A218836 -id:A218836 - cp's OEIS frontend

A219410 T(n,k)=Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal or antidiagonal neighbor in a random 0..2 nXk array.

1, 2, 1, 4, 7, 1, 7, 28, 21, 1, 12, 98, 181, 65, 1, 21, 351, 1199, 1180, 200, 1, 37, 1261, 8173, 14737, 7687, 616, 1, 65, 4523, 57097, 193116, 181089, 50077, 1897, 1, 114, 16233, 398375, 2633596, 4560446, 2225293, 326233, 5842, 1, 200, 58268, 2773933

Offset: 1

R. H. Hardin Nov 19 2012

Table starts
.1......2..........4............7.............12..............21
.1......7.........28...........98............351............1261
.1.....21........181.........1199...........8173...........57097
.1.....65.......1180........14737.........193116.........2633596
.1....200.......7687.......181089........4560446.......121641579
.1....616......50077......2225293......107701719......5618116265
.1...1897.....326233.....27345143.....2543481662....259468310384
.1...5842....2125270....336026564....60067211485..11983486642214
.1..17991...13845268...4129209727..1418553120783.553453107750115
.1..55405...90196219..50741147949.33500701298909
.1.170625..587591326.623524656508
.1.525456.3827916001

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

R. H. Hardin, Table of n, a(n) for n = 1..110

Column 2 is A218836

Row 1 is A005251(n+2)

A220386 T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal and antidiagonal neighbors in a random 0..1 nXk array.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 7, 25, 21, 1, 12, 79, 136, 65, 1, 21, 278, 757, 753, 200, 1, 37, 966, 5114, 7462, 4160, 616, 1, 65, 3362, 33247, 96772, 73066, 22989, 1897, 1, 114, 11642, 211944, 1191846, 1829169, 715412, 127037, 5842, 1, 200, 40375, 1358599, 14300020

Offset: 1

R. H. Hardin Dec 13 2012

Table starts
.1......2.........4...........7............12.............21..............37
.1......7........25..........79...........278............966............3362
.1.....21.......136.........757..........5114..........33247..........211944
.1.....65.......753........7462.........96772........1191846........14300020
.1....200......4160.......73066.......1829169.......42415575.......953142266
.1....616.....22989......715412......34521416.....1509240923.....63512364783
.1...1897....127037.....7003040.....651568437....53692543412...4230913273170
.1...5842....702009....68557435...12297040231..1910141303899.281832304026762
.1..17991...3879313...671141189..232084431547.67952729355483
.1..55405..21437148..6570141318.4380156648582
.1.170625.118462034.64318370539
.1.525456.654623158

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

R. H. Hardin, Table of n, a(n) for n = 1..110

Column 2 is A218836

Row 1 is A005251(n+2)

A221035 T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal and antidiagonal neighbors in a random 0..2 nXk array.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 7, 33, 21, 1, 12, 119, 228, 65, 1, 21, 457, 1733, 1561, 200, 1, 37, 1710, 14277, 24485, 10648, 616, 1, 65, 6466, 110506, 419506, 345755, 72625, 1897, 1, 114, 24433, 870027, 6637755, 12239631, 4882030, 495329, 5842, 1, 200, 92196, 6717882

Offset: 1

R. H. Hardin Dec 29 2012

Table starts
.1......2.........4...........7...........12............21............37
.1......7........33.........119..........457..........1710..........6466
.1.....21.......228........1733........14277........110506........870027
.1.....65......1561.......24485.......419506.......6637755.....106761517
.1....200.....10648......345755.....12239631.....393788081...12843452050
.1....616.....72625.....4882030....356411031...23287585899.1537391091350
.1...1897....495329....68933905..10373626389.1376083439034
.1...5842...3378333...973340015.301896920812
.1..17991..23041525.13743460075
.1..55405.157152036
.1.170625
.1

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

R. H. Hardin, Table of n, a(n) for n = 1..82

Column 2 is A218836

Row 1 is A005251(n+2)

A232047 T(n,k)=Number of nXk 0..1 arrays with no element less than a strict majority of its horizontal and antidiagonal neighbors.

