A245995 -id:A245995 - cp's OEIS frontend

A245996 Number of length 1+2 0..n arrays with no pair in any consecutive three terms totaling exactly n.

2, 8, 28, 64, 126, 216, 344, 512, 730, 1000, 1332, 1728, 2198, 2744, 3376, 4096, 4914, 5832, 6860, 8000, 9262, 10648, 12168, 13824, 15626, 17576, 19684, 21952, 24390, 27000, 29792, 32768, 35938, 39304, 42876, 46656, 50654, 54872, 59320, 64000, 68922

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

From Pontus von Brömssen, Jan 10 2022: (Start)
Proof of the empirical observations in the Formula section:
For k = 1, 2, 3, let N_k be the number of triples (x, y, z) with x, y, and z in 0..n, that satisfy x+y = n (if k=1), x+y = y+z = n (if k=2), or x+y = y+z = z+x = n (if k=3).
By inclusion-exclusion (and symmetry between x, y, and z), a(n) = (n+1)^3 - 3*N_1 + 3*N_2 - N_3. The unique solution to x+y = y+z = z+x = n is x = y = z = n/2, so N_3 = 1 if n is even, otherwise N_3 = 0. We write this as N_3 = [n even]. It is easily seen that N_1 = (n+1)^2 (x and z can be chosen freely and y = n-x) and that N_2 = n+1 (y can be chosen freely and x = z = n-y), so a(n) = (n+1)^3 - 3*(n+1)^2 + 3*(n+1) - [n even] = n^3 + [n odd] = 2*ceiling(n^3/2) = 2*A036486(n).
The recurrence and the generating function follow from this. (End)

			Some solutions for n=10:
  6   9   5   8   0   5   8   6   9   8   5   0   4   8   5   2
  3   8   3   0   0   7   9   5   0   4   7   5   2   4   7   6
  6   9   6   9   5   9   7   3   7   4   1   7  10   0   2   6

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Row 1 of A245995.

Cf. A036486.

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
From R. J. Mathar, Aug 10 2014: (Start)
Empirical: a(n) = 2*A036486(n).
G.f.: 2*x*(1+x+4*x^2) / ( (1+x)*(x-1)^4 ). (End)

A245990 Number of length n+2 0..3 arrays with no pair in any consecutive three terms totalling exactly 3.

Original entry on oeis.org

28, 68, 164, 396, 956, 2308, 5572, 13452, 32476, 78404, 189284, 456972, 1103228, 2663428, 6430084, 15523596, 37477276, 90478148, 218433572, 527345292, 1273124156, 3073593604, 7420311364, 17914216332, 43248744028, 104411704388

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

			Some solutions for n=10:
  3  2  1  1  2  1  1  2  0  2  3  0  1  3  3  0
  3  2  1  0  2  3  0  2  2  3  2  0  3  1  2  2
  3  3  3  0  3  1  0  2  0  2  2  0  1  1  2  0
  3  2  1  1  3  1  0  2  0  3  2  1  3  3  3  2
  3  3  1  1  1  1  0  0  0  3  0  1  1  1  3  2
  2  3  0  0  3  3  2  2  1  2  2  1  1  3  2  2
  2  2  1  0  1  1  2  2  1  3  0  1  1  3  2  2
  2  3  1  1  3  3  3  2  1  3  2  0  3  3  3  0
  3  3  1  1  3  1  3  2  3  3  0  0  1  2  2  0
  2  1  0  3  3  1  2  3  3  2  2  0  1  3  2  1
  3  1  0  1  3  0  3  3  2  2  2  0  0  3  3  1
  2  0  1  3  3  0  2  1  3  0  3  1  0  1  2  3

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

Column 3 of A245995.

Essentially 4 times A001333.

Empirical: a(n) = 2*a(n-1) + a(n-2).
Conjectures from Colin Barker, Nov 05 2018: (Start)
G.f.: 4*x*(7 + 3*x) / (1 - 2*x - x^2).
a(n) = sqrt(2)*((1-sqrt(2))^n*(-4+3*sqrt(2)) + (1+sqrt(2))^n*(4+3*sqrt(2))).
(End)

A245989 Number of length n 0..2 arrays with no pair in any consecutive three terms totalling exactly 2.

