A228285 -id:A228285 - cp's OEIS frontend

A228279 Number of n X 4 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

3, 6, 35, 133, 587, 2448, 10414, 44024, 186414, 789100, 3340345, 14140347, 59858152, 253389483, 1072638232, 4540650778, 19221306410, 81366888278, 344439152622, 1458066449898, 6172230293325, 26128045670722, 110604228640954

Offset: 1

R. H. Hardin, Aug 19 2013

nonn

Column 4 of A228285.

			Some solutions for n=4:
..1..0..1..0....1..0..0..1....1..0..0..1....1..0..1..0....1..0..0..0
..0..0..0..0....0..0..0..0....0..0..1..0....0..0..0..0....0..0..0..1
..1..0..1..0....0..0..0..0....0..0..0..0....0..0..1..0....0..1..0..0
..0..0..0..0....0..0..1..0....1..0..0..1....1..0..0..0....0..0..0..0

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

See A228277-A228285, especially the latter.

a(n) = a(n-1) + 10*a(n-2) + 15*a(n-3) + 4*a(n-4) - 6*a(n-5) - a(n-6) + 3*a(n-7) - a(n-8).

G.f.: x*(1 - x)*(3 + 6*x + 5*x^2 - 2*x^3) / (1 - x - 10*x^2 - 15*x^3 - 4*x^4 + 6*x^5 + x^6 - 3*x^7 + x^8). - Colin Barker, Mar 16 2018

Edited by N. J. A. Sloane, Aug 22 2013

A228280 Number of nX5 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

Original entry on oeis.org

5, 13, 112, 587, 3631, 21166, 126119, 745178, 4416695, 26150120, 154877307, 917205757, 5431915952, 32169045631, 190512481196, 1128258633821, 6681806858103, 39571194265886, 234349700556332, 1387872742075595

Offset: 1

R. H. Hardin Aug 19 2013

nonn

Column 5 of A228285.

			Some solutions for n=4
..1..0..1..0..0....1..0..0..0..0....1..0..1..0..0....1..0..0..0..0
..0..0..0..0..0....0..0..0..1..0....0..0..0..0..0....0..0..0..0..1
..0..0..0..1..0....0..0..0..0..0....0..1..0..1..0....0..1..0..1..0
..0..1..0..0..0....0..0..1..0..1....1..0..0..0..0....1..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (1, 21, 48, 14, -69, -38, 68, 13, -57, 37, -8, -2, 1).

See A228277-A228285, especially the latter.

See also the k=5 variants of A228390, A228476, A228506 etc.

a(n) = a(n-1) +21*a(n-2) +48*a(n-3) +14*a(n-4) -69*a(n-5) -38*a(n-6) +68*a(n-7) +13*a(n-8) -57*a(n-9) +37*a(n-10) -8*a(n-11) -2*a(n-12) +a(n-13)

G.f.: x*(5 + 8*x - 6*x^2 - 38*x^3 - 2*x^4 - 5*x^5 + 45*x^6 - 51*x^7 + 26*x^8 - 4*x^9 - 2*x^10 + x^11)/((1 + 2*x - 2*x^3 + x^4)*(1 - 3*x - 15*x^2 - 16*x^3 + 11*x^4 + 20*x^5 - 19*x^6 + 8*x^7 - x^9)). - M. F. Hasler, Apr 27 2014

Edited by N. J. A. Sloane, Aug 22 2013

A228281 Number of nX6 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

Original entry on oeis.org

8, 28, 337, 2448, 21166, 172082, 1428523, 11771298, 97268701, 802886174, 6629901197, 54739811878, 451976078779, 3731849749697, 30812948919061, 254414847888742, 2100639733295629, 17344457600010491, 143208852222784259

Offset: 1

R. H. Hardin Aug 19 2013

nonn

Column 6 of A228285

			Some solutions for n=4
..1..0..0..0..0..0....1..0..0..0..1..0....1..0..0..0..1..0....1..0..1..0..0..1
..0..0..0..1..0..0....0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0
..0..1..0..0..0..1....0..0..0..0..0..0....0..0..1..0..0..1....0..0..0..0..0..0
..0..0..0..0..1..0....0..0..0..1..0..1....0..0..0..0..0..0....0..0..0..1..0..0

