cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A203286 Number of arrays of 2n nondecreasing integers in -3..3 with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

4, 12, 28, 57, 104, 176, 280, 425, 620, 876, 1204, 1617, 2128, 2752, 3504, 4401, 5460, 6700, 8140, 9801, 11704, 13872, 16328, 19097, 22204, 25676, 29540, 33825, 38560, 43776, 49504, 55777, 62628, 70092, 78204, 87001, 96520, 106800, 117880, 129801
Offset: 1

Views

Author

R. H. Hardin, Dec 31 2011

Keywords

Comments

Column 3 of A203291.
a(n-4) seems to be the number of face-magic cubes or order 2 with magic sum n, which means the sum of the 4 numbers at the 4 corners of each of the 6 faces equals n. (The 8 integers at the corners do not need to be distinct; copies by the 48 operations of rotations and flips are counted separately. All 8 integers are positive.). E.g., 4 =a(5-4) is the number of cubes with magic sum 5 obtained by placing 1 at 6 of the 8 corners but 2 at two corners opposite to each other along a space diagonal (with 4 different space diagonals available). See also A381589 and A115264. - R. J. Mathar, Mar 11 2025

Examples

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

Links

Crossrefs

Cf. A203291, A005994.

Formula

Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6).
Conjectures from Colin Barker, Jun 04 2018: (Start)
G.f.: x*(4 - 4*x + 5*x^3 - 4*x^4 + x^5) / ((1 - x)^5*(1 + x)).
a(n) = (48 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n even.
a(n) = (42 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n odd.
(End)

A203287 Number of arrays of 2n nondecreasing integers in -4..4 with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

5, 21, 69, 188, 444, 944, 1844, 3369, 5825, 9621, 15285, 23492, 35080, 51084, 72756, 101601, 139401, 188257, 250613, 329304, 427584, 549176, 698304, 879749, 1098881, 1361721, 1674977, 2046108, 2483364, 2995856, 3593596, 4287573, 5089797, 6013377
Offset: 1

Views

Author

R. H. Hardin, Dec 31 2011

Keywords

Comments

Column 4 of A203291.

Examples

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

Links

Crossrefs

Cf. A203291.

Formula

Empirical: a(n) = 4*a(n-1) -4*a(n-2) -3*a(n-3) +6*a(n-4) -6*a(n-7) +3*a(n-8) +4*a(n-9) -4*a(n-10) +a(n-11).
Empirical g.f.: x*(5 + x + 5*x^2 + 11*x^3 + x^4 + x^5 - 6*x^6 + 3*x^7 + 4*x^8 - 4*x^9 + x^10) / ((1 - x)^7*(1 + x)^2*(1 + x + x^2)). - Colin Barker, Jun 04 2018

A203288 Number of arrays of 2n nondecreasing integers in -5..5 with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

6, 35, 154, 544, 1614, 4206, 9880, 21363, 43136, 82267, 149456, 260438, 437670, 712512, 1127794, 1741011, 2628022, 3887553, 5646312, 8065104, 11345704, 15738920, 21553576, 29166921, 39036102, 51711323, 67850258, 88234392
Offset: 1

Views

Author

R. H. Hardin Dec 31 2011

Keywords

Comments

Column 5 of A203291

Examples

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

Links

Formula

Empirical: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +3*a(n-4) +7*a(n-6) -10*a(n-7) +a(n-8) -a(n-10) +10*a(n-11) -7*a(n-12) -3*a(n-14) +2*a(n-15) +4*a(n-16) -4*a(n-17) +a(n-18)

A203289 Number of arrays of 2n nondecreasing integers in -6..6 with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

7, 54, 310, 1380, 5036, 15798, 43958, 111053, 259023, 565010, 1164062, 2283094, 4289830, 7762060, 13582916, 23070101, 38147395, 61569710, 97213302, 150446614, 228597390, 341536652, 502400112, 728473703, 1042269641, 1472826974
Offset: 1

Views

Author

R. H. Hardin Dec 31 2011

Keywords

Comments

Column 6 of A203291

Examples

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

Links

Formula

Empirical: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -3*a(n-5) +7*a(n-6) -4*a(n-7) -4*a(n-9) -4*a(n-10) +9*a(n-11) +2*a(n-13) -2*a(n-14) -9*a(n-16) +4*a(n-17) +4*a(n-18) +4*a(n-20) -7*a(n-21) +3*a(n-22) -4*a(n-23) +2*a(n-24) +4*a(n-25) -4*a(n-26) +a(n-27)

A203290 Number of arrays of 2n nondecreasing integers in -7..7 with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

8, 80, 580, 3172, 13934, 51768, 168314, 491187, 1310474, 3241912, 7518232, 16487840, 34435060, 68887342, 132636714, 246789735, 445262312, 781274626, 1336560456, 2234216814, 3656356722, 5868037410, 9249277184, 14337411833
Offset: 1

