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proposed
approved
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Revision History for A000204
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.
(history; published version)
(history; published version)
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editing
proposed
COMMENTS
This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m - 1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n - 1 - (m - 1)*i, i - 1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m = 2: A000204, m = 3: A001609, m = 4: A014097, m = 5: A058368, m = 6: A058367, m = 7: A058366, m = 8: A058365, m = 9: A058364.
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proposed
editing
STATUS
editing
proposed
FORMULA
a(n) = 2*fibonacciFibonacci(n-1) + fibonacciFibonacci(n), n >= 1. - Zerinvary Lajos, Oct 05 2007
L(n) is the term (1, 1) in the 1 x X 2 matrix [2, -1].[1, 1; 1, 0]^n. - Alois P. Heinz, Jul 25 2008
a(n) = ((1 + sqrt5sqrt(5))^n - (1 - sqrt5sqrt(5))^n)/(2^n*sqrt5sqrt(5)) + ((1 + sqrt5sqrt(5))^(n - 1) - (1 - sqrt5sqrt(5))^(n - 1))/(2^(n - 2)*sqrt5sqrt(5)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009, Jan 14 2009
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approved
editing
LINKS
Sergio Falcon, <a href="httpshttp://www.saspublisherssaspublisher.com/articlewp-content/uploads/2014/255406/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
STATUS
proposed
approved
STATUS
editing
proposed
LINKS
Sergio Falcon, <a href="httphttps://saspublisherwww.saspublishers.com/wp-content/uploads/2014article/062554/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
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approved
editing
STATUS
reviewed
approved
STATUS
proposed
reviewed