A269604 Number of length-n 0..6 arrays with no repeated value differing from the previous repeated value by one or less.

7, 49, 336, 2298, 15630, 105892, 714874, 4811578, 32300252, 216337084, 1446056046, 9648789758, 64281141440, 427655897226, 2841661493142, 18861464959350, 125070420653458, 828618463551536, 5485481885293294, 36288577806336542

Offset: 1

Author

R. H. Hardin, Mar 01 2016

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Column 6 of A269606.

Programs

Maple

with(LinearAlgebra):
T:= Matrix(56,56):
for x from 0 to 6 do
  for v from 0 to 7 do
    i:= 1 + x + 7*v;
    for y in {$0..6} minus {x} do
      T[i,1+y+7*v]:= 1;
    od:
    if abs(x-v) > 1 or v=7 then T[i,1+x+7*x]:= 1 fi
od od:
u:= Vector([0$49,1$7]): v:= Vector(56,1):
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) = 26*a(n-1) - 243*a(n-2) + 833*a(n-3) + 567*a(n-4) - 7567*a(n-5) - 1006*a(n-6) + 27361*a(n-7) + 31306*a(n-8) + 9984*a(n-9).

Empirical g.f.: x*(7 - 133*x + 763*x^2 - 362*x^3 - 7256*x^4 + 3224*x^5 + 32851*x^6 + 34133*x^7 + 10511*x^8) / ((1 - 6*x)*(1 - 20*x + 123*x^2 - 95*x^3 - 1137*x^4 + 745*x^5 + 5476*x^6 + 5495*x^7 + 1664*x^8)). - Colin Barker, Jan 24 2019

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