Views

Author

R. H. Hardin Dec 31 2011

Keywords

Comments

Column 7 of A203291

Examples

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

Links

Formula

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

A203292 Number of arrays of 4 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

3, 6, 12, 21, 35, 54, 80, 113, 155, 206, 268, 341, 427, 526, 640, 769, 915, 1078, 1260, 1461, 1683, 1926, 2192, 2481, 2795, 3134, 3500, 3893, 4315, 4766, 5248, 5761, 6307, 6886, 7500, 8149, 8835, 9558, 10320, 11121, 11963, 12846, 13772, 14741, 15755
Offset: 1

Views

Author

R. H. Hardin, Dec 31 2011

Keywords

Comments

Row 2 of A203291.

Examples

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

Links

Crossrefs

Cf. A203291.

Formula

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, Jun 04 2018: (Start)
G.f.: x*(3 - 3*x + 3*x^3 - x^4) / ((1 - x)^4*(1 + x)).
a(n) = (24 + 32*n + 6*n^2 + 4*n^3)/24 for n even.
a(n) = (30 + 32*n + 6*n^2 + 4*n^3)/24 for n odd.
(End)

A203293 Number of arrays of 6 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

4, 10, 28, 69, 154, 310, 580, 1013, 1680, 2662, 4064, 6005, 8634, 12114, 16644, 22441, 29760, 38878, 50116, 63817, 80374, 100206, 123784, 151609, 184240, 222266, 266340, 317149, 375446, 442022, 517740, 603501, 700284, 809110, 931080, 1067341, 1219126
Offset: 1

Views

Author

R. H. Hardin, Dec 31 2011

Keywords

Comments

Row 3 of A203291.

Examples

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

Links

Crossrefs

Cf. A203291.

Formula

Empirical: a(n) = 3*a(n-1) -a(n-2) -4*a(n-3) +2*a(n-4) +2*a(n-5) +2*a(n-6) -4*a(n-7) -a(n-8) +3*a(n-9) -a(n-10).
Empirical g.f.: x*(4 - 2*x + 2*x^2 + 11*x^3 + 7*x^4 + x^5 - 4*x^6 + x^7 + 3*x^8 - x^9) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)). - Colin Barker, Jun 04 2018

A203294 Number of arrays of 8 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

5, 15, 57, 188, 544, 1380, 3172, 6687, 13163, 24435, 43195, 73194, 119616, 189346, 291486, 437723, 642977, 925889, 1309647, 1822576, 2499166, 3380806, 4517000, 5966265, 7797585, 10091447, 12941561, 16456084, 20759580, 25994452, 32323204
Offset: 1

Views

Author

R. H. Hardin, Dec 31 2011

Keywords

Comments

Row 4 of A203291.

Examples

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

Links

Crossrefs

Cf. A203291.

Formula

Empirical: a(n) = 3*a(n-1) -a(n-2) -3*a(n-3) +7*a(n-6) -3*a(n-7) -2*a(n-8) -2*a(n-9) -3*a(n-10) +7*a(n-11) -3*a(n-14) -a(n-15) +3*a(n-16) -a(n-17).

A203295 Number of arrays of 10 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

6, 21, 104, 444, 1614, 5036, 13934, 34847, 80258, 172405, 349182, 672246, 1238648, 2196082, 3763716, 6258579, 10130162, 16003183, 24731962, 37466168, 55732478, 81531582, 117456262, 166829725, 233871176, 323887385, 443498894, 600898968
Offset: 1

Views

Author

R. H. Hardin Dec 31 2011

Keywords

Comments

Row 5 of A203291

Examples

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

Links

Formula

Empirical: a(n) = 3*a(n-1) -a(n-2) -3*a(n-3) +a(n-4) -2*a(n-5) +5*a(n-6) +a(n-7) +a(n-8) -3*a(n-9) -7*a(n-10) +2*a(n-11) +2*a(n-12) +4*a(n-13) +2*a(n-14) +2*a(n-15) -7*a(n-16) -3*a(n-17) +a(n-18) +a(n-19) +5*a(n-20) -2*a(n-21) +a(n-22) -3*a(n-23) -a(n-24) +3*a(n-25) -a(n-26)

A203296 Number of arrays of 12 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

7, 28, 176, 944, 4206, 15798, 51768, 151393, 403131, 991692, 2280620, 4948566, 10208256, 20143302, 38215998, 70007951, 124283183, 214475760, 360744276, 592751998, 953388836, 1503671490, 2329136950, 3548069499, 5322005825
Offset: 1

Views

Author

R. H. Hardin Dec 31 2011

Keywords

Comments

Row 6 of A203291

Examples

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

Links

Formula

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