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

See A228277-A228285, especially the latter.

a(n) = a(n-1) +42*a(n-2) +147*a(n-3) +70*a(n-4) -478*a(n-5) -449*a(n-6) +1199*a(n-7) +732*a(n-8) -2727*a(n-9) +659*a(n-10) +3827*a(n-11) -5776*a(n-12) +3926*a(n-13) -1152*a(n-14) -148*a(n-15) +154*a(n-16) +32*a(n-17) -29*a(n-18) -6*a(n-19) +3*a(n-20) +a(n-21)

Edited by N. J. A. Sloane, Aug 22 2013

A228282 Number of nX7 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

Original entry on oeis.org

13, 60, 1034, 10414, 126119, 1428523, 16566199, 190540884, 2197847780, 25325358687, 291935092921, 3364727410265, 38782728207101, 447011297075966, 5152298718205939, 59385855860517581, 684486816741728022

Offset: 1

R. H. Hardin Aug 19 2013

nonn

Column 7 of A228285

			Some solutions for n=4
..1..0..1..0..1..0..1....1..0..1..0..1..0..1....1..0..0..0..0..0..0
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0
..0..0..0..0..0..0..1....0..0..0..0..0..0..1....0..0..1..0..1..0..1
..0..0..1..0..0..0..0....1..0..1..0..1..0..0....0..1..0..0..0..0..0

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

See A228277-A228285, especially the latter.

a(n) = a(n-1) +85*a(n-2) +432*a(n-3) +192*a(n-4) -3711*a(n-5) -5096*a(n-6) +21164*a(n-7) +27340*a(n-8) -112654*a(n-9) -37244*a(n-10) +477721*a(n-11) -464722*a(n-12) -897815*a(n-13) +3102284*a(n-14) -4149918*a(n-15) +2761082*a(n-16) -138325*a(n-17) -1353257*a(n-18) +942033*a(n-19) +64683*a(n-20) -365483*a(n-21) +80904*a(n-22) +92350*a(n-23) -27097*a(n-24) -23292*a(n-25) +2585*a(n-26) +5635*a(n-27) +1405*a(n-28) -561*a(n-29) -545*a(n-30) -173*a(n-31) -14*a(n-32) +5*a(n-33) +a(n-34)

Edited by N. J. A. Sloane, Aug 22 2013

A228283 Number of nX8 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

Original entry on oeis.org

21, 129, 3154, 44024, 745178, 11771298, 190540884, 3057290265, 49208639399, 791176762937, 12725363193829, 204647839919537, 3291296999329711, 52931964429099939, 851279732514835940, 13690693829364308175

Offset: 1

R. H. Hardin Aug 19 2013

nonn

Column 8 of A228285

			Some solutions for n=4
..1..0..0..0..1..0..1..0....1..0..0..1..0..0..1..0....1..0..0..0..1..0..0..0
..0..0..0..1..0..0..0..0....0..0..1..0..0..1..0..0....0..0..1..0..0..0..1..0
..0..0..0..0..0..1..0..0....0..0..0..0..1..0..0..0....0..0..0..0..0..0..0..0
..1..0..1..0..1..0..0..1....0..1..0..1..0..0..1..0....0..0..0..0..1..0..1..0

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

See A228277-A228285, especially the latter.