Original entry on oeis.org

1, 3, 6, 8, 12, 18, 26, 38, 56, 82, 120, 176, 258, 378, 554, 812, 1190, 1744, 2556, 3746, 5490, 8046, 11792, 17282, 25328, 37120, 54402, 79730, 116850, 171252, 250982, 367832, 539084, 790066, 1157898, 1696982, 2487048, 3644946, 5341928, 7828976, 11473922

Offset: 0

R. H. Hardin, Aug 09 2014

Also, number of length n ternary words with no pair of equal consecutive letters and avoiding the subwords 010, 101, 020, 202. - Miquel A. Fiol, Dec 22 2023

			Some solutions for n=12:
  0  1  0  1  1  0  2  2  0  2  0  2  0  0  0  1
  0  2  1  2  0  0  1  2  1  1  0  2  0  0  1  2
  0  2  0  2  0  1  2  1  0  2  0  2  0  0  0  2
  0  2  0  1  0  0  2  2  0  2  1  1  1  1  0  1
  0  1  1  2  0  0  2  2  0  2  0  2  0  0  0  2
  0  2  0  2  1  1  2  1  1  1  0  2  0  0  0  2
  0  2  0  2  0  0  1  2  0  2  0  2  0  0  1  2
  0  2  0  2  0  0  2  2  0  2  0  2  0  0  0  2
  0  1  1  2  1  1  2  1  0  2  0  1  1  0  0  1
  0  2  0  1  0  0  2  2  0  2  0  2  0  0  0  2
  0  2  0  2  0  0  2  2  0  1  1  2  0  0  0  2
  0  2  0  2  1  0  1  1  0  2  0  1  0  0  1  2

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (210 terms from R. H. Hardin)

Column 2 of A245995.

gf=(x^4 + x^3 + 3*x^2 + 2*x + 1) / (1 - x - x^3);Table[SeriesCoefficient[gf, {x, 0, n}], {n, 0, 40}]  (* James C. McMahon, Dec 30 2023 *)

a(n) = a(n-1) + a(n-3) for n>=5.

G.f.: (x^4 + x^3 + 3*x^2 + 2*x + 1) / (1 - x - x^3). - Colin Barker, Nov 05 2018

Edited by Alois P. Heinz, Dec 30 2023

A245991 Number of length n+2 0..4 arrays with no pair in any consecutive three terms totalling exactly 4.

Original entry on oeis.org

64, 208, 676, 2196, 7132, 23168, 75260, 244464, 794096, 2579500, 8379052, 27217860, 88412560, 287192948, 932896352, 3030352272, 9843575108, 31975148500, 103865731612, 337389844512, 1095952482460, 3560011839440, 11564081956784

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

			Some solutions for n=7:
  3  3  4  0  2  2  3  2  2  3  4  3  4  0  4  0
  4  4  1  3  1  1  4  3  4  2  2  0  1  3  2  1
  4  4  1  0  1  1  3  4  4  3  4  2  2  2  1  1
  4  2  2  2  0  4  4  3  2  3  3  0  0  3  4  2
  3  4  0  0  2  2  3  3  1  0  2  1  3  4  4  4
  3  3  0  3  0  1  2  3  4  0  4  2  3  3  3  4
  2  3  0  0  1  1  0  0  2  2  1  1  0  2  4  4
  4  0  3  0  1  2  3  0  1  3  2  4  2  3  4  4
  4  0  2  1  1  4  0  3  0  4  1  1  1  0  2  4

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

Column 4 of A245995.

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

Empirical g.f.: 4*x*(16 + 20*x + 49*x^2 + 15*x^3) / (1 - 2*x - x^2 - 9*x^3 - 3*x^4). - Colin Barker, Nov 05 2018

A245992 Number of length n+2 0..5 arrays with no pair in any consecutive three terms totalling exactly 5.

Original entry on oeis.org

126, 534, 2262, 9582, 40590, 171942, 728358, 3085374, 13069854, 55364790, 234529014, 993480846, 4208452398, 17827290438, 75517614150, 319897747038, 1355108602302, 5740332156246, 24316437227286, 103006081065390

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

Column 5 of A245995

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

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

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

Empirical: G.f.: -6*x*(21+5*x) / ( -1+4*x+x^2 ). - R. J. Mathar, Aug 10 2014

A245993 Number of length n+2 0..6 arrays with no pair in any consecutive three terms totalling exactly 6.

Original entry on oeis.org

216, 1116, 5766, 29790, 153906, 795144, 4108062, 21223992, 109652160, 566509902, 2926831878, 15121261374, 78122885724, 403616150730, 2085253182384, 10773307330236, 55659500700198, 287560720444278, 1485661331645130

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

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

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

Column 6 of A245995.

Empirical: a(n) = 4*a(n-1) + a(n-2) + 25*a(n-3) + 5*a(n-4).

Empirical g.f.: 6*x*(36 + 42*x + 181*x^2 + 35*x^3) / (1 - 4*x - x^2 - 25*x^3 - 5*x^4). - Colin Barker, Nov 06 2018

A245994 Number of length n+2 0..7 arrays with no pair in any consecutive three terms totalling exactly 7.

Original entry on oeis.org

344, 2120, 13064, 80504, 496088, 3057032, 18838280, 116086712, 715358552, 4408238024, 27164786696, 167396958200, 1031546535896, 6356676173576, 39171603577352, 241386297637688, 1487489389403480, 9166322634058568

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

Column 7 of A245995

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

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

Empirical: a(n) = 6*a(n-1) +a(n-2).

