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

1,1

Links

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

Robert Israel, Maple-assisted proof for empirical recursion

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

Example

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

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

Crossrefs

Column 5 of A269606.

Sequence in context: A269534 A269675 A269406 * A027910 A075848 A096979

Adjacent sequences: A269600 A269601 A269602 * A269604 A269605 A269606

Keyword

nonn

Author

R. H. Hardin, Mar 01 2016

Status

approved