a(n) = a(n-1) +170*a(n-2) +1243*a(n-3) +452*a(n-4) -25328*a(n-5) -47829*a(n-6) +329977*a(n-7) +677669*a(n-8) -3899000*a(n-9) -4142351*a(n-10) +40598266*a(n-11) -22625542*a(n-12) -280040412*a(n-13) +733571937*a(n-14) +225285263*a(n-15) -5406913431*a(n-16) +15244941445*a(n-17) -23143116063*a(n-18) +18064627845*a(n-19) +2246759550*a(n-20) -21472294755*a(n-21) +18258761486*a(n-22) +5172378119*a(n-23) -19704882159*a(n-24) +6931675953*a(n-25) +12628340471*a(n-26) -10036097232*a(n-27) -6485990080*a(n-28) +8290578844*a(n-29) +3493944338*a(n-30) -5396581563*a(n-31) -2482388885*a(n-32) +2807416030*a(n-33) +2023351213*a(n-34) -877760575*a(n-35) -1366907685*a(n-36) -211758176*a(n-37) +511379314*a(n-38) +405881894*a(n-39) +67901250*a(n-40) -105839886*a(n-41) -110007288*a(n-42) -59224072*a(n-43) -20510652*a(n-44) -4214944*a(n-45) -234085*a(n-46) +87637*a(n-47) -2782*a(n-48) -10138*a(n-49) -671*a(n-50) +608*a(n-51) +45*a(n-52) -30*a(n-53) -a(n-54) +a(n-55)

Edited by N. J. A. Sloane, Aug 22 2013

A228284 Number of nX9 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

Original entry on oeis.org

34, 277, 9637, 186414, 4416695, 97268701, 2197847780, 49208639399, 1105411581741, 24801939723742, 556713719650007, 12494370307905104, 280426447918993931, 6293836975716291709, 141258681814255369288

Offset: 1

R. H. Hardin Aug 19 2013

nonn

Column 9 of A228285

			Some solutions for n=4
..1..0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..0..0
..0..0..0..1..0..0..1..0..0....0..0..0..1..0..0..1..0..0
..0..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0..0
..0..0..1..0..1..0..1..0..1....0..0..1..0..1..0..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 89

See A228277-A228285, especially the latter.

Satisfies a recurrence of order 89 (see link above) - see Zeilberger links in A228285 for a proof that this is correct, and for a g.f.

Edited by N. J. A. Sloane, Aug 22 2013

A228481 Number of nX7 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

Original entry on oeis.org

13, 88, 1181, 13050, 152373, 1748684, 20185842, 232542935, 2680777055, 30897168815, 356129525933, 4104762592860, 47311923541451, 545321432832844, 6285425006257311, 72446383973495105, 835023655383918595

Offset: 1

R. H. Hardin Aug 22 2013

nonn

Column 7 of A228482

Same recurrences as A228285 except in addition a smaller one for column 5

			Some solutions for n=4
..1..0..0..0..0..0..1....1..0..1..0..0..0..0....1..0..0..0..0..1..0
..0..0..0..0..0..0..0....0..0..0..0..0..1..0....0..0..0..1..0..0..0
..1..0..0..0..0..0..0....1..0..0..0..0..0..0....1..0..0..0..1..0..0
..0..0..1..0..0..1..0....0..1..0..1..0..0..1....0..0..0..0..0..0..0

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

a(n) = a(n-1) +85*a(n-2) +432*a(n-3) +192*a(n-4) -3711*a(n-5) -5096*a(n-6) +21164*a(n-7) +27340*a(n-8) -112654*a(n-9) -37244*a(n-10) +477721*a(n-11) -464722*a(n-12) -897815*a(n-13) +3102284*a(n-14) -4149918*a(n-15) +2761082*a(n-16) -138325*a(n-17) -1353257*a(n-18) +942033*a(n-19) +64683*a(n-20) -365483*a(n-21) +80904*a(n-22) +92350*a(n-23) -27097*a(n-24) -23292*a(n-25) +2585*a(n-26) +5635*a(n-27) +1405*a(n-28) -561*a(n-29) -545*a(n-30) -173*a(n-31) -14*a(n-32) +5*a(n-33) +a(n-34)

A228477 Number of nX3 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

Original entry on oeis.org

2, 4, 14, 41, 127, 386, 1181, 3605, 11013, 33635, 102734, 313780, 958385, 2927208, 8940618, 27307464, 83405606, 254747013, 778077691, 2376494562, 7258563605, 22169941573, 67713990833, 206819875427, 631693101322, 1929389878184

Offset: 1

R. H. Hardin Aug 22 2013

nonn

Column 3 of A228482

Same recurrences as A228285 except in addition a smaller one for column 5

			Some solutions for n=4
..1..0..1....1..0..1....1..0..0....1..0..0....1..0..0....1..0..0....1..0..0
..0..0..0....0..0..0....0..1..0....0..0..1....0..0..0....0..0..0....0..0..1
..1..0..1....0..0..0....0..0..0....1..0..0....1..0..1....0..0..0....1..0..0
..0..0..0....0..0..0....1..0..0....0..0..0....0..0..0....0..0..0....0..0..1

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

a(n) = a(n-1) +5*a(n-2) +4*a(n-3) -a(n-5).

