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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

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 October 26 05:58 EDT 2021. Contains 348257 sequences. (Running on oeis4.)