A196700 -id:A196700 - cp's OEIS frontend

A196856 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,1,0,2,4 for x=0,1,2,3,4.

1, 2, 2, 4, 11, 4, 6, 38, 38, 6, 12, 136, 206, 136, 12, 22, 496, 1370, 1370, 496, 22, 40, 1792, 8767, 16876, 8767, 1792, 40, 74, 6440, 56470, 199125, 199125, 56470, 6440, 74, 136, 23306, 363685, 2369631, 4326720, 2369631, 363685, 23306, 136, 250, 84180

Offset: 1

R. H. Hardin Oct 06 2011

Every 0 is next to 0 3's, every 1 is next to 1 1's, every 2 is next to 2 0's, every 3 is next to 3 2's, every 4 is next to 4 4's
Table starts
...1......2........4...........6.............12...............22
...2.....11.......38.........136............496.............1792
...4.....38......206........1370...........8767............56470
...6....136.....1370.......16876.........199125..........2369631
..12....496.....8767......199125........4326720.........94652804
..22...1792....56470.....2369631.......94652804.......3805929995
..40...6440...363685....28194191.....2069281146.....152929723254
..74..23306..2343584...335586650....45250654402....6148984299163
.136..84180.15108610..3993458371...989582452566..247223213832440
.250.303664.97376923.47516896629.21641093339471.9939022894905148

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

R. H. Hardin, Table of n, a(n) for n = 1..144

Column 1 is A196700

A196950 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 4,1,0,2,3 for x=0,1,2,3,4.

Original entry on oeis.org

1, 2, 2, 4, 11, 4, 6, 38, 38, 6, 12, 136, 210, 136, 12, 22, 496, 1416, 1416, 496, 22, 40, 1792, 9199, 18060, 9199, 1792, 40, 74, 6440, 60206, 220433, 220433, 60206, 6440, 74, 136, 23306, 393585, 2711711, 5039158, 2711711, 393585, 23306, 136, 250, 84180

Offset: 1

R. H. Hardin Oct 08 2011

Every 0 is next to 0 4's, every 1 is next to 1 1's, every 2 is next to 2 0's, every 3 is next to 3 2's, every 4 is next to 4 3's
Table starts
...1......2.........4...........6.............12................22
...2.....11........38.........136............496..............1792
...4.....38.......210........1416...........9199.............60206
...6....136......1416.......18060.........220433...........2711711
..12....496......9199......220433........5039158.........115982718
..22...1792.....60206.....2711711......115982718........4998342821
..40...6440....393585....33344883.....2668162530......215125975535
..74..23306...2575161...410312728....61397041229.....9264434681345
.136..84180..16853437..5047886243..1412800280934...398999272140802
.250.303664.110272832.62090427845.32510004942226.17183027304906341

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

R. H. Hardin, Table of n, a(n) for n = 1..144

Column 1 is A196700

Column 2 is A196850

A242239 T(n,k)=Number of length n+k+1 0..k arrays with every value 0..k appearing at least once in every consecutive k+2 elements, and new values 0..k introduced in order.

Original entry on oeis.org

3, 6, 5, 10, 12, 8, 15, 22, 22, 13, 21, 35, 43, 40, 21, 28, 51, 71, 82, 74, 34, 36, 70, 106, 139, 157, 136, 55, 45, 92, 148, 211, 271, 304, 250, 89, 55, 117, 197, 298, 416, 531, 586, 460, 144, 66, 145, 253, 400, 592, 821, 1047, 1129, 846, 233, 78, 176, 316, 517, 799

Offset: 1

R. H. Hardin, May 08 2014

Table starts
...3....6...10...15....21....28....36....45....55....66....78....91...105
...5...12...22...35....51....70....92...117...145...176...210...247...287
...8...22...43...71...106...148...197...253...316...386...463...547...638
..13...40...82..139...211...298...400...517...649...796...958..1135..1327
..21...74..157..271...416...592...799..1037..1306..1606..1937..2299..2692
..34..136..304..531...821..1174..1590..2069..2611..3216..3884..4615..5409
..55..250..586.1047..1626..2332..3165..4125..5212..6426..7767..9235.10830
..89..460.1129.2059..3231..4642..6308..8229.10405.12836.15522.18463.21659
.144..846.2176.4047..6411..9256.12587.16429.20782.25646.31021.36907.43304
.233.1556.4195.7955.12716.18442.25138.32821.41527.51256.62008.73783.86581

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

R. H. Hardin, Table of n, a(n) for n = 1..731

Column 1 is A000045(n+3)
Column 2 is A196700(n+3)
Row 1 is A000217(n+1)
Row 2 is A000326(n+1)
Row 3 is A069099(n+1)
Row 4 is A220083