Original entry on oeis.org

2, 2, 4, 4, 7, 8, 7, 15, 21, 16, 12, 34, 80, 65, 32, 21, 79, 318, 446, 200, 64, 37, 184, 1315, 3082, 2477, 616, 128, 65, 426, 5364, 22063, 29974, 13752, 1897, 256, 114, 984, 21680, 153562, 377676, 290672, 76375, 5842, 512, 200, 2274, 87452, 1060850, 4588174

Offset: 1

R. H. Hardin, Nov 17 2013

Table starts
....2.....2........4..........7...........12..............21................37
....4.....7.......15.........34...........79.............184...............426
....8....21.......80........318.........1315............5364.............21680
...16....65......446.......3082........22063..........153562...........1060850
...32...200.....2477......29974.......377676.........4588174..........55505057
...64...616....13752.....290672......6430408.......136134243........2882322121
..128..1897....76375....2821630....109609484......4041385884......149582129861
..256..5842...424115...27382537...1868028342....119990644449.....7766282047395
..512.17991..2355221..265752221..31836538191...3562337669985...403179428472169
.1024.55405.13079032.2579134666.542586883485.105762437152368.20931014633412316

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

R. H. Hardin, Table of n, a(n) for n = 1..478

Column 1 is A000079
Column 2 is A218836
Row 1 is A005251(n+2)

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1) +3*a(n-2) +a(n-3)
k=3: a(n) = 4*a(n-1) +9*a(n-2) -a(n-3) -6*a(n-4) for n>5
k=4: [order 8] for n>9
k=5: [order 14] for n>15
k=6: [order 24] for n>26
k=7: [order 44] for n>47
Empirical for row n:
n=1: a(n) = 2*a(n-1) -a(n-2) +a(n-3) for n>4
n=2: a(n) = 4*a(n-1) -6*a(n-2) +7*a(n-3) -6*a(n-4) +3*a(n-5) -a(n-6) -a(n-7) for n>8
n=3: [order 15] for n>18
n=4: [order 33] for n>36
n=5: [order 78] for n>84

A297314 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 1 or 2 neighboring 1s.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 7, 23, 21, 1, 12, 66, 117, 65, 1, 21, 207, 497, 609, 200, 1, 37, 654, 2577, 3808, 3159, 616, 1, 65, 2049, 13937, 35476, 29212, 16389, 1897, 1, 114, 6422, 72541, 340825, 484808, 223995, 85041, 5842, 1, 200, 20119, 375054, 2997197, 8273245

Offset: 1

R. H. Hardin, Dec 28 2017

Table starts
.1.....2.......4.........7..........12............21..............37
.1.....7......23........66.........207...........654............2049
.1....21.....117.......497........2577.........13937...........72541
.1....65.....609......3808.......35476........340825.........2997197
.1...200....3159.....29212......484808.......8273245.......121339476
.1...616...16389....223995.....6623719.....200646607......4893232934
.1..1897...85041...1717882....90535227....4869858862....197589351469
.1..5842..441225..13174266..1237278512..118156684121...7976248015498
.1.17991.2289339.101033369.16909630099.2867120332406.322003901582689

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

R. H. Hardin, Table of n, a(n) for n = 1..447

Column 2 is A218836.

Row 1 is A005251(n+2).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 2*a(n-1) +3*a(n-2) +a(n-3)
k=3: a(n) = 3*a(n-1) +11*a(n-2) +3*a(n-3) -6*a(n-4)
k=4: [order 8] for n>9
k=5: [order 12] for n>14
k=6: [order 22] for n>25
k=7: [order 35] for n>39
Empirical for row n:
n=1: a(n) = 2*a(n-1) -a(n-2) +a(n-3)
n=2: [order 9]
n=3: [order 23]
n=4: [order 61]

