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

0,3

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.

Links

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

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Sequence in context: A102471 A243490 A094566 * A217349 A329574 A087278

Adjacent sequences: A190015 A190016 A190017 * A190019 A190020 A190021

Keyword

nonn

Author

Reinhard Zumkeller, May 04 2011

Status

approved