login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A153281 Triangle read by rows, A130321 * A127647. Also, number of subsets of [n+2] with consecutive integers that start at k. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A316477 A104733 A201703 * A338654 A130584 A339046

Adjacent sequences:  A153278 A153279 A153280 * A153282 A153283 A153284

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Dec 23 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 16 15:37 EDT 2021. Contains 345063 sequences. (Running on oeis4.)