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A206455 T(n,k) = number of 0..k arrays of length n avoiding the consecutive pattern 0..k. 12
2, 3, 3, 4, 9, 4, 5, 16, 26, 5, 6, 25, 64, 75, 6, 7, 36, 125, 255, 216, 7, 8, 49, 216, 625, 1016, 622, 8, 9, 64, 343, 1296, 3124, 4048, 1791, 9, 10, 81, 512, 2401, 7776, 15615, 16128, 5157, 10, 11, 100, 729, 4096, 16807, 46655, 78050, 64257, 14849, 11, 12, 121, 1000 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..10000

Robert Israel, Proof of recurrence for column k

FORMULA

Empirical: T(n,k) = sum{i=0..floor(n/(k+1))} ( (-1)^i * (k+1)^(n-(k+1)*i) * binomial(n-k*i,i) ) (after A076264)

Empirical for column k: a(n) = (k+1)*a(n-1) - a(n-(k+1)).

Formula for column k verified by Robert Israel, Dec 17 2017 (see link).

EXAMPLE

Table starts

2 3 4 5 6 7 8 9 10 11 ...

3 9 16 25 36 49 64 81 100 121 ...

4 26 64 125 216 343 512 729 1000 1331 ...

5 75 255 625 1296 2401 4096 6561 10000 14641 ...

6 216 1016 3124 7776 16807 32768 59049 100000 161051 ...

7 622 4048 15615 46655 117649 262144 531441 1000000 1771561 ...

8 1791 16128 78050 279924 823542 2097152 4782969 10000000 19487171 ...

9 5157 64257 390125 1679508 5764787 16777215 43046721 100000000 214358881 ...

...

MAPLE

N:= 20: # for the first N antidiagonals

for k from 1 to N-1 do

F[k]:= gfun:-rectoproc({a(n)=(k+1)*a(n-1) - a(n-k-1), seq(a(j)=(k+1)^j, j=1..k), a(k+1)=(k+1)^(k+1)-1}, a(n), remember)

od:

seq(seq(F[m-j](j), j=1..m-1), m=1..N); # Robert Israel, Dec 17 2017

MATHEMATICA

nmax = 20;

col[k_] := col[k] = Module[{a}, a[n_ /; n>2] := a[n] = (k+1)*a[n-1]-a[n-k-1]; a[0]=1; a[1]=k+1; a[2]=(k+1)^2; a[_?Negative]=0; Array[a, nmax]];

T[n_, k_] := If[k == 1, n+1, col[k][[n]]];

Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 22 2022 *)

CROSSREFS

Columns 2, 3... are A076264, A206450, A206451, A206452.

Subdiagonal 1 is A048861(n+1)

Sequence in context: A193821 A130743 A263775 * A227263 A111574 A330510

Adjacent sequences: A206452 A206453 A206454 * A206456 A206457 A206458

KEYWORD

nonn,tabl,look

AUTHOR

R. H. Hardin, Feb 07 2012

STATUS

approved

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Last modified December 5 10:23 EST 2022. Contains 358586 sequences. (Running on oeis4.)