A212868 -id:A212868 - cp's OEIS frontend

A212864 Number of nondecreasing sequences of n 1..4 integers with no element dividing the sequence sum.

0, 2, 3, 5, 6, 9, 12, 15, 17, 22, 26, 30, 34, 40, 45, 51, 56, 63, 70, 77, 83, 92, 100, 108, 116, 126, 135, 145, 154, 165, 176, 187, 197, 210, 222, 234, 246, 260, 273, 287, 300, 315, 330, 345, 359, 376, 392, 408, 424, 442, 459, 477, 494, 513, 532, 551, 569, 590, 610, 630, 650

Offset: 1

R. H. Hardin, May 29 2012

nonn

Column 4 of A212868.

			All solutions for n=8:
..2....2....2....2....2....2....3....2....2....3....2....2....2....3....3
..2....3....2....2....2....2....3....2....3....3....2....3....2....4....3
..3....3....2....2....2....2....3....3....4....3....2....3....2....4....3
..3....3....2....2....2....2....3....3....4....3....3....3....3....4....4
..3....3....3....2....2....2....3....3....4....3....4....3....3....4....4
..3....3....4....2....2....3....3....4....4....3....4....3....3....4....4
..3....3....4....3....2....3....4....4....4....3....4....4....4....4....4
..4....3....4....4....3....3....4....4....4....4....4....4....4....4....4

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

Cf. A212868.

Empirical: a(n) = a(n-1) + a(n-3) - a(n-5) - a(n-7) + a(n-8).

Empirical g.f.: x^2*(2 + x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2)). - Colin Barker, Jul 21 2018

A212865 Number of nondecreasing sequences of n 1..5 integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 5, 9, 15, 22, 32, 40, 59, 74, 97, 124, 159, 188, 229, 260, 301, 347, 415, 477, 559, 630, 715, 801, 897, 987, 1106, 1214, 1342, 1471, 1623, 1760, 1934, 2099, 2287, 2475, 2683, 2878, 3116, 3334, 3581, 3832, 4115, 4377, 4681, 4968, 5283, 5605, 5965, 6310, 6707

Offset: 1

R. H. Hardin, May 29 2012

nonn

Column 5 of A212868.

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

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

Empirical: a(n) = 2*a(n-1) -2*a(n-3) +a(n-5) +2*a(n-6) -2*a(n-7) -2*a(n-8) +2*a(n-9) +a(n-10) -a(n-12) -2*a(n-13) +2*a(n-14) +2*a(n-15) -2*a(n-16) -a(n-17) +2*a(n-19) +a(n-20) -4*a(n-21) +a(n-22) +2*a(n-23) -a(n-25) -2*a(n-26) +2*a(n-27) +2*a(n-28) -2*a(n-29) -a(n-30) +a(n-32) +2*a(n-33) -2*a(n-34) -2*a(n-35) +2*a(n-36) +a(n-37) -2*a(n-39) +2*a(n-41) -a(n-42).

If the above empirical recurrence by R. H. Hardin is correct, then the denominator of the g.f. (that determines the above recurrence) equals (1-x)^2*(1-x^2)*(1-x^12)*(1-x^15)*(1-x^20)/((1-x^4)*(1-x^5)). - Petros Hadjicostas, Sep 09 2019

A212866 Number of nondecreasing sequences of n 1..6 integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 7, 16, 29, 52, 82, 122, 182, 259, 363, 492, 648, 816, 1018, 1268, 1586, 1973, 2419, 2904, 3452, 4063, 4762, 5543, 6421, 7393, 8487, 9700, 11052, 12543, 14183, 15960, 17915, 20023, 22303, 24760, 27422, 30279, 33373, 36697, 40284, 44131, 48250, 52614

Offset: 1

R. H. Hardin, May 29 2012

nonn

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

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

Column 6 of A212868.

S6:= combinat:-powerset({$2..6}):
f:= proc(n) local s,t,G,S,i,j,T;
  t:= 0:
  for S in S6 do
    G:= coeff(mul(add(x^i*y^(i*j),i=0..n),j=S),x,n);
    T:= select(s -> S = select(k -> s mod k <> 0, {$2..6}), [$2*n..6*n]);
    t:= t + add(coeff(G,y,s),s= T);
  od;
  t
end proc:
map(f, [$1..50]); # Robert Israel, Nov 23 2023

Empirical: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +2*a(n-4) +a(n-5) +a(n-6) -4*a(n-7) +4*a(n-9) -a(n-10) -a(n-11) -a(n-12) -a(n-13) +4*a(n-14) -4*a(n-16) +a(n-17) +a(n-18) +2*a(n-19) -a(n-20) -5*a(n-21) +5*a(n-22) +a(n-23) -2*a(n-24) -a(n-25) -a(n-26) +4*a(n-27) -4*a(n-29) +a(n-30) +a(n-31) +a(n-32) +a(n-33) -4*a(n-34) +4*a(n-36) -a(n-37) -a(n-38) -2*a(n-39) +2*a(n-40) +2*a(n-41) -3*a(n-42) +a(n-43).