Empirical: G.f.: -8*x*(43+7*x) / ( -1+6*x+x^2 ). - R. J. Mathar, Aug 10 2014

A245997 Number of length 2+2 0..n arrays with no pair in any consecutive three terms totalling exactly n.

Original entry on oeis.org

2, 12, 68, 208, 534, 1116, 2120, 3648, 5930, 9100, 13452, 19152, 26558, 35868, 47504, 61696, 78930, 99468, 123860, 152400, 185702, 224092, 268248, 318528, 375674, 440076, 512540, 593488, 683790, 783900, 894752, 1016832, 1151138, 1298188

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

Row 2 of A245995

			Some solutions for n=10
..5....2....2....2....8....9....2....4....6....8....8....5....3....0....1....6
..0....2....7....7....7....4....9...10....7....6...10....0....6....4....2...10
..1....1...10....2....5....3....6....4....2....6....7....7....0....4....6....9
..8....1....5....7....0....3....9....5....6....3...10....7....0...10....2....3

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

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

Empirical: G.f.: -2*x*(1+3*x+17*x^2+13*x^3+14*x^4) / ( (1+x)^2*(x-1)^5 ). - R. J. Mathar, Aug 10 2014

A245998 Number of length 3+2 0..n arrays with no pair in any consecutive three terms totalling exactly n.

Original entry on oeis.org

2, 18, 164, 676, 2262, 5766, 13064, 25992, 48170, 82810, 135852, 212268, 320894, 468846, 668432, 929296, 1267794, 1696482, 2236340, 2903220, 3723302, 4716118, 5913624, 7339416, 9031802, 11018826, 13345724, 16045372, 19170510, 22759230

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

Row 3 of A245995

			Some solutions for n=10
..6....9....2....7....1...10....9....7....9....9....3....5....1....5....5....6
..8....9....7...10....4....8....8....9...10....6....9....6....5....6....8....1
..3...10...10....9....3....4...10....0....3....2....2....1....2....7....3....5
..6....2....5....4....2....9...10....3....2...10....4....2....1....2....5....1
..3....2...10....4....9...10....6....0....9....9...10....3....7....1....8....0

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

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

Empirical: G.f.: 2*x*(1+6*x+55*x^2+100*x^3+183*x^4+86*x^5+49*x^6) / ( (1+x)^3*(x-1)^6 ). - R. J. Mathar, Aug 10 2014

A245999 Number of length 4+2 0..n arrays with no pair in any consecutive three terms totalling exactly n.

Original entry on oeis.org

2, 26, 396, 2196, 9582, 29790, 80504, 185192, 391290, 753570, 1371972, 2352636, 3877286, 6128486, 9405552, 13997520, 20363634, 28934442, 40377980, 55306340, 74651742, 99252846, 130367976, 169112376, 217138922, 275894450, 347501364

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

Row 4 of A245995.

			Some solutions for n=8
..1....2....1....3....2....0....0....1....1....0....4....0....2....5....5....3
..4....7....3....2....0....2....2....5....1....7....1....2....2....2....2....7
..5....5....4....1....7....1....5....5....5....6....6....3....7....1....5....7
..6....5....1....4....2....8....4....7....8....5....3....4....7....8....7....5
..8....0....5....0....2....1....8....7....4....4....6....1....4....5....0....2
..4....1....6....1....7....8....7....4....6....7....1....1....7....5....6....2

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

Cf. A245995.

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

Empirical: G.f.: -2*x*(1 +10*x +158*x^2 +502*x^3 +1436*x^4 +1510*x^5 +1498*x^6 +474*x^7 +171*x^8) / ( (1+x)^4*(x-1)^7 ). - R. J. Mathar, Aug 10 2014

cp's OEIS Frontend

A245996 Number of length 1+2 0..n arrays with no pair in any consecutive three terms totaling exactly n.

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A245990 Number of length n+2 0..3 arrays with no pair in any consecutive three terms totalling exactly 3.

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A245989 Number of length n 0..2 arrays with no pair in any consecutive three terms totalling exactly 2.

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A245991 Number of length n+2 0..4 arrays with no pair in any consecutive three terms totalling exactly 4.

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A245992 Number of length n+2 0..5 arrays with no pair in any consecutive three terms totalling exactly 5.

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A245993 Number of length n+2 0..6 arrays with no pair in any consecutive three terms totalling exactly 6.

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A245994 Number of length n+2 0..7 arrays with no pair in any consecutive three terms totalling exactly 7.

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A245997 Number of length 2+2 0..n arrays with no pair in any consecutive three terms totalling exactly n.

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A245998 Number of length 3+2 0..n arrays with no pair in any consecutive three terms totalling exactly n.

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A245999 Number of length 4+2 0..n arrays with no pair in any consecutive three terms totalling exactly n.

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