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Number of nX1 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor
+10
1
1, 5, 25, 95, 325, 1121, 3985, 14288, 50995, 181336, 644721, 2294193, 8166441, 29066618, 103444256, 368138471, 1310164527, 4662787112, 16594519920, 59058487061, 210183969235, 748026706926, 2662163892493, 9474416502527
FORMULA
Empirical: a(n) = 5*a(n-1) -10*a(n-2) +20*a(n-3) -15*a(n-4) +21*a(n-5) -7*a(n-6) +8*a(n-7) -a(n-8) +a(n-9).
Empirical g.f.: -x*(1+x^2)*(x^6+6*x^4+9*x^2+1) / ( -1+5*x-10*x^2+20*x^3-15*x^4+21*x^5-7*x^6+8*x^7-x^8+x^9 ). - R. J. Mathar, Jul 02 2013
EXAMPLE
Some solutions for n=4
..2....0....4....0....4....0....2....4....0....0....2....1....0....1....3....2
..2....1....4....4....4....1....2....4....3....4....4....3....1....2....3....2
..1....2....1....4....4....4....1....1....4....4....4....4....4....2....2....3
..0....2....1....4....4....4....0....0....4....4....4....4....4....0....0....3
Number of n X 2 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor.
+10
1
5, 165, 3209, 53233, 904175, 15703631, 271942421, 4694586685, 81072693659, 1400687275175, 24198485401075, 418031304578759, 7221573611135109, 124755169399163453, 2155187028114381493, 37231528687477204117
FORMULA
Empirical: a(n) = 15*a(n-1) -15*a(n-2) +821*a(n-3) +1430*a(n-4) +9500*a(n-5) +11557*a(n-6) +27197*a(n-7) +15316*a(n-8) +14400*a(n-9).
EXAMPLE
Some solutions for n=4
..0..3....4..3....3..3....3..2....2..1....0..4....4..1....1..3....0..3....4..3
..4..3....4..0....2..0....3..1....2..2....0..4....4..3....1..4....3..3....4..0
..4..2....0..4....0..0....2..2....0..0....3..4....4..2....1..4....4..4....1..2
..2..0....0..4....2..2....4..4....0..0....3..0....4..4....2..2....0..0....4..4
Number of n X 3 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor.
+10
1
25, 3209, 263725, 19617781, 1508941377, 117061422325, 9045835238617, 698574745175731, 53971916518600687, 4170021237063241785, 322172335579673190661, 24890749591184617620689, 1923046620328708053554417
FORMULA
Empirical: a(n) = 95*a(n-1) -2335*a(n-2) +88226*a(n-3) -1401023*a(n-4) +27475918*a(n-5) -129518265*a(n-6) +3400204242*a(n-7) -8711692058*a(n-8) +135640160693*a(n-9) -1855897876716*a(n-10) -5857278804400*a(n-11) -207877345476359*a(n-12) -658851172326065*a(n-13) -2982964537519341*a(n-14) -1083968424653882*a(n-15) +175197610451185704*a(n-16) +687892185774751661*a(n-17) +749598878295789128*a(n-18) -1959929535011513006*a(n-19) -101370860705854696394*a(n-20) -340081565613577375756*a(n-21) +730110761278751037036*a(n-22) +3920118189702812561672*a(n-23) +12619795867291127254976*a(n-24) +27464586017442664267488*a(n-25) -245687729406837440328288*a(n-26) -909390951510003354547680*a(n-27) +1548025686942659823298048*a(n-28) +8058128637358990922686656*a(n-29) -2828697130837882118085760*a(n-30) -34619981240886667486806528*a(n-31) -13877583110751726094654208*a(n-32) +62093718052951543212177920*a(n-33) +85983078233799603752551424*a(n-34) +83941563134487710369197056*a(n-35) -150134157540885554603827200*a(n-36) -718538527715802110878629888*a(n-37) -125076595984856461470449664*a(n-38) +1768257902572823829865512960*a(n-39) +995241004870815671157964800*a(n-40) -2384512233033125505637416960*a(n-41) -1921544137783350730214277120*a(n-42) +1893287364220246513759027200*a(n-43) +1987808550619888781399162880*a(n-44) -836639937477404072843673600*a(n-45) -1213996168239086214394675200*a(n-46) +146320063422714906830438400*a(n-47) +433459533137700803995238400*a(n-48) +26235006449466816331776000*a(n-49) -82818213393016855461888000*a(n-50) -14649656375672234311680000*a(n-51) +6466292717028177346560000*a(n-52) +1643415634622545920000000*a(n-53).
