A190018 Union of A000045, A007598, and A059929.

0, 1, 2, 3, 4, 5, 8, 9, 10, 13, 21, 24, 25, 34, 55, 64, 65, 89, 144, 168, 169, 233, 377, 441, 442, 610, 987, 1155, 1156, 1597, 2584, 3025, 3026, 4181, 6765, 7920, 7921, 10946, 17711, 20736, 20737, 28657, 46368, 54288, 54289, 75025, 121393, 142129, 142130

Offset: 0

Author

Reinhard Zumkeller, May 04 2011

Keywords

• nonn

Comments

Each term is F(k) or F(k)^2 or F(k-1)*F(k+1) for appropriate k, F=A000045, the Fibonacci numbers.

Examples

			a(10) = F(8) = 21;
a(11) = F(4) * F(6) = 3 * 8 = 24;
a(12) = F(5)^2 = 5^2 = 25;
a(13) = F(9) = 34;
a(14) = F(10) = 55;
a(15) = F(6)^2 = 8^2 = 64;
a(16) = F(5) * F(7) = 5 * 13 = 65;
a(17) = F(11) = 89;
a(18) = F(12) = 144;
a(19) = F(6) * F(8) = 8 * 21 = 168;
a(20) = F(7)^2 = 13^2 = 169.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Programs

Haskell

a190018 n = a190018_list !! n
a190018_list = 0 : drop 2 (merge (merge fibs $
    map (^ 2) fibs) $ zipWith (*) fibs (drop 2 fibs))
    where fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
          merge xs'@(x:xs) ys'@(y:ys)
             | x < y     = x : merge xs ys'
             | x == y    = x : merge xs ys
             | otherwise = y : merge xs' ys

Maple

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

Mathematica

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 *)

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

Formula

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