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A054451
Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).
10
1, 1, 4, 5, 12, 17, 33, 50, 88, 138, 232, 370, 609, 979, 1596, 2575, 4180, 6755, 10945, 17700, 28656, 46356, 75024, 121380, 196417, 317797, 514228, 832025, 1346268, 2178293, 3524577, 5702870, 9227464, 14930334, 24157816, 39088150, 63245985, 102334135
OFFSET
0,3
COMMENTS
Equals triangle A173284 * [1, 2, 3, ...]. - Gary W. Adamson, Mar 03 2010
FORMULA
a(n) = A054450(n+2, 2).
G.f.: Fib(x)/(1-x^2)^2, with Fib(x)=1/(1-x-x^2) = g.f. A000045 (Fibonacci numbers without 0).
a(2*k) = A027941(k)= F(2*k+3)-1; a(2*k+1)= F(2*(k+2))-(k+2)= A054452(k), k >= 0.
a(n-2) = Fibonacci(n+1) - binomial(n-floor(n/2), floor(n/2)), or a(n-2) = Sum_{i=0..floor(n/2)-1} binomial(n-i, i). - Jon Perry, Mar 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, k). - Paul Barry, Oct 23 2004
MAPLE
BB:=1/(1-k^2)^2/(1-k-k^2): seq(coeff(series(BB, k, n+1), k, n), n=0..50); # Zerinvary Lajos, May 16 2007
MATHEMATICA
LinearRecurrence[{1, 3, -2, -3, 1, 1}, {1, 1, 4, 5, 12, 17}, 40] (* Harvey P. Dale, Oct 06 2024 *)
PROG
(PARI) Vec(-1/((x-1)^2*(x+1)^2*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Jun 14 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Apr 27 2000
EXTENSIONS
More terms from James A. Sellers, Apr 28 2000
STATUS
approved