A257062 -id:A257062 - cp's OEIS frontend

A257059 Number of length n 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

1, 4, 18, 81, 1024, 7353, 106056, 1249592, 22143847, 282475249, 8589934592, 133642251792, 3531308221901, 71789886480657, 2171807721018968, 45949729863572161, 2218611106740436992, 53241469411989452625

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

Diagonal of A257062

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

R. H. Hardin, Table of n, a(n) for n = 1..93

Cf. A257062

A257060 Number of length n 1..(6+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

4, 18, 81, 364, 1636, 7353, 33048, 148534, 667585, 3000456, 13485528, 60610609, 272413948, 1224362538, 5502888657, 24732693652, 111160914460, 499611933801, 2245502257776, 10092393813166, 45360191702017, 203871056692176

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

			Some solutions for n=4:
..4....4....3....4....2....2....3....4....3....2....4....6....6....3....4....6
..4....2....3....2....6....2....7....2....5....4....5....4....4....3....5....2
..6....2....3....2....2....4....5....3....7....3....7....6....5....6....7....4
..2....1....6....4....6....1....3....3....7....1....6....2....7....3....4....4

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

Column 6 of A257062.

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

Empirical g.f.: x*(4 + 2*x + x^2) / (1 - 4*x - 2*x^2 - x^3). - Colin Barker, Dec 20 2018

A257061 Number of length n 1..(7+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

5, 27, 141, 738, 3866, 20249, 106056, 555483, 2909419, 15238479, 79813616, 418034724, 2189514005, 11467878868, 60064583029, 314596463387, 1647741976789, 8630273820766, 45202238742834, 236752903766237, 1240025693431636

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

			Some solutions for n=4:
..4....6....4....8....3....6....6....6....2....2....2....3....4....4....3....3
..8....8....6....8....5....2....8....3....2....1....7....3....8....2....7....5
..4....1....5....6....8....4....2....1....4....7....6....8....6....6....5....8
..6....1....7....5....2....3....4....4....6....4....1....1....3....3....1....4

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

Column 7 of A257062.

Empirical: a(n) = 4*a(n-1) + 5*a(n-2) + 7*a(n-3) + 4*a(n-4).

Empirical g.f.: x*(5 + 7*x + 8*x^2 + 4*x^3) / (1 - 4*x - 5*x^2 - 7*x^3 - 4*x^4). - Colin Barker, Dec 20 2018

A257063 Number of length 1 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 23, 24, 24, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 40, 41, 42, 43, 43, 44, 44, 45, 46, 47, 47, 48, 48, 49, 50, 51, 51

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

			All solutions for n=4:
..2....4....3

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

Row 1 of A257062.

Empirical: a(n) = a(n-1) + a(n-6) - a(n-7).
Empirical for n mod 6 = 0: a(n) = (2/3)*n
Empirical for n mod 6 = 1: a(n) = (2/3)*n + (1/3)
Empirical for n mod 6 = 2: a(n) = (2/3)*n + (2/3)
Empirical for n mod 6 = 3: a(n) = (2/3)*n + 1
Empirical for n mod 6 = 4: a(n) = (2/3)*n + (1/3)
Empirical for n mod 6 = 5: a(n) = (2/3)*n + (2/3).
Empirical g.f.: x*(1 + x + x^2 + x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)). - Colin Barker, Dec 20 2018

A257064 Number of length 2 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

2, 4, 7, 9, 16, 18, 27, 35, 45, 49, 64, 68, 84, 98, 115, 121, 144, 150, 173, 193, 217, 225, 256, 264, 294, 320, 351, 361, 400, 410, 447, 479, 517, 529, 576, 588, 632, 670, 715, 729, 784, 798, 849, 893, 945, 961, 1024, 1040, 1098, 1148, 1207, 1225, 1296, 1314, 1379

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

			All solutions for n=4:
..2....4....4....3....3....2....3....2....4
..1....2....4....1....3....2....5....4....5

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

Row 2 of A257062.