A212867 Number of nondecreasing sequences of n 1..7 integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 12, 29, 59, 112, 199, 319, 503, 733, 1067, 1508, 2104, 2827, 3801, 4974, 6404, 8054, 10101, 12573, 15680, 19309, 23586, 28399, 33966, 40235, 47554, 55866, 65509, 76417, 88971, 103077, 119201, 137210, 157549, 180024, 205015, 232280, 262333, 294930

Offset: 1

R. H. Hardin May 29 2012

nonn

Column 7 of A212868

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

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

A212870 Number of nondecreasing sequences of 3 1..n integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 0, 1, 3, 9, 16, 29, 43, 64, 92, 127, 168, 219, 281, 355, 435, 531, 638, 762, 901, 1057, 1230, 1420, 1628, 1858, 2107, 2380, 2674, 2992, 3338, 3707, 4099, 4524, 4976, 5465, 5975, 6516, 7092, 7704, 8348, 9025, 9744, 10495, 11289, 12125, 12994, 13907, 14859

Offset: 1

R. H. Hardin, May 29 2012

nonn

Row 3 of A212868.

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

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

Cf. A212868.

A212871 Number of nondecreasing sequences of 4 1..n integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 0, 1, 5, 15, 29, 59, 103, 168, 259, 386, 553, 772, 1043, 1401, 1832, 2356, 2980, 3729, 4622, 5680, 6872, 8263, 9872, 11716, 13767, 16122, 18765, 21707, 25010, 28650, 32686, 37173, 42036, 47466, 53375, 59783, 66728, 74355, 82632, 91519, 101150

Offset: 1

R. H. Hardin May 29 2012

nonn

Row 4 of A212868

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

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

A212872 Number of nondecreasing sequences of 5 1..n integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 0, 2, 6, 22, 52, 112, 212, 376, 640, 1011, 1560, 2293, 3328, 4711, 6524, 8765, 11703, 15273, 19886, 25459, 32224, 40193, 50022, 61521, 75059, 90825, 109479, 130520, 155415, 183229, 215594, 252070, 293377, 340266, 393232, 451522, 517139, 590227

Offset: 1

R. H. Hardin, May 29 2012

nonn

Row 5 of A212868.

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

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

A212873 Number of nondecreasing sequences of 6 1..n integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 0, 2, 9, 32, 82, 199, 407, 796, 1424, 2407, 3948, 6166, 9456, 14171, 20556, 29007, 40714, 55528, 75323, 100720, 132454, 171336, 221156, 281727, 355410, 445061, 553660, 680365, 835511, 1013730, 1226556, 1475591, 1764220, 2101833, 2493659

Offset: 1

R. H. Hardin May 29 2012

nonn

Row 6 of A212868

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

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

A212874 Number of nondecreasing sequences of 7 1..n integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 0, 2, 12, 40, 122, 319, 722, 1503, 2872, 5159, 9087, 15030, 24441, 38349, 58701, 86682, 127439, 180934, 255292, 354781, 484863, 649910, 870659, 1146208, 1495708, 1935479, 2485685, 3146345, 3979502, 4964688, 6177498, 7637218, 9375304

Offset: 1

R. H. Hardin May 29 2012

nonn

Row 7 of A212868

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

R. H. Hardin, Table of n, a(n) for n = 1..192

A212869 Number of nondecreasing sequences of n 1..n+1 integers with no element dividing the sequence sum.

Original entry on oeis.org

0, 1, 3, 15, 52, 199, 722, 2693, 10052, 37344, 141943, 528323, 2020746, 7720299, 29535575, 112041465, 434810382, 1666519344, 6447232492, 24895996003, 96654652273

Offset: 1

R. H. Hardin May 29 2012

nonn

Superdiagonal 1 of A212868

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

cp's OEIS Frontend

A212864 Number of nondecreasing sequences of n 1..4 integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212865 Number of nondecreasing sequences of n 1..5 integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

Formula

A212866 Number of nondecreasing sequences of n 1..6 integers with no element dividing the sequence sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A212867 Number of nondecreasing sequences of n 1..7 integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

A212870 Number of nondecreasing sequences of 3 1..n integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A212871 Number of nondecreasing sequences of 4 1..n integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

A212872 Number of nondecreasing sequences of 5 1..n integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

A212873 Number of nondecreasing sequences of 6 1..n integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

A212874 Number of nondecreasing sequences of 7 1..n integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples

Links

A212869 Number of nondecreasing sequences of n 1..n+1 integers with no element dividing the sequence sum.

Views

Author

Keywords

Comments

Examples