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A141539 Square array A(n,k) of numbers of length n binary words with at least k "0" between any two "1" digits (n,k >= 0), read by antidiagonals. 4
1, 1, 2, 1, 2, 4, 1, 2, 3, 8, 1, 2, 3, 5, 16, 1, 2, 3, 4, 8, 32, 1, 2, 3, 4, 6, 13, 64, 1, 2, 3, 4, 5, 9, 21, 128, 1, 2, 3, 4, 5, 7, 13, 34, 256, 1, 2, 3, 4, 5, 6, 10, 19, 55, 512, 1, 2, 3, 4, 5, 6, 8, 14, 28, 89, 1024, 1, 2, 3, 4, 5, 6, 7, 11, 19, 41, 144, 2048, 1, 2, 3, 4, 5, 6, 7, 9, 15, 26, 60, 233, 4096 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A(n,k+1) = A(n,k) - A143291(n,k).

From Gary W. Adamson, Dec 19 2009: (Start)

Alternative method generated from variants of an infinite lower triangle T(n) = A000012 = (1; 1,1; 1,1,1;...) such that T(n) has the leftmost column shifted up n times. Then take Lim_{k=1..inf.} (T(n))^k, obtaining a left-shifted vector considered as rows of an array (deleting the first 1) as follows:

1, 2, 4, 8, 16, 32, 64, 128, 256,... = the doubling series.

1, 1, 2, 3, .5, .8, 13, .21, .34,... = the Fibonacci numbers.

1, 1, 1, 2, .3, .4, .6, ..9, .13,... = A000930

1, 1, 1, 1, .2, .3, .4, ..5, ..7,... = A003269

...with the next rows A003520, A005708, A005709,... such that beginning with the Fibonacci row, the succession of rows are recursive sequences generated from a(n) = a(n-1) + a(n-2); a(n) = a(n-1) + a(n-3),...a(n) = a(n-1) + a(n-k); k = 2,3,4,... Last, columns going up from the topmost 1 become rows of triangle A141539. (End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

FORMULA

G.f. of column k: x^(-k)/(1-x-x^(k+1)). A(n,k) = 2^n if k=0, else A(n,k) = n+1 if n<=k, else A(n,k) = A(n-1,k) + A(n-k-1,k).

EXAMPLE

A(4,2) = 6, because 6 binary words of length 4 have at least 2 "0" between any two "1" digits: 0000, 0001, 0010, 0100, 1000, 1001.

Square array A(n,k) begins:

    1,  1,  1,  1,  1,  1,  1,  1, ...

    2,  2,  2,  2,  2,  2,  2,  2, ...

    4,  3,  3,  3,  3,  3,  3,  3, ...

    8,  5,  4,  4,  4,  4,  4,  4, ...

   16,  8,  6,  5,  5,  5,  5,  5, ...

   32, 13,  9,  7,  6,  6,  6,  6, ...

   64, 21, 13, 10,  8,  7,  7,  7, ...

  128, 34, 19, 14, 11,  9,  8,  8, ...

MAPLE

A:= proc(n, k) option remember;

      if k=0 then 2^n

    elif n<=k and n>=0 then n+1

    elif n>0 then A(n-1, k) +A(n-k-1, k)

    else          A(n+1+k, k) -A(n+k, k)

      fi

    end:

seq(seq(A(n, d-n), n=0..d), d=0..15);

MATHEMATICA

a[n_, k_] := a[n, k] = Which[k == 0, 2^n, n <= k && n >= 0, n+1, n > 0, a[n-1, k] + a[n-k-1, k], True, a[n+1+k, k] - a[n+k, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-Fran├žois Alcover, Dec 17 2013, translated from Maple *)

CROSSREFS

Cf. column k=0: A000079, k=1: A000045(n+2), k=2: A000930(n+2), A068921, A078012(n+5), k=3: A003269(n+4), A017898(n+7), k=4: A003520(n+4), A017899(n+9), k=5: A005708(n+5), A017900(n+11), k=6: A005709(n+6), A017901(n+13), k=7: A005710(n+7), A017902(n+15), k=8: A005711(n+7), A017903(n+17), k=9: A017904(n+19), k=10: A017905(n+21), k=11: A017906(n+23), k=12: A017907(n+25), k=13: A017908(n+27), k=14: A017909(n+29).

Diagonal: A000027.

See also A143291.

Sequence in context: A081532 A174843 A253572 * A243851 A168266 A059250

Adjacent sequences:  A141536 A141537 A141538 * A141540 A141541 A141542

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 15 2008

STATUS

approved

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Last modified February 22 05:17 EST 2019. Contains 320385 sequences. (Running on oeis4.)