Empirical: a(n) = a(n-1) + 2*a(n-6) - 2*a(n-7) - a(n-12) + a(n-13).
Empirical for n mod 6 = 0: a(n) = (4/9)*n^2 + (1/3)*n
Empirical for n mod 6 = 1: a(n) = (4/9)*n^2 + (11/18)*n + (17/18)
Empirical for n mod 6 = 2: a(n) = (4/9)*n^2 + (13/18)*n + (7/9)
Empirical for n mod 6 = 3: a(n) = (4/9)*n^2 + 1*n
Empirical for n mod 6 = 4: a(n) = (4/9)*n^2 + (4/9)*n + (1/9)
Empirical for n mod 6 = 5: a(n) = (4/9)*n^2 + (8/9)*n + (4/9).
Empirical g.f.: x*(2 + 2*x + 3*x^2 + 2*x^3 + 7*x^4 + 2*x^5 + 5*x^6 + 4*x^7 + 4*x^8 + x^10) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2). - Colin Barker, Dec 20 2018

A257065 Number of length 3 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

2, 6, 18, 27, 64, 81, 141, 200, 293, 343, 512, 578, 776, 954, 1208, 1331, 1728, 1875, 2291, 2652, 3147, 3375, 4096, 4356, 5070, 5678, 6494, 6859, 8000, 8405, 9497, 10416, 11633, 12167, 13824, 14406, 15956, 17250, 18948, 19683, 21952, 22743, 24831, 26564

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

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

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

Row 3 of A257062.

Empirical: a(n) = a(n-1) + 3*a(n-6) - 3*a(n-7) - 3*a(n-12) + 3*a(n-13) + a(n-18) - a(n-19).
Empirical for n mod 6 = 0: a(n) = (8/27)*n^3 + (4/9)*n^2 + (1/6)*n
Empirical for n mod 6 = 1: a(n) = (8/27)*n^3 + (2/3)*n^2 + (17/18)*n + (5/54)
Empirical for n mod 6 = 2: a(n) = (8/27)*n^3 + (2/3)*n^2 + (7/9)*n - (16/27)
Empirical for n mod 6 = 3: a(n) = (8/27)*n^3 + (8/9)*n^2 + (1/2)*n + (1/2)
Empirical for n mod 6 = 4: a(n) = (8/27)*n^3 + (4/9)*n^2 + (2/9)*n + (1/27)
Empirical for n mod 6 = 5: a(n) = (8/27)*n^3 + (8/9)*n^2 + (8/9)*n + (8/27).
Empirical g.f.: x*(2 + 4*x + 12*x^2 + 9*x^3 + 37*x^4 + 17*x^5 + 54*x^6 + 47*x^7 + 57*x^8 + 23*x^9 + 58*x^10 + 15*x^11 + 24*x^12 + 13*x^13 + 11*x^14 + x^16) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^3*(1 + x + x^2)^3). - Colin Barker, Dec 20 2018

A257066 Number of length 4 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

2, 11, 45, 81, 256, 364, 738, 1149, 1905, 2401, 4096, 4912, 7172, 9297, 12685, 14641, 20736, 23436, 30344, 36455, 45633, 50625, 65536, 71872, 87438, 100767, 120141, 130321, 160000, 172300, 201782, 226521, 261745, 279841, 331776, 352944, 402848

