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

1,1

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Robert Israel, Maple-assisted proof of empirical recursion

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

Example

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

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

Crossrefs

Column 6 of A269606.

Sequence in context: A269535 A269676 A269407 * A269465 A268455 A323559

Adjacent sequences: A269601 A269602 A269603 * A269605 A269606 A269607

Keyword

nonn

Author

R. H. Hardin, Mar 01 2016

Status

approved