004901179392*a(n-32) +42178458483591906497874886656*a(n-34) -4502039752244879046605799424*a(n-36)

cp's OEIS Frontend

A223345 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of n X n 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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A223346 3 X 3 X 3 triangular graph without horizontal edges coloring a rectangular array: number of n X 1 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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A223347 3 X 3 X 3 triangular graph without horizontal edges coloring a rectangular array: number of n X 3 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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A223348 3 X 3 X 3 triangular graph without horizontal edges coloring a rectangular array: number of n X 4 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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A223349 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nX5 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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A223350 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nX6 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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A223351 3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nX7 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

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