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

Row 4 of A257062

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

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

Cf. A257062

Empirical: a(n) = a(n-1) +4*a(n-6) -4*a(n-7) -6*a(n-12) +6*a(n-13) +4*a(n-18) -4*a(n-19) -a(n-24) +a(n-25)
Empirical for n mod 6 = 0: a(n) = (16/81)*n^4 + (4/9)*n^3 + (1/3)*n^2
Empirical for n mod 6 = 1: a(n) = (16/81)*n^4 + (50/81)*n^3 + (107/108)*n^2 + (44/81)*n - (113/324)
Empirical for n mod 6 = 2: a(n) = (16/81)*n^4 + (46/81)*n^3 + (83/108)*n^2 - (7/162)*n + (25/81)
Empirical for n mod 6 = 3: a(n) = (16/81)*n^4 + (20/27)*n^3 + (7/9)*n^2 + (2/3)*n
Empirical for n mod 6 = 4: a(n) = (16/81)*n^4 + (32/81)*n^3 + (8/27)*n^2 + (8/81)*n + (1/81)
Empirical for n mod 6 = 5: a(n) = (16/81)*n^4 + (64/81)*n^3 + (32/27)*n^2 + (64/81)*n + (16/81)

A257067 Number of length 5 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

3, 20, 113, 243, 1024, 1636, 3866, 6599, 12387, 16807, 32768, 41744, 66291, 90598, 133205, 161051, 248832, 292932, 401910, 501113, 661703, 759375, 1048576, 1185856, 1507979, 1788296, 2222649, 2476099, 3200000, 3532100, 4287258, 4926235, 5889323

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

Row 5 of A257062

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

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

Cf. A257062

Empirical: a(n) = a(n-2) +a(n-3) -a(n-5) +4*a(n-6) -4*a(n-8) -4*a(n-9) +4*a(n-11) -6*a(n-12) +6*a(n-14) +6*a(n-15) -6*a(n-17) +4*a(n-18) -4*a(n-20) -4*a(n-21) +4*a(n-23) -a(n-24) +a(n-26) +a(n-27) -a(n-29)
Empirical for n mod 6 = 0: a(n) = (32/243)*n^5 + (32/81)*n^4 + (4/9)*n^3 + (1/9)*n^2
Empirical for n mod 6 = 1: a(n) = (32/243)*n^5 + (128/243)*n^4 + (487/486)*n^3 + (853/972)*n^2 + (34/243)*n + (313/972)
Empirical for n mod 6 = 2: a(n) = (32/243)*n^5 + (112/243)*n^4 + (355/486)*n^3 + (145/486)*n^2 + (125/486)*n + (209/243)
Empirical for n mod 6 = 3: a(n) = (32/243)*n^5 + (16/27)*n^4 + (8/9)*n^3 + (8/9)*n^2 + (1/3)*n
Empirical for n mod 6 = 4: a(n) = (32/243)*n^5 + (80/243)*n^4 + (80/243)*n^3 + (40/243)*n^2 + (10/243)*n + (1/243)
Empirical for n mod 6 = 5: a(n) = (32/243)*n^5 + (160/243)*n^4 + (320/243)*n^3 + (320/243)*n^2 + (160/243)*n + (32/243)

A257068 Number of length 6 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

4, 33, 284, 729, 4096, 7353, 20249, 37893, 80545, 117649, 262144, 354756, 612724, 882855, 1398784, 1771561, 2985984, 3661425, 5323339, 6888321, 9595049, 11390625, 16777216, 19566096, 26006996, 31736589, 41119756, 47045881, 64000000, 72407025

