A190041 -id:A190041 - cp's OEIS frontend

A190034 Number of nondecreasing arrangements of n+2 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

2, 6, 10, 27, 33, 107, 80, 349, 264, 735, 357, 2809, 736, 2565, 3262, 10997, 1258, 17921, 2313, 34880, 12649, 17448, 4348, 211004, 15839, 42957, 49372, 169716, 10846, 430082, 11210, 1004561, 85127, 101536, 102904, 2715826, 20183, 190249, 208100

Offset: 1

R. H. Hardin May 04 2011

nonn

Diagonal of A190041

			All solutions for n=3
..3....0....1....1....1....1....0....1....2....1
..3....3....3....1....1....1....1....2....3....2
..3....3....3....1....2....2....1....2....3....3
..3....3....3....2....2....3....2....3....3....3
..3....3....3....3....3....3....3....3....3....3

R. H. Hardin, Table of n, a(n) for n = 1..142

A190035 Number of nondecreasing arrangements of n+2 numbers in 0..3 with the last equal to 3 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

5, 7, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230, 234, 238

Offset: 1

R. H. Hardin, May 04 2011

Column 3 of A190041.

			All solutions for n=4:
..1....1....0....3....2....0....1....0....1....1....0....1....1....1
..1....1....3....3....3....1....1....1....2....1....1....2....3....1
..2....1....3....3....3....1....2....1....3....1....1....2....3....1
..2....1....3....3....3....2....3....2....3....2....1....3....3....2
..3....2....3....3....3....2....3....3....3....3....2....3....3....2
..3....3....3....3....3....3....3....3....3....3....3....3....3....3

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041, A073760, A016825.

Empirical: a(n) = 4*n - 2 for n>2.
Conjectures from Colin Barker, May 03 2018: (Start)
G.f.: x*(5 - 3*x + x^2 + x^3) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>4.
(End)

A190036 Number of nondecreasing arrangements of n+2 numbers in 0..4 with the last equal to 4 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

7, 12, 18, 27, 39, 53, 69, 87, 107, 129, 153, 179, 207, 237, 269, 303, 339, 377, 417, 459, 503, 549, 597, 647, 699, 753, 809, 867, 927, 989, 1053, 1119, 1187, 1257, 1329, 1403, 1479, 1557, 1637, 1719, 1803, 1889, 1977, 2067, 2159, 2253, 2349, 2447, 2547, 2649

Offset: 1

R. H. Hardin, May 04 2011

nonn

Column 4 of A190041.

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

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

Empirical: a(n) = n^2 + 3*n - 1 for n>3.
Conjectures from Colin Barker, May 04 2018: (Start)
G.f.: x*(7 - 9*x + 3*x^2 + 2*x^3 - x^5) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
(End)

A190037 Number of nondecreasing arrangements of n+2 numbers in 0..5 with the last equal to 5 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

8, 12, 16, 23, 33, 45, 57, 69, 81, 93, 105, 117, 129, 141, 153, 165, 177, 189, 201, 213, 225, 237, 249, 261, 273, 285, 297, 309, 321, 333, 345, 357, 369, 381, 393, 405, 417, 429, 441, 453, 465, 477, 489, 501, 513, 525, 537, 549, 561, 573, 585, 597, 609, 621, 633, 645

Offset: 1

R. H. Hardin, May 04 2011

nonn

Column 5 of A190041.

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

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

Empirical: a(n) = 12*n - 27 for n>4.

Empirical: G.f.: x*(8 - 4*x + 3*x^3 + 3*x^4 + 2*x^5) / (1 - x)^2. a(n) = 2*a(n-1) - a(n-2) for n>2. - Colin Barker, May 04 2018

A190038 Number of nondecreasing arrangements of n+2 numbers in 0..6 with the last equal to 6 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

10, 18, 30, 47, 72, 107, 151, 203, 263, 331, 407, 491, 583, 683, 791, 907, 1031, 1163, 1303, 1451, 1607, 1771, 1943, 2123, 2311, 2507, 2711, 2923, 3143, 3371, 3607, 3851, 4103, 4363, 4631, 4907, 5191, 5483, 5783, 6091, 6407, 6731, 7063, 7403, 7751, 8107

Offset: 1

R. H. Hardin, May 04 2011

nonn

Column 6 of A190041.

