A245869 -id:A245869 - cp's OEIS frontend

A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).

1, 6, 19, 36, 61, 90, 127, 168, 217, 270, 331, 396, 469, 546, 631, 720, 817, 918, 1027, 1140, 1261, 1386, 1519, 1656, 1801, 1950, 2107, 2268, 2437, 2610, 2791, 2976, 3169, 3366, 3571, 3780, 3997, 4218, 4447, 4680, 4921, 5166, 5419, 5676, 5941, 6210, 6487, 6768, 7057, 7350, 7651, 7956, 8269

Offset: 0

N. J. A. Sloane, Jan 30 2004

Also degree of toric ideal associated with path with n+2 nodes [Eriksson].
Also number of triples (t_1, t_2, t_3) with all t_i in the range 0 <= t_i <= n such that at least one t_i + t_j = n (with i != j). - R. H. Hardin, Aug 04 2014
Conjecture: a(n) is the maximum number of areas that are defined by the 3n angle (n+1)-sectors in a triangle. - Nicolas Nagel, Sep 09 2016

			Some triples for n=10 (from _R. H. Hardin_, Aug 04 2014):
..3....1....2....1....7....9....5....8....5....6....9....4...10....8....6....2
..3....3....8....9....3....3....7....2....9....4....3...10....9....1....8....7
..7....7...10....5....2....1....3....7....1....3....7....0....1....9....4....8

R. H. Hardin and N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [First 210 terms from Hardin]
N. Eriksson, Toric ideals of homogeneous phylogenetic models, arXiv:math/0401175 [math.CO], 2004.
Nicolas Nagel, Example picture for angle (n+1)-sectors
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A090382, A090383, A090384, A090385.
Row 1 of A245869.
Central spine of triangle in A245556. Cf. also A245557.

[3*n*(n+1)+(1+(-1)^n)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2016

f:=n-> if n mod 2 = 0 then t:=n/2; 12*t^2+6*t+1 else
t:=(n-1)/2; 12*t^2+18*t+6; fi;
[seq(f(n), n=0..100)];

CoefficientList[Series[(1 + 4 x + 7 x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 52}], x] (* Michael De Vlieger, May 07 2016 *)
Table[3 n (n + 1) + (1 + (-1)^n)/2, {n, 0, 52}] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {1, 6, 19, 36}, 53] (* Michael De Vlieger, Sep 12 2016 *)

x='x+O('x^99); Vec((1+4*x+7*x^2)/((1-x)^2*(1-x^2))) \\ Altug Alkan, May 12 2016

G.f.: (1+4x+7x^2)/((1-x)^2*(1-x^2)).
a(2t) = 12t^2+6t+1, a(2t+1) = 12t^2+18t+6 (t >= 0).
The defining g.f. implies the recurrence a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), an empirical discovery of R. H. Hardin.
a(n) = 3*n*(n+1)+(1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016
E.g.f.: 3*x*(2 + x)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016

Edited by N. J. A. Sloane, Aug 04 2014 (merging the old A090381 and A245870).

A245864 Number of length n+2 0..2 arrays with some pair in every consecutive three terms totalling exactly 2.

Original entry on oeis.org

19, 45, 103, 239, 553, 1281, 2967, 6873, 15921, 36881, 85435, 197911, 458463, 1062035, 2460217, 5699123, 13202089, 30582803, 70845443, 164114349, 380172929, 880675315, 2040095313, 4725906149, 10947620333, 25360298571, 58747446847

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Column 2 of A245869.

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

Empirical g.f.: x*(19 + 7*x - 6*x^2 - 12*x^3 - 9*x^4) / ((1 - x)*(1 - x - 2*x^2 - 2*x^3 - x^4)). - Colin Barker, Nov 03 2018

A245865 Number of length n+2 0..4 arrays with some pair in every consecutive three terms totalling exactly 4.

