A245869 T(n,k)=Number of length n+2 0..k arrays with some pair in every consecutive three terms totalling exactly k.

6, 19, 10, 36, 45, 16, 61, 100, 103, 26, 90, 193, 256, 239, 42, 127, 318, 549, 676, 553, 68, 168, 493, 960, 1629, 1764, 1281, 110, 217, 712, 1579, 3102, 4753, 4624, 2967, 178, 270, 993, 2368, 5515, 9726, 13961, 12100, 6873, 288, 331, 1330, 3433, 8840, 18505, 30900

Offset

1,1

Comments

Table starts

.....6.......19........36.........61..........90..........127..........168

....10.......45.......100........193.........318..........493..........712

....16......103.......256........549.........960.........1579.........2368

....26......239.......676.......1629........3102.........5515.........8840

....42......553......1764.......4753........9726........18505........31176

....68.....1281......4624......13961.......30900........63241.......113024

...110.....2967.....12100......40901.......97602.......214315.......404264

...178.....6873.....31684.....119953......309078.......729097......1455496

...288....15921.....82944.....351649......977664......2475985......5223552

...466....36881....217156....1031057.....3094038......8415217.....18775816

...754....85435....568516....3022933.....9789654.....28590415.....67437448

..1220...197911...1488400....8863117....30977796.....97151683....242306240

..1974...458463...3896676...25986061....98020170....330100459....870461352

..3194..1062035..10201636...76189749...310161870...1121650903...3127322696

..5168..2460217..26708224..223384017...981426624...3811203385..11235107264

..8362..5699123..69923044..654949861..3105480558..12950003383..40363689352

.13530.13202089.183060900.1920277409..9826505742..44002376953.145010699592

.21892.30582803.479259664.5630150189.31093507092.149514426895.520968428032

Links

R. H. Hardin, Table of n, a(n) for n = 1..9999

Robert Israel, Proof of recurrences for columns

Robert Israel et al, A Pattern for OEIS Sequence A245869, Mathematics StackExchange, Jul 29-30, 2024.

Formula

Empirical for column k:

k=1: a(n) = a(n-1) +a(n-2)

k=2: a(n) = 2*a(n-1) +a(n-2) -a(n-4) -a(n-5)

k=3: a(n) = 2*a(n-1) +2*a(n-2) -a(n-3)

k=4: a(n) = 3*a(n-1) +a(n-2) -a(n-3) -5*a(n-4) -8*a(n-5) +3*a(n-6)

k=5: a(n) = 2*a(n-1) +4*a(n-2) -a(n-3)

k=6: a(n) = 3*a(n-1) +3*a(n-2) -a(n-3) -9*a(n-4) -24*a(n-5) +5*a(n-6)

k=7: a(n) = 2*a(n-1) +6*a(n-2) -a(n-3)

k=8: a(n) = 3*a(n-1) +5*a(n-2) -a(n-3) -13*a(n-4) -48*a(n-5) +7*a(n-6)

k=9: a(n) = 2*a(n-1) +8*a(n-2) -a(n-3)

Empirical for row n:

n=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)

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

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

n=4: a(n) = 3*a(n-1) -a(n-2) -5*a(n-3) +5*a(n-4) +a(n-5) -3*a(n-6) +a(n-7)

n=5: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)

n=6: a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9)

n=7: a(n) = 2*a(n-1) +3*a(n-2) -8*a(n-3) -2*a(n-4) +12*a(n-5) -2*a(n-6) -8*a(n-7) +3*a(n-8) +2*a(n-9) -a(n-10)

From Robert Israel, Aug 06 2024: (Start) For odd k, T(n,k) = 2 T(n-1,k) + (k-1) T(n-2,k) - T(n-3,k).

For even k, T(n,k) = 3 T(n-1,k) + (k-3) T(n-2,k) - T(n-3,k) + (2 k - 3) T(n-4,k) - k (k-2) T(n-5,k) + (k-1) T(n-6,k).

See links. (End)

Example

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

Crossrefs

Column 1 is A006355(n+4)

Column 3 is A206981(n+2)

Row 1 is A090381.

Sequence in context: A119813 A370716 A119986 * A370155 A184197 A173568

Adjacent sequences: A245866 A245867 A245868 * A245870 A245871 A245872

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 04 2014

Status

approved