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

Author

R. H. Hardin, Aug 04 2014

Keywords

• nonn
• tabl

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

Examples

			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
		

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.

Crossrefs

Column 1 is A006355(n+4)

Column 3 is A206981(n+2)

Row 1 is A090381.

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)