%I #27 Jul 02 2025 16:01:59

%S 1,1,4,5,12,17,33,50,88,138,232,370,609,979,1596,2575,4180,6755,10945,

%T 17700,28656,46356,75024,121380,196417,317797,514228,832025,1346268,

%U 2178293,3524577,5702870,9227464,14930334,24157816,39088150,63245985,102334135

%N Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

%C Equals triangle A173284 * [1, 2, 3, ...]. - _Gary W. Adamson_, Mar 03 2010

%H Colin Barker, <a href="/A054451/b054451.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-3,1,1).

%F a(n) = A054450(n+2, 2).

%F G.f.: Fib(x)/(1-x^2)^2, with Fib(x)=1/(1-x-x^2) = g.f. A000045 (Fibonacci numbers without 0).

%F a(2*k) = A027941(k)= F(2*k+3)-1; a(2*k+1)= F(2*(k+2))-(k+2)= A054452(k), k >= 0.

%F 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

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, k). - _Paul Barry_, Oct 23 2004

%p 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

%t LinearRecurrence[{1,3,-2,-3,1,1},{1,1,4,5,12,17},40] (* _Harvey P. Dale_, Oct 06 2024 *)

%o (PARI) Vec(-1/((x-1)^2*(x+1)^2*(x^2+x-1)) + O(x^100)) \\ _Colin Barker_, Jun 14 2015

%Y Cf. A054450, A049310, A000045, A052952.

%Y Cf. A007382.

%Y Cf. A173284.

%K easy,nonn

%O 0,3

%A _Wolfdieter Lang_, Apr 27 2000

%E More terms from _James Sellers_, Apr 28 2000