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +a(n-2) +a(n-3)
k=3: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4)
k=4: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5)
k=5: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6)
k=6: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7)
k=7: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8)
k=8: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9)
k=9: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9) +a(n-10)
Empirical for row n:
n=1: a(n) = (1/2)*n^2 + (3/2)*n + 1
n=2: a(n) = (3/2)*n^2 + (5/2)*n + 1
n=3: a(n) = (7/2)*n^2 + (7/2)*n + 1
n=4: a(n) = (15/2)*n^2 + (9/2)*n + 1
n=5: a(n) = (31/2)*n^2 + (11/2)*n + 1 for n>1
n=6: a(n) = (63/2)*n^2 + (13/2)*n + 1 for n>2
n=7: a(n) = (127/2)*n^2 + (15/2)*n + 1 for n>3
n=8: a(n) = (255/2)*n^2 + (17/2)*n + 1 for n>4
n=9: a(n) = (511/2)*n^2 + (19/2)*n + 1 for n>5
n=10: a(n) = (1023/2)*n^2 + (21/2)*n + 1 for n>6
n=11: a(n) = (2047/2)*n^2 + (23/2)*n + 1 for n>7
n=12: a(n) = (4095/2)*n^2 + (25/2)*n + 1 for n>8
n=13: a(n) = (8191/2)*n^2 + (27/2)*n + 1 for n>9
n=14: a(n) = (16383/2)*n^2 + (29/2)*n + 1 for n>10
n=15: a(n) = (32767/2)*n^2 + (31/2)*n + 1 for n>11
Empirical large-k generalization, for k>n-4: T(n,k) = ((2^n-1)/2)*k^2 + ((2*n+1)/2)*k + 1
Empirical recurrence generalization, for column k: a(n) = sum {i in 1..k+1} a(n-i)

A197001 Decimal expansion of the slope of the line y=mx which meets the curve y=1+cos(x) orthogonally over the interval [0, 2*Pi] (as in A197000).

Original entry on oeis.org

1, 0, 5, 4, 1, 7, 8, 4, 4, 2, 6, 5, 6, 8, 4, 2, 1, 7, 5, 1, 5, 7, 4, 7, 7, 3, 4, 3, 0, 5, 6, 7, 3, 4, 8, 3, 7, 4, 6, 1, 4, 2, 1, 0, 4, 5, 8, 9, 1, 6, 0, 6, 6, 4, 5, 3, 6, 7, 7, 2, 1, 8, 5, 0, 7, 8, 2, 3, 8, 0, 7, 2, 5, 6, 7, 6, 3, 2, 7, 7, 7, 9, 0, 9, 4, 3, 3, 8, 4, 5, 0, 3, 2, 0, 5, 7, 5, 4, 6, 9, 3

Offset: 1

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=1.2488014367215508560475125020128381535587614...
yo=1.3164595537507515212878992732671186100622603...
m=1.05417844265684217515747734305673483746142104...
|OP|=1.81454423617045980814297669595599066552030...

Cf. A196700, A196996, A197002.

c = 1;
xo = x /.
  FindRoot[x == Sin[x] (c + Cos[x]), {x, 1, 1.3}, WorkingPrecision -> 100]
RealDigits[xo] (* A197000 *)
m = 1/Sin[xo]
RealDigits[m]  (* A197001 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{c + Cos[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi}],  ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

A248959 Number of ternary words of length n in which all digits 0..2 occur in every subword of 4 consecutive digits.

Original entry on oeis.org

1, 3, 9, 27, 36, 72, 132, 240, 444, 816, 1500, 2760, 5076, 9336, 17172, 31584, 58092, 106848, 196524, 361464, 664836, 1222824, 2249124, 4136784, 7608732, 13994640, 25740156, 47343528, 87078324, 160162008, 294583860, 541824192, 996570060, 1832978112

Offset: 0

Andrew Woods, Jan 12 2015

For n < 4 the constraint is voidly satisfied: each of the n-digit words satisfies the definition since there is no subword of length 4. - M. F. Hasler, Jan 13 2015

Colin Barker, Table of n, a(n) for n = 0..1000 (corrected by _Georg Fischer_, Jan 20 2019)
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A001590, A196700.

Join[{1,3,9,27},LinearRecurrence[{1,1,1},{36,72,132},30]] (* Harvey P. Dale, Mar 12 2015 *)

Vec((1+2*x+5*x^2+14*x^3-3*x^4-3*x^6)/(1-x-x^2-x^3) + O(x^100)) \\ Colin Barker, Jan 12 2015; extended to indices 0..3 by M. F. Hasler, Jan 13 2015

G.f.: (1 + 2*x + 5*x^2 + 14*x^3 - 3*x^4 - 3*x^6)/(1 - x - x^2 - x^3). - Corrected by Colin Barker, Jan 12 2015
a(n) = a(n-1) + a(n-2) + a(n-3).
a(n) = A001590(n+1) * 12, for n>=4.
a(n) = A196700(n) * 6, for n>=4. - Alois P. Heinz, Jan 12 2015

a(0)-a(3) from M. F. Hasler, Jan 13 2015

cp's OEIS Frontend

A196856 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,1,0,2,4 for x=0,1,2,3,4.

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A196950 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 4,1,0,2,3 for x=0,1,2,3,4.

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A242239 T(n,k)=Number of length n+k+1 0..k arrays with every value 0..k appearing at least once in every consecutive k+2 elements, and new values 0..k introduced in order.

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Comments

Examples

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A197001 Decimal expansion of the slope of the line y=mx which meets the curve y=1+cos(x) orthogonally over the interval [0, 2*Pi] (as in A197000).

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Mathematica

A248959 Number of ternary words of length n in which all digits 0..2 occur in every subword of 4 consecutive digits.

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Programs

Mathematica

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