A253337 Number of (n+2)X(3+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 3, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

1928, 11581, 100512, 722826, 5136108, 32653458, 237047904, 1348380871, 8419903755, 34660596075, 237311962480, 870344555637, 3971606587854, 8571859069436, 48627137874733, 107516598517821, 328229525693772

Offset

1,1

Comments

Column 3 of A253342.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Formula

Empirical: a(n) = a(n-1) +12*a(n-4) -12*a(n-5) -66*a(n-8) +66*a(n-9) +220*a(n-12) -220*a(n-13) -495*a(n-16) +495*a(n-17) +792*a(n-20) -792*a(n-21) -924*a(n-24) +924*a(n-25) +792*a(n-28) -792*a(n-29) -495*a(n-32) +495*a(n-33) +220*a(n-36) -220*a(n-37) -66*a(n-40) +66*a(n-41) +12*a(n-44) -12*a(n-45) -a(n-48) +a(n-49) for n>71.

Empirical for n mod 4 = 0: a(n) = (1540096/467775)*n^12 - (7542784/155925)*n^11 + (108720928/42525)*n^10 - (453584308/2835)*n^9 + (1688010882377/453600)*n^8 - (38365151849041/806400)*n^7 + (826965583000531727/1393459200)*n^6 - (177348516650145373/23224320)*n^5 + (3324474156765521413/87091200)*n^4 + (3655946568356028397/7257600)*n^3 - (6889324950876403213/831600)*n^2 + (148601409077722168/3465)*n - 76338916329333 for n>32.

Empirical for n mod 4 = 1: a(n) = (1540096/467775)*n^12 - (2000896/51975)*n^11 + (2934368/1215)*n^10 - (20814436/135)*n^9 + (219838768751/64800)*n^8 - (11126652794539/268800)*n^7 + (151347083951906659/278691840)*n^6 - (112830681771625133/15482880)*n^5 + (7025769392932947439/199065600)*n^4 + (2695111383331559479/5529600)*n^3 - (378402709350579141439/48660480)*n^2 + (252331545669785791283/6307840)*n - (2355042957414789949/32768) for n>32.

Empirical for n mod 4 = 2: a(n) = (1540096/467775)*n^12 - (12163072/155925)*n^11 + (140347168/42525)*n^10 - (175136764/945)*n^9 + (2270678296457/453600)*n^8 - (194367562500739/2419200)*n^7 + (1489082590893077807/1393459200)*n^6 - (299952198050090239/23224320)*n^5 + (31829307458540304637/348364800)*n^4 + (795696519935387869/4838400)*n^3 - (879463200527799781589/106444800)*n^2 + (10907905965331721363/197120)*n - (62807878440696537/512) for n>32.

Empirical for n mod 4 = 3: a(n) = (1540096/467775)*n^12 - (2048/231)*n^11 + (57215776/42525)*n^10 - (68511332/567)*n^9 + (879319947977/453600)*n^8 - (16229707133/13824)*n^7 - (136576134639067441/1393459200)*n^6 - (1373621259681677/3096576)*n^5 - (31143022700068321667/1393459200)*n^4 + (3741645534691964251/4644864)*n^3 - (12898132224286711537733/1703116800)*n^2 + (42037452007127774261/1622016)*n - (622613756095641673/32768) for n>32.

Example

Some solutions for n=3
..0..1..1..2..0....0..3..2..2..1....0..1..1..1..1....0..1..1..2..1
..1..2..0..1..1....2..2..1..2..3....1..2..0..1..1....1..2..1..1..1
..1..1..2..1..1....2..1..3..2..1....0..0..1..1..0....1..1..2..1..2
..2..1..1..1..2....2..1..2..2..2....1..0..1..0..1....1..1..2..1..1
..1..2..1..2..1....1..2..2..2..2....0..1..1..1..1....2..2..1..2..2
Knight distance matrix for n=3
..0..3..2..3..2
..3..4..1..2..3
..2..1..4..3..2
..3..2..3..2..3
..2..3..2..3..4

Crossrefs

Sequence in context: A133301 A258841 A099482 * A200435 A252108 A220717

Adjacent sequences: A253334 A253335 A253336 * A253338 A253339 A253340

Keyword

nonn

Author

R. H. Hardin, Dec 30 2014

Status

approved