Original entry on oeis.org

61, 193, 549, 1629, 4753, 13961, 40901, 119953, 351649, 1031057, 3022933, 8863117, 25986061, 76189749, 223384017, 654949861, 1920277409, 5630150189, 16507298221, 48398515249, 141901859897, 416048676085, 1219832512513, 3576483842281

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

			Some solutions for n=8:
  1   1   0   4   1   3   2   0   0   0   1   4   2   2   1   3
  2   4   1   1   0   0   2   1   2   4   0   0   3   2   3   0
  3   0   3   3   4   4   0   3   2   4   3   1   1   0   1   4
  2   4   1   0   3   0   4   1   1   0   1   4   2   4   2   2
  2   4   1   4   1   3   2   2   2   4   1   0   3   0   2   2
  0   0   3   0   3   1   2   2   3   2   3   3   1   1   2   0
  2   4   3   4   2   0   0   4   1   2   4   1   0   3   2   4
  2   3   1   0   1   3   2   0   3   1   0   4   4   1   4   3
  0   1   3   4   2   1   2   2   4   3   1   0   0   4   0   0
  2   4   1   3   2   3   4   2   1   3   4   1   1   0   4   1

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

Column 4 of A245869.

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

Empirical g.f.: x*(61 + 10*x - 91*x^2 - 150*x^3 - 185*x^4 + 75*x^5) / (1 - 3*x - x^2 + x^3 + 5*x^4 + 8*x^5 - 3*x^6). - Colin Barker, Nov 04 2018

A245866 Number of length n+2 0..5 arrays with some pair in every consecutive three terms totalling exactly 5.

Original entry on oeis.org

90, 318, 960, 3102, 9726, 30900, 97602, 309078, 977664, 3094038, 9789654, 30977796, 98020170, 310161870, 981426624, 3105480558, 9826505742, 31093507092, 98387556594, 311322635814, 985101990912, 3117106968486, 9863299264806

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Column 5 of A245869.

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

Empirical g.f.: 6*x*(15 + 23*x - 6*x^2) / (1 - 2*x - 4*x^2 + x^3). - Colin Barker, Nov 04 2018

A245867 Number of length n+2 0..6 arrays with some pair in every consecutive three terms totalling exactly 6.

Original entry on oeis.org

127, 493, 1579, 5515, 18505, 63241, 214315, 729097, 2475985, 8415217, 28590415, 97151683, 330100459, 1121650903, 3811203385, 12950003383, 44002376953, 149514426895, 508030458319, 1726221621517, 5865476355769, 19930126601527

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Column 6 of A245869.

Empirical: a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3) - 9*a(n-4) - 24*a(n-5) + 5*a(n-6).

Empirical g.f.: x*(127 + 112*x - 281*x^2 - 574*x^3 - 1141*x^4 + 245*x^5) / (1 - 3*x - 3*x^2 + x^3 + 9*x^4 + 24*x^5 - 5*x^6). - Colin Barker, Nov 04 2018

A245868 Number of length n+2 0..7 arrays with some pair in every consecutive three terms totalling exactly 7.

Original entry on oeis.org

168, 712, 2368, 8840, 31176, 113024, 404264, 1455496, 5223552, 18775816, 67437448, 242306240, 870461352, 3127322696, 11235107264, 40363689352, 145010699592, 520968428032, 1871637364264, 6724074597128, 24157004951808, 86786820122120

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted derivation of recurrence

Column 7 of A245869.

Empirical: a(n) = 2*a(n-1) + 6*a(n-2) - a(n-3).
Empirical g.f.: 8*x*(21 + 47*x - 8*x^2) / (1 - 2*x - 6*x^2 + x^3). - Colin Barker, Nov 04 2018
Empirical recurrence verified: see link. - Robert Israel, May 13 2020

A245871 Number of length 2+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

Original entry on oeis.org

10, 45, 100, 193, 318, 493, 712, 993, 1330, 1741, 2220, 2785, 3430, 4173, 5008, 5953, 7002, 8173, 9460, 10881, 12430, 14125, 15960, 17953, 20098, 22413, 24892, 27553, 30390, 33421, 36640, 40065, 43690, 47533, 51588, 55873, 60382, 65133, 70120, 75361

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Row 2 of A245869.