G.f.: -x*(-2-2*x+x^3) / ( (1+x)*(x^4-x^3-3*x^2-2*x+1) ). - R. J. Mathar, Aug 25 2013

A228478 Number of n X 4 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

Original entry on oeis.org

3, 9, 41, 172, 728, 3084, 13050, 55252, 233875, 990055, 4191028, 17741339, 75101906, 317918500, 1345800258, 5696989354, 24116274286, 102088075950, 432155280235, 1829382955772, 7744072911728, 32781908823977, 138771103836595

Offset: 1

R. H. Hardin, Aug 22 2013

nonn

Same recurrences as A228285 except in addition a smaller one for column 5.

			Some solutions for n=4:
..1..0..1..0....1..0..1..0....1..0..0..0....1..0..0..1....1..0..0..0
..0..0..0..1....0..0..0..0....0..1..0..1....0..0..0..0....0..0..0..0
..1..0..0..0....0..0..0..1....0..0..0..0....1..0..0..1....0..1..0..1
..0..1..0..0....0..1..0..0....0..1..0..1....0..1..0..0....0..0..0..0

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

Column 4 of A228482.

a(n) = a(n-1) + 10*a(n-2) + 15*a(n-3) + 4*a(n-4) - 6*a(n-5) - a(n-6) + 3*a(n-7) - a(n-8).

Empirical g.f.: x*(3 + 6*x + 2*x^2 - 4*x^3 - x^4 + 3*x^5 - x^6) / (1 - x - 10*x^2 - 15*x^3 - 4*x^4 + 6*x^5 + x^6 - 3*x^7 + x^8). - Colin Barker, Sep 11 2018

A228479 Number of n X 5 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

Original entry on oeis.org

5, 19, 127, 728, 4354, 25699, 152373, 902042, 5342712, 31639786, 187379548, 1109702480, 6571916787, 38920392611, 230495519461, 1365047364511, 8084124028133, 47876038862427, 283532896714060, 1679146925634733

Offset: 1

R. H. Hardin, Aug 22 2013

nonn

Same recurrences as A228285 except in addition this smaller one for this column.

			Some solutions for n=4:
..1..0..1..0..0....1..0..0..0..0....1..0..0..0..0....1..0..1..0..0
..0..0..0..1..0....0..0..0..0..1....0..0..0..0..0....0..0..0..1..0
..1..0..0..0..1....1..0..0..0..0....1..0..0..0..0....1..0..0..0..0
..0..0..1..0..0....0..0..0..0..0....0..0..0..1..0....0..1..0..0..1

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

Column 5 of A228482.

Empirical: a(n) = 3*a(n-1) + 15*a(n-2) + 16*a(n-3) - 11*a(n-4) - 20*a(n-5) + 19*a(n-6) - 8*a(n-7) + a(n-9).

Empirical g.f.: x*(5 + 4*x - 5*x^2 - 18*x^3 + 16*x^4 - 6*x^5 + x^7) / (1 - 3*x - 15*x^2 - 16*x^3 + 11*x^4 + 20*x^5 - 19*x^6 + 8*x^7 - x^9). - Colin Barker, Sep 11 2018

cp's OEIS Frontend

A228279 Number of n X 4 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A228280 Number of nX5 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A228281 Number of nX6 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A228282 Number of nX7 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A228283 Number of nX8 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A228284 Number of nX9 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A228481 Number of nX7 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

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A228477 Number of nX3 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

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A228478 Number of n X 4 binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

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