A269603 Number of length-n 0..5 arrays with no repeated value differing from the previous repeated value by one or less.

6, 36, 210, 1220, 7030, 40288, 229754, 1304934, 7385898, 41679780, 234601902, 1317558578, 7385249086, 41325945826, 230904832646, 1288466651340, 7181415415962, 39985405920156, 222432351559566, 1236355637246456

Offset: 1

Author

R. H. Hardin, Mar 01 2016

Keywords

• nonn

Examples

			Some solutions for n=6:
..4. .4. .1. .3. .0. .3. .5. .1. .1. .5. .5. .1. .3. .0. .2. .0
..3. .5. .5. .2. .3. .1. .1. .3. .1. .5. .3. .2. .2. .0. .5. .3
..4. .3. .3. .1. .3. .5. .2. .2. .3. .2. .4. .1. .5. .2. .4. .5
..2. .2. .1. .3. .2. .1. .1. .0. .0. .5. .3. .4. .2. .3. .1. .0
..3. .5. .2. .0. .1. .3. .4. .4. .3. .2. .1. .0. .2. .1. .4. .4
..0. .1. .5. .3. .2. .3. .1. .4. .5. .2. .2. .2. .4. .3. .1. .4
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Robert Israel, Maple-assisted proof for empirical recursion

Crossrefs

Column 5 of A269606.

Programs

Maple

T:= Matrix(42,42):
for x from 0 to 5 do
  for v from 0 to 6 do
    i:= 1 + x + 6*v;
    for y in {$0..5} minus {x} do
      T[i,1+y+6*v]:= 1;
    od:
    if abs(x-v) > 1 or v=6 then T[i,1+x+6*x]:= 1 fi
od od:
u:= Vector([0$36,1$6]): v:= Vector(42,1):
Tv[1]:= v:
for n from 2 to 50 do Tv[n]:= T . Tv[n-1] od:
seq(u^%T . Tv[n], n=1..50); # Robert Israel, Jan 24 2019

Formula

Empirical: a(n) = 17*a(n-1) - 91*a(n-2) + 83*a(n-3) + 542*a(n-4) - 550*a(n-5) - 1651*a(n-6) - 745*a(n-7).

Empirical g.f.: 2*x*(3 - 33*x + 72*x^2 + 214*x^3 - 420*x^4 - 922*x^5 - 393*x^6) / ((1 - 5*x)*(1 - 12*x + 31*x^2 + 72*x^3 - 182*x^4 - 360*x^5 - 149*x^6)). - Colin Barker, Jan 24 2019

Empirical recursion verified - see link. - Robert Israel, Jan 24 2019