A200057 -id:A200057 - cp's OEIS frontend

A200051 Number of -2..2 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

1, 4, 10, 26, 68, 178, 472, 1276, 3462, 9496, 26024, 71956, 198740, 552814, 1535556, 4290252, 11968194, 33553214, 93917400, 264020106, 741024426, 2087799972, 5872999754, 16577458520, 46720454112, 132081262316, 372843051320

Offset: 1

R. H. Hardin Nov 13 2011

nonn

Column 2 of A200057

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

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

A200052 Number of -3..3 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

1, 6, 22, 78, 288, 1098, 4224, 16432, 64310, 253692, 1003374, 3992358, 15908668, 63684848, 255153798, 1026057576, 4127971346, 16658240722, 67237254862, 272092306164, 1101134446908, 4466186295698, 18113540529096, 73607652658268, 299074725301088, 1217290931946978

Offset: 1

R. H. Hardin, Nov 13 2011

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

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

Column 3 of A200057.

A200053 Number of -4..4 arrays X (0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

1, 8, 36, 172, 840, 4172, 20978, 106674, 545698, 2811236, 14534258, 75522854, 393338058, 2056376914, 10767639532, 56550307652, 297322835298, 1567022163228, 8265441146830, 43685281805084, 231022736833454, 1223830782531260

Offset: 1

R. H. Hardin, Nov 13 2011

nonn

Column 4 of A200057.

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

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

Cf. A200057.

T:= proc(a,n,s)
   option remember;
   if n = 1 then
      if s = a then 1
      else 0
      fi
   else
     add(procname(-j,n-1,a-s), j=a+1..4)
   fi
end proc:
A:= proc(n)  2*add(T(a,n,0),a=-4..4) end proc: A(1):= 1:
seq(A(n), n=1..30); # Robert Israel, Nov 19 2014

A200054 Number of -5..5 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

1, 10, 56, 324, 1948, 11962, 74338, 466548, 2947742, 18746754, 119701782, 767860824, 4938868628, 31876070432, 206122332340, 1336425077996, 8676994345132, 56457974059982, 367738443114696, 2399500033305708

Offset: 1

R. H. Hardin Nov 13 2011

nonn

Column 5 of A200057

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

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

A200055 Number of -6..6 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

1, 12, 78, 546, 3914, 28554, 211242, 1577878, 11867186, 89815404, 682642050, 5211283212, 39897460856, 306446600454, 2358644800468, 18199380449274, 140644480480176, 1089068472750998, 8443172123261506, 65564035662489652

Offset: 1

R. H. Hardin Nov 13 2011

nonn

Column 6 of A200057

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

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

A200056 Number of -7..7 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

1, 14, 106, 850, 7074, 59910, 514168, 4453946, 38855488, 341052122, 3006680636, 26619825378, 236386648574, 2105717447626, 18798362729604, 168222072625668, 1507848061916274, 13541292792666004, 121763260320351938

Offset: 1

R. H. Hardin Nov 13 2011

nonn

Column 7 of A200057

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

R. H. Hardin, Table of n, a(n) for n = 1..189

A200058 Number of -n..n arrays x(0..3) of 4 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

4, 26, 78, 172, 324, 546, 850, 1252, 1764, 2398, 3170, 4092, 5176, 6438, 7890, 9544, 11416, 13518, 15862, 18464, 21336, 24490, 27942, 31704, 35788, 40210, 44982, 50116, 55628, 61530, 67834, 74556, 81708, 89302, 97354, 105876, 114880, 124382, 134394

Offset: 1

R. H. Hardin, Nov 13 2011

nonn

Row 4 of A200057.

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

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

Cf. A200057.

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

Empirical g.f.: 2*x*(2 + x + x^2)*(1 + 3*x + x^2) / ((1 - x)^4*(1 + x + x^2)). - Colin Barker, May 17 2018

A200059 Number of -n..n arrays x(0..4) of 5 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

6, 68, 288, 840, 1948, 3914, 7074, 11862, 18732, 28244, 40970, 57598, 78816, 105444, 138284, 178282, 226362, 283598, 351026, 429852, 521230, 626492, 746910, 883944, 1038982, 1213616, 1409348, 1627896, 1870884, 2140158, 2437454, 2764750, 3123900

Offset: 1

R. H. Hardin, Nov 13 2011

nonn

Row 5 of A200057.

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

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

Cf. A200057.

Empirical: a(n) = 2*a(n-1) -a(n-3) -2*a(n-5) +2*a(n-6) +a(n-8) -2*a(n-10) +a(n-11).

Empirical g.f.: 2*x*(3 + 28*x + 76*x^2 + 135*x^3 + 168*x^4 + 159*x^5 + 105*x^6 + 51*x^7 + 10*x^8 + x^9) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)). - Colin Barker, May 17 2018

A200060 Number of -n..n arrays x(0..5) of 6 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

10, 178, 1098, 4172, 11962, 28554, 59910, 114232, 202314, 337902, 538054, 823496, 1218978, 1753638, 2461350, 3381092, 4557298, 6040218, 7886274, 10158420, 12926498, 16267598, 20266414, 25015604, 30616142, 37177686, 44818926, 53667948

Offset: 1

R. H. Hardin, Nov 13 2011

nonn

Row 6 of A200057.

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

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

Cf. A200057.

Empirical: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +2*a(n-5) -a(n-6) -a(n-7) +2*a(n-8) -a(n-10) -2*a(n-11) +3*a(n-12) -a(n-13).

Empirical g.f.: 2*x*(5 + 74*x + 292*x^2 + 622*x^3 + 910*x^4 + 1045*x^5 + 999*x^6 + 782*x^7 + 452*x^8 + 162*x^9 + 24*x^10 + x^11) / ((1 - x)^6*(1 + x)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, May 17 2018

A200061 Number of -n..n arrays x(0..6) of 7 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

14, 472, 4224, 20978, 74338, 211242, 514168, 1115572, 2215290, 4099888, 7165376, 11941380, 19119208, 29580842, 44432366, 65037696, 93057524, 130487666, 179703518, 243503094, 325156994, 428455996, 557766736, 718083722, 915090720

Offset: 1

R. H. Hardin Nov 13 2011

nonn

Row 7 of A200057

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

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

Empirical: a(n) = 2*a(n-1) -a(n-3) -a(n-5) +a(n-6) -2*a(n-7) +2*a(n-8) +a(n-9) -a(n-13) -2*a(n-14) +2*a(n-15) -a(n-16) +a(n-17) +a(n-19) -2*a(n-21) +a(n-22)

cp's OEIS Frontend

A200051 Number of -2..2 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200052 Number of -3..3 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200053 Number of -4..4 arrays X (0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200054 Number of -5..5 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200055 Number of -6..6 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200056 Number of -7..7 arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200058 Number of -n..n arrays x(0..3) of 4 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200059 Number of -n..n arrays x(0..4) of 5 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200060 Number of -n..n arrays x(0..5) of 6 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A200061 Number of -n..n arrays x(0..6) of 7 elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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