cp's OEIS Frontend

A221596 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 1 or less.

%I A221596 #6 Jul 23 2025 02:02:09
%S A221596 0,0,4,0,7,8,0,10,17,16,0,13,26,49,32,0,16,35,100,139,64,0,19,44,169,
%T A221596 342,393,128,0,22,53,256,651,1210,1113,256,0,25,62,361,1068,2715,4240,
%U A221596 3151,512,0,28,71,484,1593,5082,11011,14898,8921,1024,0,31,80,625,2226,8475
%N A221596 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 1 or less.
%C A221596 Table starts
%C A221596 ....0......0.......0........0........0.........0.........0.........0..........0
%C A221596 ....4......7......10.......13.......16........19........22........25.........28
%C A221596 ....8.....17......26.......35.......44........53........62........71.........80
%C A221596 ...16.....49.....100......169......256.......361.......484.......625........784
%C A221596 ...32....139.....342......651.....1068......1593......2226......2967.......3816
%C A221596 ...64....393....1210.....2715.....5082......8475.....13056.....18987......26430
%C A221596 ..128...1113....4240....11011....22912.....41401.....67936....103975.....150976
%C A221596 ..256...3151...14898....45099...105586....210101....374342....615965.....954572
%C A221596 ..512...8921...52306...184063...482204...1047967...2006006...3504371....5714456
%C A221596 .1024..25257..183684...752155..2210256...5267759..10894988..20352239...35218688
%C A221596 .2048..71507..645006..3072247.10115926..26387005..58789204.116958723..213700742
%C A221596 .4096.202449.2264978.12550859.46327024.132384353.318224626.675761541.1307528098
%H A221596 R. H. Hardin, <a href="/A221596/b221596.txt">Table of n, a(n) for n = 1..2080</a>
%F A221596 Empirical for column k:
%F A221596 k=1: a(n) = 2*a(n-1) for n>2
%F A221596 k=2: a(n) = 2*a(n-1) +2*a(n-2) +a(n-3) for n>4
%F A221596 k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
%F A221596 k=4: a(n) = 3*a(n-1) +4*a(n-2) +6*a(n-4) +4*a(n-5) +4*a(n-6)
%F A221596 k=5: a(n) = 4*a(n-1) +3*a(n-2) -6*a(n-3) +19*a(n-4) +5*a(n-5) +a(n-6)
%F A221596 k=6: a(n) = 4*a(n-1) +5*a(n-2) -7*a(n-3) +33*a(n-4) +17*a(n-5) +24*a(n-6) -5*a(n-7) +2*a(n-8)
%F A221596 k=7: a(n) = 5*a(n-1) +3*a(n-2) -16*a(n-3) +65*a(n-4) -14*a(n-5) +23*a(n-6) +2*a(n-7) +8*a(n-8)
%F A221596 Empirical for row n:
%F A221596 n=2: a(n) = 3*n + 1
%F A221596 n=3: a(n) = 9*n - 1
%F A221596 n=4: a(n) = 9*n^2 + 6*n + 1
%F A221596 n=5: a(n) = 54*n^2 - 69*n + 63 for n>2
%F A221596 n=6: a(n) = 27*n^3 + 108*n^2 - 252*n + 267 for n>3
%F A221596 n=7: a(n) = 243*n^3 - 351*n^2 + 237*n + 127 for n>2
%e A221596 Some solutions for n=6 k=4
%e A221596 ..0....2....3....3....2....2....1....1....2....4....4....2....0....2....4....1
%e A221596 ..0....1....4....2....3....3....2....1....3....3....4....2....1....3....3....2
%e A221596 ..2....4....4....2....1....0....2....3....4....0....4....1....0....2....4....2
%e A221596 ..2....4....1....4....1....0....1....2....0....1....3....4....0....2....4....1
%e A221596 ..0....1....1....3....0....2....2....3....1....4....3....3....1....1....0....0
%e A221596 ..0....2....1....2....1....3....1....2....2....3....2....3....0....1....0....0
%Y A221596 Column 3 is A221568
%Y A221596 Row 2 is A016777
%Y A221596 Row 3 is A017257(n-1)
%Y A221596 Row 4 is A016778
%K A221596 nonn,tabl
%O A221596 1,3
%A A221596 _R. H. Hardin_ Jan 20 2013