A219421 T(n,k)=Unchanging value maps: number of nXk binary arrays indicating the locations of corresponding elements unequal to no horizontal or antidiagonal neighbor in a random 0..2 nXk array.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 7, 20, 21, 1, 12, 57, 93, 65, 1, 21, 160, 453, 436, 200, 1, 37, 454, 2121, 3617, 2043, 616, 1, 65, 1292, 9926, 28154, 28888, 9573, 1897, 1, 114, 3676, 46776, 218838, 372560, 230726, 44857, 5842, 1, 200, 10452, 220655, 1709703, 4775307

Offset: 1

R. H. Hardin Nov 19 2012

Table starts
.1......2.........4...........7.............12................21
.1......7........20..........57............160...............454
.1.....21........93.........453...........2121..............9926
.1.....65.......436........3617..........28154............218838
.1....200......2043.......28888.........372560...........4775307
.1....616......9573......230726........4934366.........104434119
.1...1897.....44857.....1842766.......65352789........2284814260
.1...5842....210190....14717828......865523957.......49979133672
.1..17991....984904...117548611....11462977378.....1093274545336
.1..55405...4615043...938839259...151815415029....23915155162485
.1.170625..21625074..7498337380..2010639231109...523138192294728
.1.525456.101330329.59887849061.26628853182336.11443519356879492

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

R. H. Hardin, Table of n, a(n) for n = 1..178

Column 2 is A218836
Column 3 is A218837
Column 4 is A218838
Row 1 is A005251(n+2)

A221499 T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal and antidiagonal neighbors in a random 0..3 nXk array.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 7, 33, 21, 1, 12, 119, 228, 65, 1, 21, 457, 1733, 1561, 200, 1, 37, 1710, 14297, 24485, 10648, 616, 1, 65, 6466, 111042, 420022, 345755, 72625, 1897, 1, 114, 24433, 874106, 6665056, 12253352, 4882030, 495329, 5842, 1, 200, 92196, 6765307

Offset: 1

R. H. Hardin Jan 18 2013

Table starts
.1......2.........4...........7...........12............21............37
.1......7........33.........119..........457..........1710..........6466
.1.....21.......228........1733........14297........111042........874106
.1.....65......1561.......24485.......420022.......6665056.....107190767
.1....200.....10648......345755.....12253352.....395300442...12890161742
.1....616.....72625.....4882030....356799776...23379869304.1542965772979
.1...1897....495329....68933905..10385011060.1381866811158
.1...5842...3378333...973340015.302233979821
.1..17991..23041525.13743460075
.1..55405.157152036
.1.170625
.1

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

R. H. Hardin, Table of n, a(n) for n = 1..83

Column 2 is A218836
Column 3 is A221030
Column 4 is A221031
Row 1 is A005251(n+2)
Row 2 is A221036

A097472 Number of different candle trees having a total of m edges.

Original entry on oeis.org

1, 3, 10, 31, 96, 296, 912, 2809, 8651, 26642, 82047, 252672, 778128, 2396320, 7379697, 22726483, 69988378, 215535903, 663763424, 2044122936, 6295072048, 19386276329, 59701891739, 183857684514, 566207320575, 1743689586432

Offset: 0

Alexander Malkis, Sep 18 2004

A candle tree is a graph on the plane square lattice Z X Z whose edges have length one with the following properties: (a) It contains a line segment ("trunk") of length from 0 to m on the vertical axis, its lowest node is at the origin. (b) It contains horizontal line segments ("branches"); each of them intersects the trunk. (c) Each branch is allowed to have "candles", which are vertical edges of length 1, whose lower node is on a branch.

Row sums of triangle in A238241. - Philippe Deléham, Feb 21 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Malkis, Polyedges, polyominoes and the 'Digit' game, diploma thesis in computer science, Universität des Saarlandes, 2003, Saarbrücken.
Index entries for linear recurrences with constant coefficients, signature (3,1,-2,-1).

Bisection of A060945 and |A077930|.

