A250561 -id:A250561 - cp's OEIS frontend

A250554 Number of length n+2 0..1 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

2, 8, 14, 32, 62, 128, 254, 512, 1022, 2048, 4094, 8192, 16382, 32768, 65534, 131072, 262142, 524288, 1048574, 2097152, 4194302, 8388608, 16777214, 33554432, 67108862, 134217728, 268435454, 536870912, 1073741822, 2147483648, 4294967294

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

			Some solutions for n=6:
..1....1....0....0....0....0....1....0....1....1....0....1....0....1....0....0
..0....1....1....1....0....0....1....1....0....0....1....1....0....1....1....0
..0....1....1....0....1....1....0....1....1....0....1....1....1....0....0....1
..1....1....0....0....0....0....1....1....0....0....0....1....1....0....0....0
..1....0....1....0....0....1....1....1....0....1....0....0....1....0....1....1
..1....1....0....1....1....0....1....1....1....1....0....0....1....1....0....1
..0....1....0....0....1....0....0....1....0....1....0....0....0....0....0....1
..1....1....1....1....1....0....0....0....1....0....1....0....0....0....1....1

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

Column 1 of A250561.

Empirical: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3).
Empirical: a(n) = 2^(n+1) for even n, 2^(n+1)-2 for odd n.
Empirical g.f.: 2*x*(1 + 2*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)). - Colin Barker, Nov 14 2018

A250555 Number of length n+2 0..2 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

5, 25, 83, 297, 989, 3113, 9611, 29257, 88503, 266769, 802699, 2412545, 7246311, 21755889, 65301163, 195969945, 588042095, 1764390025, 5293696363, 15882140601, 47648522543, 142949767457, 428857697715, 1286589881041, 3859803210479

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Column 2 of A250561

			Some solutions for n=6
..2....1....2....0....0....2....0....2....0....2....2....2....1....2....1....0
..1....0....2....2....0....0....2....2....0....2....0....1....0....2....1....0
..2....1....2....1....1....2....2....1....0....0....2....0....1....1....1....2
..1....0....0....0....2....2....0....2....2....2....0....0....0....0....2....2
..1....0....0....0....1....1....0....0....1....1....1....0....0....1....2....0
..1....0....0....0....0....2....0....1....1....2....1....2....1....0....1....2
..2....1....0....1....1....1....2....1....2....2....0....0....1....0....0....2
..1....2....0....1....1....1....0....1....2....0....0....1....0....2....2....2

R. H. Hardin, Table of n, a(n) for n = 1..106

Empirical: a(n) = 4*a(n-1) +2*a(n-2) -17*a(n-3) -3*a(n-4) +24*a(n-5) +16*a(n-6) -13*a(n-7) -26*a(n-8) +2*a(n-9) +12*a(n-10) for n>15

A250556 Number of length n+2 0..3 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

8, 60, 302, 1516, 7126, 30780, 127586, 518052, 2085808, 8367220, 33513408, 134137736, 536713774, 2147172564, 8589316642, 34358507208, 137436497326, 549750908948, 2199013454674, 8796073430888, 35184332920270, 140737410034420

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Column 3 of A250561.

			Some solutions for n=6
  3    1    1    0    3    3    3    1    1    1    3    0    1    2    0    2
  2    2    0    0    0    3    0    0    0    2    3    3    1    2    1    2
  3    2    2    1    2    1    0    2    1    2    2    2    3    1    1    2
  2    2    2    3    0    1    0    3    1    2    0    1    3    3    3    0
  1    0    1    2    1    3    1    3    2    2    0    2    1    3    1    1
  0    3    0    1    1    2    3    3    0    2    0    1    3    3    0    2
  3    1    2    0    3    3    3    0    2    1    3    1    0    0    3    2
  2    0    3    0    0    1    0    3    1    2    3    2    2    0    0    2

Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 1..1000 (terms 1..47 from R. H. Hardin).
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

From Manuel Kauers and Christoph Koutschan, Mar 01 2023: (Start)
Generating function: 2*x*(4 + 2*x - 3*x^2 + 73*x^3 + 115*x^4 - 139*x^5 - 453*x^6 - 1231*x^7 + 38*x^8 + 406*x^9 + 3597*x^10 + 2087*x^11 + 1666*x^12 - 3614*x^13 - 4178*x^14 - 4504*x^15 + 903*x^16 + 1985*x^17 + 4173*x^18 + 403*x^19 - 202*x^20 - 1324*x^21 - 1296*x^22 + 684*x^23 - 300*x^24 + 508*x^25 - 56*x^26 + 32*x^27)/((1 - 4*x)*(1 - 2*x)*(1 - x)^3*(1 + x)^2*(1 + x^2)^2*(1 - 2*x^3)^2).
Recurrence equation: 32*a(n) - 56*a(n + 1) + 28*a(n + 2) - 36*a(n + 3) - 8*a(n + 4) + 84*a(n + 5) - 44*a(n + 6) + 58*a(n + 7) - 73*a(n + 8) - a(n + 9) + 4*a(n + 10) - 8*a(n + 11) + 42*a(n + 12) - 26*a(n + 13) + 12*a(n + 14) - 14*a(n + 15) + 7*a(n + 16) - a(n + 17) = 0 for n>11. (End)

