A153281 - OEIS

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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 (list; table; graph; refs; listen; history; 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 consective integers and there are 2^(n+1-k) subsets of {k+2,...,n+2}. [From Dennis P. Walsh, Dec 21 2011]
	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. [From 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}.` `[From Dennis Walsh, Dec 21 2011]`
	MAPLE	`>with(combinat, fibonacci):` `>seq(seq(2^(n+1-k)*fibonacci(k), k=1..(n+1)), n=0..10);`
	CROSSREFS	`Cf. A130321, A127647, A008466` `Sequence in context: A094571 A104733 A201703 * A130584 A078458 A033317` `Adjacent sequences: A153278 A153279 A153280 * A153282 A153283 A153284`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008`

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Last modified February 15 23:53 EST 2012. Contains 205860 sequences.