A212782 -id:A212782 - cp's OEIS frontend

A212776 Half the number of 0..2 arrays of length n+2 with second differences nonzero.

11, 27, 66, 162, 397, 973, 2385, 5846, 14329, 35122, 86088, 211011, 517211, 1267741, 3107372, 7616509, 18668898, 45759514, 112161581, 274920321, 673859821, 1651704234, 4048508001, 9923336574, 24323184920, 59618790539, 146132186103

Offset: 1

R. H. Hardin May 27 2012

nonn

Column 2 of A212782

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

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

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

Empirical: G.f. -x*(11+16*x+17*x^2+9*x^3) / ( -1+x+2*x^2+3*x^3+2*x^4 ). - R. J. Mathar, Jul 01 2013

A212777 Half the number of 0..3 arrays of length n+2 with second differences nonzero.

Original entry on oeis.org

28, 99, 350, 1238, 4379, 15490, 54793, 193821, 685609, 2425226, 8578828, 30346157, 107344412, 379712753, 1343169821, 4751236703, 16806698494, 59450861306, 210297394895, 743891565706, 2631390949017, 9308101672049, 32925839761497

Offset: 1

R. H. Hardin, May 27 2012

nonn

Column 3 of A212782.

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

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

Cf. A212782.

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

Empirical g.f.: x*(28 + 43*x + 12*x^2 - 41*x^3 - 32*x^4) / (1 - 2*x - 5*x^2 - 3*x^3 + 4*x^4 + 4*x^5). - Colin Barker, Jul 21 2018

A212778 Half the number of 0..4 arrays of length n+2 with second differences nonzero.

Original entry on oeis.org

56, 252, 1134, 5104, 22972, 103391, 465336, 2094360, 9426184, 42424863, 190943548, 859388488, 3867889651, 17408390437, 78350750657, 352636859298, 1587129076523, 7143265484325, 32150026443551, 144699115914141, 651254025656788

Offset: 1

R. H. Hardin, May 27 2012

nonn

Column 4 of A212782.

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

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

Cf. A212782.

Empirical: a(n) = 2*a(n-1) + 7*a(n-2) + 14*a(n-3) + 19*a(n-4) + 18*a(n-5) + 5*a(n-6) - 5*a(n-7) - 2*a(n-8).

Empirical g.f.: x*(56 + 140*x + 238*x^2 + 288*x^3 + 234*x^4 + 47*x^5 - 68*x^6 - 25*x^7) / (1 - 2*x - 7*x^2 - 14*x^3 - 19*x^4 - 18*x^5 - 5*x^6 + 5*x^7 + 2*x^8). - Colin Barker, Jul 21 2018

A212779 Half the number of 0..5 arrays of length n+2 with second differences nonzero.

Original entry on oeis.org

99, 546, 3010, 16594, 91482, 504337, 2780392, 15328203, 84503842, 465866699, 2568306672, 14158984054, 78057979457, 430330886291, 2372398990909, 13078951921338, 72103800421506, 397505707375713, 2191434937861300

Offset: 1

R. H. Hardin, May 27 2012

nonn

Column 5 of A212782.

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

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

Cf. A212782.

Empirical: a(n) = 3*a(n-1) +12*a(n-2) +12*a(n-3) -6*a(n-4) -18*a(n-5) -14*a(n-6) -11*a(n-7) -16*a(n-8) -11*a(n-9) -a(n-10).

A212780 Half the number of 0..6 arrays of length n+2 with second differences nonzero.

Original entry on oeis.org

159, 1034, 6724, 43727, 284360, 1849215, 12025596, 78203437, 508563345, 3307229006, 21507180645, 139862954332, 909540228150, 5914814473647, 38464522156756, 250137932701244, 1626667429300337, 10578351299907331

Offset: 1

R. H. Hardin May 27 2012

nonn

Column 6 of A212782

			Some solutions for n=5
..0....0....3....0....0....0....0....0....0....4....0....4....0....0....0....0
..4....2....0....0....0....3....5....4....0....0....6....0....5....5....1....6
..6....2....4....2....2....0....0....1....3....1....6....0....0....1....3....4
..3....5....6....0....3....1....4....4....0....0....2....5....2....1....6....4
..2....1....1....3....2....3....4....6....6....5....1....1....5....0....4....1
..6....2....6....1....6....3....3....1....0....6....4....2....2....1....3....1
..2....0....3....1....4....1....6....5....6....1....3....1....2....4....0....0

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

Empirical: a(n) = 3*a(n-1) +15*a(n-2) +39*a(n-3) +66*a(n-4) +65*a(n-5) -3*a(n-6) -107*a(n-7) -162*a(n-8) -136*a(n-9) -96*a(n-10) -71*a(n-11) -31*a(n-12) -4*a(n-13)

A212781 Half the number of 0..7 arrays of length n+2 with second differences nonzero.

