cp's OEIS Frontend

A221524 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 2 or more.

%I A221524 #6 Jul 23 2025 02:01:07
%S A221524 0,0,0,0,2,0,0,6,2,0,0,12,10,4,0,0,20,30,36,6,0,0,30,68,144,94,10,0,0,
%T A221524 42,130,400,536,274,16,0,0,56,222,900,1940,2172,768,26,0,0,72,350,
%U A221524 1764,5368,9982,8544,2182,42,0,0,90,520,3136,12458,33380,50400,33960,6170,68,0,0
%N A221524 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 2 or more.
%C A221524 Table starts
%C A221524 .0...0......0.......0.........0..........0...........0...........0............0
%C A221524 .0...2......6......12........20.........30..........42..........56...........72
%C A221524 .0...2.....10......30........68........130.........222.........350..........520
%C A221524 .0...4.....36.....144.......400........900........1764........3136.........5184
%C A221524 .0...6.....94.....536......1940.......5368.......12458.......25544........47776
%C A221524 .0..10....274....2172......9982......33380.......90684......212812.......447962
%C A221524 .0..16....768....8544.....50400.....205080......654864.....1763328......4184064
%C A221524 .0..26...2182...33960....256018....1264378.....4738970....14629962.....39113752
%C A221524 .0..42...6170..134480...1297924....7787228....34274630...121342546....365574840
%C A221524 .0..68..17476..533248...6584320...47975704...247928860..1006508448...3416978176
%C A221524 .0.110..49470.2113456..33394958..295543282..1793345580..8348594292..31937713030
%C A221524 .0.178.140066.8377808.169387004.1820672982.12971955294.69248649436.298515152986
%H A221524 R. H. Hardin, <a href="/A221524/b221524.txt">Table of n, a(n) for n = 1..1518</a>
%F A221524 Empirical for column k:
%F A221524 k=2: a(n) = a(n-1) +a(n-2)
%F A221524 k=3: a(n) = a(n-1) +4*a(n-2) +3*a(n-3) +a(n-4)
%F A221524 k=4: a(n) = 2*a(n-1) +6*a(n-2) +6*a(n-3) +4*a(n-4) +4*a(n-6)
%F A221524 k=5: a(n) = 2*a(n-1) +11*a(n-2) +20*a(n-3) +17*a(n-4) -3*a(n-5) +a(n-6)
%F A221524 k=6: a(n) = 3*a(n-1) +14*a(n-2) +29*a(n-3) +28*a(n-4) +a(n-5) +27*a(n-6) +8*a(n-7) +2*a(n-8)
%F A221524 k=7: a(n) = 3*a(n-1) +21*a(n-2) +58*a(n-3) +79*a(n-4) +32*a(n-5) +23*a(n-6) +4*a(n-7) +8*a(n-8)
%F A221524 Empirical for row n:
%F A221524 n=2: a(n) = n^2 - n
%F A221524 n=3: a(n) = n^3 - 3*n^2 + 4*n - 2
%F A221524 n=4: a(n) = n^4 - 2*n^3 + n^2
%F A221524 n=5: a(n) = n^5 - n^4 - 10*n^3 + 38*n^2 - 60*n + 40 for n>2
%F A221524 n=6: a(n) = n^6 - 20*n^4 + 83*n^3 - 182*n^2 + 236*n - 148 for n>3
%F A221524 n=7: a(n) = n^7 + n^6 - 29*n^5 + 109*n^4 - 204*n^3 + 202*n^2 - 80*n for n>2
%e A221524 Some solutions for n=6 k=4
%e A221524 ..1....0....4....4....3....3....0....4....2....0....4....0....0....1....3....0
%e A221524 ..3....3....0....0....0....1....2....0....4....3....2....4....2....3....1....3
%e A221524 ..4....0....3....2....4....1....4....2....4....1....4....1....3....0....2....0
%e A221524 ..1....0....4....4....1....4....2....3....2....2....2....1....0....0....4....4
%e A221524 ..4....4....2....1....0....3....1....0....4....4....4....4....0....2....1....3
%e A221524 ..2....1....4....4....2....1....4....3....0....2....2....2....3....4....3....1
%Y A221524 Column 2 is A006355
%Y A221524 Row 2 is A002378(n-1)
%Y A221524 Row 3 is A034262(n-1)
%Y A221524 Row 4 is A035287
%K A221524 nonn,tabl
%O A221524 1,5
%A A221524 _R. H. Hardin_ Jan 19 2013