CoefficientList[Series[1/(x^4+2x^3-x^2-3x+1),{x,0,30}],x] (* or *) LinearRecurrence[{3,1,-2,-1},{1,3,10,31},30] (* Harvey P. Dale, Jun 14 2011 *)

a(n):=sum(sum(binomial(k,n-m-k+1)*binomial(k+2*m-1,2*m-1),k,1,n-m+1),m,1,n)+1; /* Vladimir Kruchinin, May 12 2011 */

a(n)=sum(m=1,n,sum(k=1,n-m+1,binomial(k,n-m-k+1)*binomial(k+2*m-1,2*m-1))) \\ Charles R Greathouse IV, Jun 17 2013

a(n) = Sum_{s, d, k>=0 with s+d+k=m} binomial(s+2d+1, s)*binomial(s, k);
generating function = 1/((1-x)*(1-2*x-3*x^2-x^3)).
a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4);
a(n) = 1 + Sum_{m=1..n} Sum_{k=1..n-m+1} binomial(k, n-m-k+1)*binomial(k+2*m-1,2*m-1). - Vladimir Kruchinin, May 12 2011
a(n) = Sum_{k=0..n} A238241(n,k). - Philippe Deléham, Feb 21 2014
a(n) - a(n-1) = A218836(n). - R. J. Mathar, Jun 17 2020

A368205 a(n) = 3a(n-1) - 2a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

Original entry on oeis.org

1, 3, 7, 14, 25, 40, 56, 63, 37, -71, -350, -945, -2064, -3952, -6783, -10381, -13625, -13330, -2359, 33208, 117672, 288959, 598325, 1099385, 1812546, 2640543, 3197152, 2497824, -1541375, -12816925, -37865849, -86422322, -170718343, -301444536, -476474600, -655816385, -713055419, -351058887, 1028750562, 4501424879, 11797832400, 25361896880, 47988600961

Offset: 0

Raul Prisacariu, Dec 18 2023

sign

Whittaker's Root Series Formula is applied to the polynomial equation -1+2x+3x^2+x^3. The following infinite series involving the Plastic Ratio (rho) is obtained: rho - 1 = 1/2 - 3/(2*7) + 7/(7*21) - 14/(21*65) + 25/(65*200) - 40/(200*616) + 56/(616*1897) - ...

The terms of the sequence appear in the numerators of the infinite sequence (with alternating signs). The denominators of the sequence are formed by multiplying consecutive terms from the sequence A218836.

			a(0) = 1,
a(1) = 3*a(0) = 3*1 = 3,
a(2) = 3*a(1) - 2*a(0) = 3*3 - 2*1 = 7,
a(3) = 3*a(2) - 2*a(1) - a(0) = 3*7 - 2*3 - 1 = 14.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A218836 (denominator), A060006.

a:=proc(n) local c1,c2,c3;
 option remember;
c1:=3; c2:=2; c3:=1;
if n=0 then 1
elif n=1 then 3
elif n=2 then 7
else c1*a(n-1)-c2*a(n-2)-c3*a(n-3); fi;
end; # N. J. A. Sloane, Dec 31 2023
[seq(a(n),n=0..30)];

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3).

a(n) = determinant of the n X n Toeplitz Matrix((3,2,-1,0,0,...,0),(3,1,0,0,0,...,0)).

cp's OEIS Frontend

A219410 T(n,k)=Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal or antidiagonal neighbor in a random 0..2 nXk array.

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A220386 T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal and antidiagonal neighbors in a random 0..1 nXk array.

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A221035 T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal and antidiagonal neighbors in a random 0..2 nXk array.

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A232047 T(n,k)=Number of nXk 0..1 arrays with no element less than a strict majority of its horizontal and antidiagonal neighbors.

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A297314 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 1 or 2 neighboring 1s.

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A219421 T(n,k)=Unchanging value maps: number of nXk binary arrays indicating the locations of corresponding elements unequal to no horizontal or antidiagonal neighbor in a random 0..2 nXk array.

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A221499 T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal and antidiagonal neighbors in a random 0..3 nXk array.

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A097472 Number of different candle trees having a total of m edges.

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Mathematica

Maxima

PARI

Formula

A368205 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

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A368205 a(n) = 3a(n-1) - 2a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.