Original entry on oeis.org

240, 1803, 13544, 101743, 764297, 5741427, 43129809, 323992699, 2433844978, 18283132290, 137343556947, 1031729811702, 7750391995200, 58221227493827, 437359985531993, 3285464171376777, 24680526747938834, 185401017567775934

Offset: 1

R. H. Hardin May 27 2012

nonn

Column 7 of A212782

			Some solutions for n=5
..1....0....1....0....1....1....1....1....0....0....0....0....1....1....0....0
..4....0....6....2....4....0....4....6....4....6....0....6....0....6....0....0
..6....7....7....6....3....6....1....1....2....5....5....5....1....2....5....4
..0....6....6....2....0....6....2....2....4....2....1....0....5....4....2....6
..0....3....0....7....1....7....0....4....0....1....4....3....1....5....5....1
..1....3....4....0....6....5....5....7....3....1....4....3....0....0....6....6
..4....6....5....6....4....5....4....4....5....7....5....0....6....7....4....4

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

Empirical: a(n) = 4*a(n-1) +22*a(n-2) +33*a(n-3) +5*a(n-4) -36*a(n-5) -40*a(n-6) -28*a(n-7) -53*a(n-8) -73*a(n-9) -37*a(n-10) +12*a(n-11) +32*a(n-12) +39*a(n-13) +28*a(n-14) +6*a(n-15)

A212783 Half the number of 0..n arrays of length 4 with second differences nonzero.

Original entry on oeis.org

5, 27, 99, 252, 546, 1034, 1803, 2925, 4517, 6670, 9528, 13204, 17869, 23655, 30763, 39344, 49626, 61782, 76067, 92673, 111885, 133914, 159072, 187592, 219813, 255987, 296483, 341572, 391650, 447010, 508075, 575157, 648709, 729062, 816696, 911964

Offset: 1

R. H. Hardin, May 27 2012

nonn

Row 2 of A212782.

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

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

Cf. A212782.

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

Empirical g.f.: x*(5 + 17*x + 40*x^2 + 42*x^3 + 29*x^4 + 9*x^5 + 2*x^6) / ((1 - x)^5*(1 + x)^2*(1 + x + x^2)). - Colin Barker, Jul 21 2018

A212784 Half the number of 0..n arrays of length 5 with second differences nonzero.

Original entry on oeis.org

8, 66, 350, 1134, 3010, 6724, 13544, 24870, 42952, 70060, 109638, 165098, 241342, 343074, 476992, 649304, 868684, 1143156, 1483630, 1900056, 2405958, 3013428, 3738744, 4596484, 5605944, 6784266, 8154174, 9735586, 11554762, 13634780

Offset: 1

R. H. Hardin May 27 2012

nonn

Row 3 of A212782

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

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

Empirical: a(n) = a(n-1) +2*a(n-2) -2*a(n-4) -4*a(n-5) +a(n-6) +3*a(n-7) +3*a(n-8) +a(n-9) -4*a(n-10) -2*a(n-11) +2*a(n-13) +a(n-14) -a(n-15)

A212785 Half the number of 0..n arrays of length 6 with second differences nonzero.

Original entry on oeis.org

13, 162, 1238, 5104, 16594, 43727, 101743, 211461, 408430, 735896, 1261599, 2064329, 3259612, 4975687, 7395946, 10715635, 15205983, 21151890, 28937100, 38956477, 51737361, 67810309, 87873466, 112625630, 142969752, 179799242

Offset: 1

R. H. Hardin May 27 2012

nonn

Row 4 of A212782

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

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

Empirical: a(n) = a(n-2) +2*a(n-3) +2*a(n-4) -a(n-5) -2*a(n-6) -5*a(n-7) -3*a(n-8) +3*a(n-10) +6*a(n-11) +5*a(n-12) +3*a(n-13) -3*a(n-14) -5*a(n-15) -6*a(n-16) -3*a(n-17) +3*a(n-19) +5*a(n-20) +2*a(n-21) +a(n-22) -2*a(n-23) -2*a(n-24) -a(n-25) +a(n-27)

A212786 Half the number of 0..n arrays of length 7 with second differences nonzero.

Original entry on oeis.org

21, 397, 4379, 22972, 91482, 284360, 764297, 1797975, 3883755, 7729696, 14517156, 25811656, 44024953, 72163603, 114677011, 176842932, 266174946, 391374778, 564396617, 798717009, 1112552471, 1525915970, 2065331568, 2759616340

Offset: 1

R. H. Hardin May 27 2012

nonn

Row 5 of A212782

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

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

Empirical: a(n) = -a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +6*a(n-5) +7*a(n-6) +a(n-7) -6*a(n-8) -16*a(n-9) -20*a(n-10) -16*a(n-11) -2*a(n-12) +18*a(n-13) +32*a(n-14) +38*a(n-15) +24*a(n-16) +2*a(n-17) -27*a(n-18) -43*a(n-19) -43*a(n-20) -27*a(n-21) +2*a(n-22) +24*a(n-23) +38*a(n-24) +32*a(n-25) +18*a(n-26) -2*a(n-27) -16*a(n-28) -20*a(n-29) -16*a(n-30) -6*a(n-31) +a(n-32) +7*a(n-33) +6*a(n-34) +4*a(n-35) +a(n-36) -a(n-37) -a(n-38) -a(n-39)

cp's OEIS Frontend

A212776 Half the number of 0..2 arrays of length n+2 with second differences nonzero.

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A212777 Half the number of 0..3 arrays of length n+2 with second differences nonzero.

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A212778 Half the number of 0..4 arrays of length n+2 with second differences nonzero.

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A212779 Half the number of 0..5 arrays of length n+2 with second differences nonzero.

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A212780 Half the number of 0..6 arrays of length n+2 with second differences nonzero.

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A212781 Half the number of 0..7 arrays of length n+2 with second differences nonzero.

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A212783 Half the number of 0..n arrays of length 4 with second differences nonzero.

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A212784 Half the number of 0..n arrays of length 5 with second differences nonzero.

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A212785 Half the number of 0..n arrays of length 6 with second differences nonzero.

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A212786 Half the number of 0..n arrays of length 7 with second differences nonzero.

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