A153281 Triangle read by rows, A130321 * A127647. Also, number of subsets of [n+2] with consecutive integers that start at k.

1, 2, 1, 4, 2, 2, 8, 4, 4, 3, 16, 8, 8, 6, 5, 32, 16, 16, 12, 10, 8, 64, 32, 32, 24, 20, 16, 13, 128, 64, 64, 48, 40, 32, 26, 21, 256, 128, 128, 96, 80, 64, 52, 42, 34, 512, 256, 256, 192, 160, 128, 104, 84, 68, 55

Offset

0,2

Comments

Row sums = A008466(k-2): (1, 3, 8, 19, 43, 94, ...).

T(n,k) is the number of subsets of {1,...,n+2} that contain consecutive integers and that have k as the first integer in the first consecutive string. (See the example below.) Hence rows sums of T(n,k) give the number of subsets of {1,...,n+2} that contain consecutive integers. Also, T(n,k) = F(k)*2^(n+1-k), where F(k) is the k-th Fibonacci number, since there are F(k) subsets of {1,...,k-2} that contain no consecutive integers and there are 2^(n+1-k) subsets of {k+2,...,n+2}. [Dennis P. Walsh, Dec 21 2011]

Links

Table of n, a(n) for n=0..54.

Formula

Triangle read by rows, A130321 * A127647. A130321 = an infinite lower triangular matrix with powers of 2: (A000079) in every column: (1, 2, 4, 8, ...).

A127647 = an infinite lower triangular matrix with the Fibonacci numbers, A000045 as the main diagonal and the rest zeros.

T(n,k)=2^(n+1-k)*F(k) where F(k) is the k-th Fibonacci number. [_Dennis Walsh_, Dec 21 2011]

Example

First few rows of the triangle:
    1;
    2,   1;
    4,   2,   2;
    8,   4,   4,   3;
   16,   8,   8,   6,   5;
   32,  16,  16,  12,  10,   8;
   64,  32,  32,  24,  20,  16,  13;
  128,  64,  64,  48,  40,  32,  26,  21;
  256, 128, 128,  96,  80,  64,  52,  42,  34;
  512, 256, 256, 192, 160, 128, 104,  84,  68,  55;
  ...
Row 4 = (16, 8, 8, 6, 5) = termwise products of (16, 8, 4, 2, 1) and (1, 1, 2, 3, 5).
For n=5 and k=3, T(5,3)=16 since there are 16 subsets of {1,2,3,4,5,6,7} containing consecutive integers with 3 as the first integer in the first consecutive string, namely,
  {1,3,4}, {1,3,4,5}, {1,3,4,6}, {1,3,4,7}, {1,3,4,5,6}, {1,3,4,5,7}, {1,3,4,6,7}, {1,3,4,5,6,7}, {3,4}, {3,4,5}, {3,4,6}, {3,4,7}, {3,4,5,6}, {3,4,5,7}, {3,4,6,7}, and {3,4,5,6,7}. [Dennis P. Walsh, Dec 21 2011]

Maple

with(combinat, fibonacci):
seq(seq(2^(n+1-k)*fibonacci(k), k=1..(n+1)), n=0..10);

Mathematica

Table[2^(n+1-k) Fibonacci[k], {n, 0, 10}, {k, n+1}]//Flatten (* Harvey P. Dale, Apr 26 2020 *)

Crossrefs

Cf. A008466, A127647, A130321.

Sequence in context: A104733 A348676 A201703 * A338654 A130584 A339046

Adjacent sequences: A153278 A153279 A153280 * A153282 A153283 A153284

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 23 2008

Status

approved