A158618 Number of gates in Ladner-Fischer prefix circuit.

0, 1, 2, 4, 5, 7, 9, 12, 13, 15, 17, 20, 22, 25, 27, 31, 32, 34, 36, 39, 41, 44, 46, 50, 52, 55, 57, 61, 64, 67, 69, 74, 75, 77, 79, 82, 84, 87, 89, 93, 95, 98, 100, 104, 107, 110, 112, 117, 119, 122, 124, 128, 131, 134, 136, 141, 143, 147, 149, 153

Offset

1,3

References

D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 0. See exercise 7.1.2.36 and solution.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

R. E. Ladner and M. J. Fischer, Parallel Prefix Computation, JACM 27 (1980) 831-838.

Formula

With s = floor(n/2), r = ceiling(n/2) and a(1) = b(1) = 0,

recurrence relation is a(n) = s + b(r) + a(s), b(n) = 2s-1 + a(r).

If n = 2^m then a(n) = 4n+1 - Fibonacci(m+5).

Prog

(PARI)
b(n)={if(n<=1, 0, 2*(n\2) - 1 + a((n+1)\2))}
a(n)={if(n<=1, 0, n\2 + b((n+1)\2) + a(n\2))} \\ Andrew Howroyd, Mar 28 2020

Crossrefs

Sequence in context: A140205 A140206 A007818 * A000788 A053039 A286753

Adjacent sequences: A158615 A158616 A158617 * A158619 A158620 A158621

Keyword

nonn

Author

Frank Ruskey, Mar 22 2009

Extensions

Terms a(41) and beyond from Andrew Howroyd, Mar 28 2020

Status

approved