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

Row 6 of A257062

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

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

Cf. A257062

Empirical: a(n) = a(n-1) +6*a(n-6) -6*a(n-7) -15*a(n-12) +15*a(n-13) +20*a(n-18) -20*a(n-19) -15*a(n-24) +15*a(n-25) +6*a(n-30) -6*a(n-31) -a(n-36) +a(n-37)
Empirical for n mod 6 = 0: a(n) = (64/729)*n^6 + (80/243)*n^5 + (40/81)*n^4 + (7/27)*n^3 + (1/36)*n^2
Empirical for n mod 6 = 1: a(n) = (64/729)*n^6 + (104/243)*n^5 + (26/27)*n^4 + (6515/5832)*n^3 + (1181/1944)*n^2 + (161/648)*n + (3197/5832)
Empirical for n mod 6 = 2: a(n) = (64/729)*n^6 + (88/243)*n^5 + (2/3)*n^4 + (2917/5832)*n^3 + (337/972)*n^2 + (50/81)*n - (1091/729)
Empirical for n mod 6 = 3: a(n) = (64/729)*n^6 + (112/243)*n^5 + (8/9)*n^4 + (29/27)*n^3 + (25/36)*n^2 + (1/6)*n + (1/4)
Empirical for n mod 6 = 4: a(n) = (64/729)*n^6 + (64/243)*n^5 + (80/243)*n^4 + (160/729)*n^3 + (20/243)*n^2 + (4/243)*n + (1/729)
Empirical for n mod 6 = 5: a(n) = (64/729)*n^6 + (128/243)*n^5 + (320/243)*n^4 + (1280/729)*n^3 + (320/243)*n^2 + (128/243)*n + (64/729)

A257069 Number of length 7 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

Original entry on oeis.org

4, 59, 713, 2187, 16384, 33048, 106056, 217603, 523733, 823543, 2097152, 3014848, 5663370, 8603223, 14688611, 19487171, 35831808, 45765000, 70508164, 94687187, 139133363, 170859375, 268435456, 322831872, 448523376, 563223955, 760729349

Offset: 1

R. H. Hardin, Apr 15 2015

nonn

Row 7 of A257062

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

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

Cf. A257062

Empirical: a(n) = a(n-1) +7*a(n-6) -7*a(n-7) -21*a(n-12) +21*a(n-13) +35*a(n-18) -35*a(n-19) -35*a(n-24) +35*a(n-25) +21*a(n-30) -21*a(n-31) -7*a(n-36) +7*a(n-37) +a(n-42) -a(n-43)
Empirical for n mod 6 = 0: a(n) = (128/2187)*n^7 + (64/243)*n^6 + (40/81)*n^5 + (32/81)*n^4 + (1/9)*n^3
Empirical for n mod 6 = 1: a(n) = (128/2187)*n^7 + (736/2187)*n^6 + (644/729)*n^5 + (2765/2187)*n^4 + (17999/17496)*n^3 + (2885/5832)*n^2 + (9121/17496)*n - (10279/17496)
Empirical for n mod 6 = 2: a(n) = (128/2187)*n^7 + (608/2187)*n^6 + (428/729)*n^5 + (1321/2187)*n^4 + (4141/8748)*n^3 + (775/1458)*n^2 - (1534/2187)*n + (1649/2187)
Empirical for n mod 6 = 3: a(n) = (128/2187)*n^7 + (256/729)*n^6 + (200/243)*n^5 + (32/27)*n^4 + (28/27)*n^3 + (17/36)*n^2 + (1/3)*n - (1/4)
Empirical for n mod 6 = 4: a(n) = (128/2187)*n^7 + (448/2187)*n^6 + (224/729)*n^5 + (560/2187)*n^4 + (280/2187)*n^3 + (28/729)*n^2 + (14/2187)*n + (1/2187)
Empirical for n mod 6 = 5: a(n) = (128/2187)*n^7 + (896/2187)*n^6 + (896/729)*n^5 + (4480/2187)*n^4 + (4480/2187)*n^3 + (896/729)*n^2 + (896/2187)*n + (128/2187)

cp's OEIS Frontend

A257059 Number of length n 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257060 Number of length n 1..(6+1) arrays with every leading partial sum divisible by 2 or 3.

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A257061 Number of length n 1..(7+1) arrays with every leading partial sum divisible by 2 or 3.

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A257063 Number of length 1 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257064 Number of length 2 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257065 Number of length 3 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257066 Number of length 4 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257067 Number of length 5 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257068 Number of length 6 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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A257069 Number of length 7 1..(n+1) arrays with every leading partial sum divisible by 2 or 3.

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