EXAMPLE
Some solutions for n=4
..1..4..2....2..3..3....4..4..2....2..3..0....0..4..2....2..0..2....1..1..3
..0..4..0....2..1..4....0..1..1....1..3..0....0..4..1....3..1..3....3..0..3
..0..3..4....4..0..4....3..3..1....3..4..4....4..0..1....4..3..3....3..2..2
..0..0..4....4..1..3....4..4..2....4..4..4....4..4..0....4..0..3....2..1..2
Number of nX4 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor
+10
1
95, 53233, 19617781, 6474088455, 2202928769005, 754868206701309, 257892549401944763, 88078489175441149541, 30090514056455942145831, 10279936119776186239585141, 3511875047988498805215621351
EXAMPLE
Some solutions for n=4
..1..0..0..3....4..0..0..0....4..0..3..2....4..3..1..0....2..3..3..4
..2..0..2..3....4..4..0..0....4..4..4..3....4..2..2..3....2..4..4..4
..2..1..0..3....4..4..1..4....0..3..4..2....2..0..1..3....2..1..3..3
..1..4..4..3....4..2..2..4....4..4..1..1....0..0..3..3....3..3..4..4
Number of nX5 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor
+10
1
325, 904175, 1508941377, 2202928769005, 3326678666649639, 5067952959051283059, 7699282272868336554723, 11691435324507478903609409, 17757482537353874208791692831, 26971137476119655323173537171423
EXAMPLE
Some solutions for n=4
..1..3..1..0..0....3..2..0..4..0....3..3..2..2..4....2..0..3..4..0
..3..3..4..4..3....3..4..0..4..1....4..2..0..1..4....4..4..3..4..2
..2..0..1..3..1....4..4..2..0..1....4..3..1..4..1....1..1..4..3..1
..4..4..0..2..0....0..3..2..2..0....2..2..0..4..1....2..2..4..2..0
Number of nX6 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor
+10
1
1121, 15703631, 117061422325, 754868206701309, 5067952959051283059, 34353493702738177260748, 232103745979086439437076471, 1567267224439734346152398392389, 10585441536020941283010401910808799
EXAMPLE
Some solutions for n=4
..2..4..3..1..1..1....3..3..4..1..3..0....3..4..0..4..3..1....2..4..3..3..2..2
..0..4..3..1..4..1....0..3..4..1..3..3....3..4..1..4..1..2....3..4..1..2..2..3
..0..4..1..0..4..1....3..3..2..1..0..2....3..1..1..0..4..3....2..2..2..0..4..3
..3..3..2..3..3..0....4..4..2..3..3..2....1..0..0..4..4..4....1..0..2..2..4..3
Number of nX7 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor
+10
1
3985, 271942421, 9045835238617, 257892549401944763, 7699282272868336554723, 232103745979086439437076471, 6970324759861265041580630860421, 209209746691129348723930919203856597
EXAMPLE
Some solutions for n=4
..4..3..2..2..1..1..4....1..1..2..3..2..0..1....1..0..4..2..3..2..4
..4..4..1..4..1..1..4....4..3..3..3..0..1..4....1..4..4..2..4..0..4
..4..1..0..4..1..1..1....4..1..3..3..1..4..4....1..1..0..3..4..3..2
..1..2..4..4..0..0..0....3..1..3..3..3..0..0....1..4..4..1..1..2..0
Number of n X n 0..4 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor
+10
0
1, 165, 263725, 6474088455, 3326678666649639, 34353493702738177260748, 6970324759861265041580630860421
EXAMPLE
Some solutions for n=4
..2..0..1..3....1..4..4..0....0..2..4..1....3..0..4..2....0..1..2..0
..2..0..4..3....4..0..1..3....1..4..4..2....3..4..4..2....4..2..3..0
..1..2..4..0....4..1..2..3....2..4..1..4....0..3..4..3....4..4..3..3
..1..4..4..1....0..1..4..4....2..0..0..4....3..3..4..3....0..4..2..3
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