A250557 Number of length n+2 0..4 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

13, 117, 761, 5105, 31525, 177421, 937817, 4803653, 24257725, 121800949, 610147053, 3053493273, 15274472475, 76391277073, 382008613455, 1910192140573, 9551389260819, 47758200502193, 238794678323345, 1193984277090533

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Column 4 of A250561

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

R. H. Hardin, Table of n, a(n) for n = 1..28

A250562 Number of length 3+2 0..n arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

14, 83, 302, 761, 1648, 3125, 5446, 8843, 13662, 20173, 28836, 39973, 54102, 71647, 93210, 119221, 150416, 187305, 230650, 281051, 339378, 406245, 482652, 569257, 667098, 776983, 900018, 1036969, 1189124, 1357341, 1542894, 1746747, 1970290

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Row 3 of A250561

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

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

Empirical: a(n) = a(n-1) -a(n-2) +2*a(n-3) +a(n-5) -2*a(n-7) -3*a(n-9) +a(n-10) -a(n-11) +3*a(n-12) +2*a(n-14) -a(n-16) -2*a(n-18) +a(n-19) -a(n-20) +a(n-21)
also a quadratic polynomial plus a linear quasipolynomial with period 60, the first 12 being:
Empirical for n mod 60 = 0: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (23/30)*n + 1
Empirical for n mod 60 = 1: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (1853/1080)*n + (251/270)
Empirical for n mod 60 = 2: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (127/270)*n + (211/270)
Empirical for n mod 60 = 3: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (337/120)*n + (13/5)
Empirical for n mod 60 = 4: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (47/270)*n + (283/135)
Empirical for n mod 60 = 5: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (2173/1080)*n + (35/54)
Empirical for n mod 60 = 6: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (23/30)*n + (13/10)
Empirical for n mod 60 = 7: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (2393/1080)*n - (77/135)
Empirical for n mod 60 = 8: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (127/270)*n + (281/135)
Empirical for n mod 60 = 9: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (277/120)*n + (33/10)
Empirical for n mod 60 = 10: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (47/270)*n + (43/54)
Empirical for n mod 60 = 11: a(n) = (329/216)*n^4 + (4747/1080)*n^3 + (1957/360)*n^2 + (2713/1080)*n - (7/135)

A250563 Number of length 4+2 0..n arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

32, 297, 1516, 5105, 13732, 31173, 63400, 117749, 204712, 336293, 529072, 800689, 1175088, 1677185, 2338728, 3193293, 4283108, 5649837, 7347520, 9429021, 11959288, 15003857, 18642524, 22951429, 28026324, 33959013, 40858932, 48834549

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Row 4 of A250561

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

R. H. Hardin, Table of n, a(n) for n = 1..75

A250558 Number of length n+2 0..5 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

18, 200, 1648, 13732, 106362, 744564, 4808120, 29723864, 180290280, 1085927844, 6524507992, 39167586296, 235053968050, 1410450413584, 8463032114236, 50779132943420

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Column 5 of A250561.

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

Cf. A250561.

A250559 Number of length n+2 0..6 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

25, 321, 3125, 31173, 290909, 2457921, 18934449, 137976845, 980389815, 6899647449, 48392977271, 339018816865, 2373951608191, 16620410187793, 116352541474531

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Column 6 of A250561

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

A250560 Number of length n+2 0..7 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

32, 480, 5446, 63400, 695890, 6924692, 62245658, 522997696, 4258085394, 34261234132, 274598186760, 2198154605100, 17589251301614, 140726940443068

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Column 7 of A250561

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

A250564 Number of length 5+2 0..n arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

62, 989, 7126, 31525, 106362, 290909, 695890, 1486139, 2927312, 5373103, 9352712, 15511735, 24776808, 38224089, 57324690, 83749019, 119721620, 167645127, 230722744, 312355167, 416892578, 548957107, 714415336, 919264381

Offset: 1

R. H. Hardin, Nov 25 2014

nonn

Row 5 of A250561

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

cp's OEIS Frontend

A250554 Number of length n+2 0..1 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250555 Number of length n+2 0..2 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250556 Number of length n+2 0..3 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250557 Number of length n+2 0..4 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250562 Number of length 3+2 0..n arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250563 Number of length 4+2 0..n arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250558 Number of length n+2 0..5 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250559 Number of length n+2 0..6 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250560 Number of length n+2 0..7 arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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A250564 Number of length 5+2 0..n arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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