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
Conjectures from Colin Barker, Nov 04 2018: (Start)
G.f.: x*(10 + 15*x - 15*x^2 + 3*x^3 - x^4) / ((1 - x)^4*(1 + x)).
a(n) = 1 + 4*n + 7*n^2 + n^3 for n even.
a(n) = -2 + 4*n + 7*n^2 + n^3 for n odd.
(End)

A245872 Number of length 3+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

Original entry on oeis.org

16, 103, 256, 549, 960, 1579, 2368, 3433, 4720, 6351, 8256, 10573, 13216, 16339, 19840, 23889, 28368, 33463, 39040, 45301, 52096, 59643, 67776, 76729, 86320, 96799, 107968, 120093, 132960, 146851, 161536, 177313, 193936, 211719, 230400, 250309

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Row 3 of A245869.

Empirical: a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
Conjectures from Colin Barker, Nov 04 2018: (Start)
G.f.: x*(16 + 71*x + 34*x^2 - 2*x^3 + 2*x^4 - x^5) / ((1 - x)^4*(1 + x)^2).
a(n) = 1 + 5*n + 13*n^2 + 5*n^3 for n even.
a(n) = -5 + 3*n + 13*n^2 + 5*n^3 for n odd.
(End)

A245873 Number of length 4+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

Original entry on oeis.org

26, 239, 676, 1629, 3102, 5515, 8840, 13625, 19810, 28071, 38316, 51349, 67046, 86339, 109072, 136305, 167850, 204895, 247220, 296141, 351406, 414459, 485016, 564649, 653042, 751895, 860860, 981765, 1114230, 1260211, 1419296, 1593569

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Row 4 of A245869.

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).
Conjectures from Colin Barker, Nov 04 2018: (Start)
G.f.: x*(26 + 161*x - 15*x^2 - 30*x^3 - 44*x^4 - 3*x^5 + x^6) / ((1 - x)^5*(1 + x)^2).
a(n) = 1 + 7*n + 20*n^2 + 16*n^3 + n^4 for n even.
a(n) = -8 - 3*n + 20*n^2 + 16*n^3 + n^4 for n odd.
(End)

A245874 Number of length 5+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

Original entry on oeis.org

42, 553, 1764, 4753, 9726, 18505, 31176, 50401, 76050, 111721, 156972, 216433, 289254, 381193, 490896, 625345, 782586, 970921, 1187700, 1442641, 1732302, 2067913, 2445144, 2876833, 3357666, 3902185, 4503996, 5179441, 5920950, 6746761

Offset: 1

R. H. Hardin, Aug 04 2014

nonn

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

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

Row 5 of A245869.

Empirical: a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6)  -2*a(n-7) + a(n-8).
Conjectures from Colin Barker, Nov 04 2018: (Start)
G.f.: x*(42 + 469*x + 574*x^2 + 371*x^3 + 10*x^4 - 121*x^5 - 2*x^6 + x^7) / ((1 - x)^5*(1 + x)^3).
a(n) = 1 + 12*n + 26*n^2 + 39*n^3 + 7*n^4 for n even.
a(n) = -9 - 18*n + 23*n^2 + 39*n^3 + 7*n^4 for n odd.
(End)

cp's OEIS Frontend

A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).

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A245864 Number of length n+2 0..2 arrays with some pair in every consecutive three terms totalling exactly 2.

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A245865 Number of length n+2 0..4 arrays with some pair in every consecutive three terms totalling exactly 4.

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A245866 Number of length n+2 0..5 arrays with some pair in every consecutive three terms totalling exactly 5.

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A245867 Number of length n+2 0..6 arrays with some pair in every consecutive three terms totalling exactly 6.

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A245868 Number of length n+2 0..7 arrays with some pair in every consecutive three terms totalling exactly 7.

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A245871 Number of length 2+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

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A245872 Number of length 3+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

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A245873 Number of length 4+2 0..n arrays with some pair in every consecutive three terms totalling exactly n.

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