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

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

Empirical: a(n) = 4*n^2 - 8*n + 11 for n>5.
Conjectures from Colin Barker, May 04 2018: (Start)
G.f.: x*(10 - 12*x + 6*x^2 + x^3 + 3*x^4 + 2*x^5 - x^6 - x^7) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>8.
(End)

A190039 Number of nondecreasing arrangements of n+2 numbers in 0..7 with the last equal to 7 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

11, 17, 22, 31, 43, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, 680, 700, 720, 740, 760, 780, 800, 820, 840, 860, 880, 900, 920, 940, 960, 980, 1000, 1020

Offset: 1

R. H. Hardin, May 04 2011

nonn

Column 7 of A190041.

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

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

Empirical: a(n) = 20*n - 60 for n>5.
Conjectures from Colin Barker, May 04 2018: (Start)
G.f.: x*(11 - 5*x - x^2 + 4*x^3 + 3*x^4 + 5*x^5 + 3*x^6) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>7.
(End)

A190040 Number of nondecreasing arrangements of n+2 numbers in 0..8 with the last equal to 8 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

13, 24, 40, 65, 105, 164, 246, 349, 472, 617, 786, 981, 1204, 1457, 1742, 2061, 2416, 2809, 3242, 3717, 4236, 4801, 5414, 6077, 6792, 7561, 8386, 9269, 10212, 11217, 12286, 13421, 14624, 15897, 17242, 18661, 20156, 21729, 23382, 25117, 26936, 28841

Offset: 1

R. H. Hardin, May 04 2011

nonn

Column 8 of A190041.

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

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

Empirical: a(n) = (1/3)*n^3 + 2*n^2 + (50/3)*n - 83 for n>6.
Conjectures from Colin Barker, May 04 2018: (Start)
G.f.: x*(13 - 28*x + 22*x^2 - 3*x^3 + 2*x^4 - 2*x^5 - 6*x^7 + x^8 + 3*x^9) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>6.
(End)

A190042 Number of nondecreasing arrangements of 5 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

2, 8, 10, 18, 16, 30, 22, 40, 31, 49, 36, 64, 41, 71, 53, 81, 55, 97, 61, 103, 74, 112, 75, 131, 80, 134, 96, 144, 94, 164, 100, 166, 117, 175, 114, 198, 119, 197, 139, 207, 133, 231, 139, 229, 160, 238, 153, 265, 158, 260, 182, 270, 172, 298, 178, 292, 203, 301, 192

Offset: 1

R. H. Hardin, May 04 2011

nonn

Row 3 of A190041.

			All solutions for n=3:
..1....1....1....3....0....1....1....0....2....1
..1....1....3....3....3....2....2....1....3....1
..1....2....3....3....3....3....2....1....3....2
..2....2....3....3....3....3....3....2....3....3
..3....3....3....3....3....3....3....3....3....3

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

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

Empirical g.f.: x*(2 + 12*x + 30*x^2 + 54*x^3 + 66*x^4 + 66*x^5 + 46*x^6 + 26*x^7 + 9*x^8 + 3*x^9) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2). - Colin Barker, May 04 2018

A190043 Number of nondecreasing arrangements of 6 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

2, 10, 14, 27, 23, 47, 31, 65, 49, 76, 52, 113, 58, 109, 92, 132, 78, 167, 87, 172, 127, 172, 108, 240, 116, 208, 167, 237, 133, 292, 143, 272, 205, 271, 169, 366, 167, 307, 244, 348, 189, 407, 198, 377, 290, 369, 218, 493, 223, 413, 322, 445, 244, 528, 263, 482, 358, 468

Offset: 1

R. H. Hardin, May 04 2011

nonn

Row 4 of A190041.

			All solutions for n=3:
..0....1....1....0....1....0....1....1....1....0....1....2....1....3
..1....2....2....1....1....3....1....3....1....1....1....3....1....3
..1....3....2....1....2....3....1....3....2....1....1....3....1....3
..2....3....3....2....2....3....1....3....3....1....2....3....2....3
..3....3....3....2....3....3....2....3....3....2....3....3....2....3
..3....3....3....3....3....3....3....3....3....3....3....3....3....3

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041.

