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A190018 Union of A000045, A007598, and A059929.

%I A190018 #18 Jan 11 2018 03:16:55
%S A190018 0,1,2,3,4,5,8,9,10,13,21,24,25,34,55,64,65,89,144,168,169,233,377,
%T A190018 441,442,610,987,1155,1156,1597,2584,3025,3026,4181,6765,7920,7921,
%U A190018 10946,17711,20736,20737,28657,46368,54288,54289,75025,121393,142129,142130
%N A190018 Union of A000045, A007598, and A059929.
%C A190018 Each term is F(k) or F(k)^2 or F(k-1)*F(k+1) for appropriate k, F=A000045, the Fibonacci numbers.
%H A190018 Reinhard Zumkeller, <a href="/A190018/b190018.txt">Table of n, a(n) for n = 0..1000</a>
%F A190018 G.f.: -x*(x^16+2*x^15+4*x^14 +5*x^13+3*x^12+x^11 -4*x^10-7*x^9-10*x^8 -12*x^7-14*x^6-14*x^5 -12*x^4-10*x^3-6*x^2-3*x-1) / ((x+1)*(x^2+1)*(x^4+1)*(x^4+x^2-1)*(x^4-x^2-1)). - _Alois P. Heinz_, May 05 2011
%e A190018 a(10) = F(8) = 21;
%e A190018 a(11) = F(4) * F(6) = 3 * 8 = 24;
%e A190018 a(12) = F(5)^2 = 5^2 = 25;
%e A190018 a(13) = F(9) = 34;
%e A190018 a(14) = F(10) = 55;
%e A190018 a(15) = F(6)^2 = 8^2 = 64;
%e A190018 a(16) = F(5) * F(7) = 5 * 13 = 65;
%e A190018 a(17) = F(11) = 89;
%e A190018 a(18) = F(12) = 144;
%e A190018 a(19) = F(6) * F(8) = 8 * 21 = 168;
%e A190018 a(20) = F(7)^2 = 13^2 = 169.
%p A190018 a:= n-> `if`(n<6, n, (Matrix(15, (i, j)-> `if`(j=i+1, 1, `if`(i=15, [-1$4,2$8,-1$3][j], 0)))^n. <<0, 1, 1, 0, 0, [1$4][], 2, 2, 3, 3, 4, 5>>)[10, 1]): seq(a(n), n=0..50);  # _Alois P. Heinz_, May 04 2011
%t A190018 CoefficientList[Series[-x*(x^16+2*x^15+4*x^14 +5*x^13+3*x^12+x^11 -4*x^10 -7*x^9-10*x^8 -12*x^7-14*x^6-14*x^5 -12*x^4-10*x^3-6*x^2-3*x-1)/((x+1)*(x^2+1)*(x^4+1)*(x^4+x^2-1)*(x^4-x^2-1)), {x, 0, 50}], x] (* _G. C. Greubel_, Jan 11 2018 *)
%o A190018 (Haskell)
%o A190018 a190018 n = a190018_list !! n
%o A190018 a190018_list = 0 : drop 2 (merge (merge fibs $
%o A190018     map (^ 2) fibs) $ zipWith (*) fibs (drop 2 fibs))
%o A190018     where fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
%o A190018           merge xs'@(x:xs) ys'@(y:ys)
%o A190018              | x < y     = x : merge xs ys'
%o A190018              | x == y    = x : merge xs ys
%o A190018              | otherwise = y : merge xs' ys
%o A190018 (PARI) x='x+O('x^50); concat([0], Vec(-x*(x^16+2*x^15+4*x^14 +5*x^13 +3*x^12+x^11 -4*x^10-7*x^9-10*x^8 -12*x^7-14*x^6-14*x^5 -12*x^4-10*x^3 -6*x^2-3*x-1)/((x+1)*(x^2+1)*(x^4+1)*(x^4+x^2-1)*(x^4-x^2-1)))) \\ _G. C. Greubel_, Jan 11 2018
%K A190018 nonn
%O A190018 0,3
%A A190018 _Reinhard Zumkeller_, May 04 2011