Empirical: a(n) = -4*a(n-1) -11*a(n-2) -23*a(n-3) -40*a(n-4) -59*a(n-5) -75*a(n-6) -81*a(n-7) -71*a(n-8) -42*a(n-9) +4*a(n-10) +59*a(n-11) +112*a(n-12) +150*a(n-13) +164*a(n-14) +150*a(n-15) +112*a(n-16) +59*a(n-17) +4*a(n-18) -42*a(n-19) -71*a(n-20) -81*a(n-21) -75*a(n-22) -59*a(n-23) -40*a(n-24) -23*a(n-25) -11*a(n-26) -4*a(n-27) -a(n-28).

Empirical g.f.: x*(2 + 18*x + 76*x^2 + 239*x^3 + 595*x^4 + 1276*x^5 + 2393*x^6 + 4053*x^7 + 6246*x^8 + 8890*x^9 + 11721*x^10 + 14448*x^11 + 16654*x^12 + 18046*x^13 + 18363*x^14 + 17598*x^15 + 15834*x^16 + 13401*x^17 + 10606*x^18 + 7850*x^19 + 5387*x^20 + 3421*x^21 + 1976*x^22 + 1037*x^23 + 476*x^24 + 189*x^25 + 59*x^26 + 14*x^27) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Colin Barker, May 04 2018

A190044 Number of nondecreasing arrangements of 7 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

2, 12, 18, 39, 33, 72, 43, 105, 69, 123, 72, 190, 83, 169, 151, 217, 108, 276, 120, 297, 206, 261, 150, 427, 179, 318, 259, 405, 185, 509, 201, 452, 326, 413, 277, 652, 228, 470, 383, 626, 263, 688, 276, 631, 501, 559, 303, 893, 323, 673, 509, 748, 341, 883, 418, 831

Offset: 1

R. H. Hardin May 04 2011

nonn

Row 5 of A190041

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

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = -4*a(n-1) -13*a(n-2) -33*a(n-3) -75*a(n-4) -152*a(n-5) -286*a(n-6) -500*a(n-7) -826*a(n-8) -1292*a(n-9) -1929*a(n-10) -2753*a(n-11) -3772*a(n-12) -4965*a(n-13) -6291*a(n-14) -7671*a(n-15) -9002*a(n-16) -10148*a(n-17) -10961*a(n-18) -11284*a(n-19) -10976*a(n-20) -9925*a(n-21) -8069*a(n-22) -5411*a(n-23) -2025*a(n-24) +1935*a(n-25) +6252*a(n-26) +10648*a(n-27) +14823*a(n-28) +18464*a(n-29) +21297*a(n-30) +23091*a(n-31) +23708*a(n-32) +23091*a(n-33) +21297*a(n-34) +18464*a(n-35) +14823*a(n-36) +10648*a(n-37) +6252*a(n-38) +1935*a(n-39) -2025*a(n-40) -5411*a(n-41) -8069*a(n-42) -9925*a(n-43) -10976*a(n-44) -11284*a(n-45) -10961*a(n-46) -10148*a(n-47) -9002*a(n-48) -7671*a(n-49) -6291*a(n-50) -4965*a(n-51) -3772*a(n-52) -2753*a(n-53) -1929*a(n-54) -1292*a(n-55) -826*a(n-56) -500*a(n-57) -286*a(n-58) -152*a(n-59) -75*a(n-60) -33*a(n-61) -13*a(n-62) -4*a(n-63) -a(n-64)

cp's OEIS Frontend

A190034 Number of nondecreasing arrangements of n+2 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

A190035 Number of nondecreasing arrangements of n+2 numbers in 0..3 with the last equal to 3 and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190036 Number of nondecreasing arrangements of n+2 numbers in 0..4 with the last equal to 4 and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190037 Number of nondecreasing arrangements of n+2 numbers in 0..5 with the last equal to 5 and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190038 Number of nondecreasing arrangements of n+2 numbers in 0..6 with the last equal to 6 and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190039 Number of nondecreasing arrangements of n+2 numbers in 0..7 with the last equal to 7 and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190040 Number of nondecreasing arrangements of n+2 numbers in 0..8 with the last equal to 8 and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190042 Number of nondecreasing arrangements of 5 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A190043 Number of nondecreasing arrangements of 6 numbers in 0..n with the last equal to n and each after the second equal to the sum of one or two of the preceding three.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula