A110428 a(1) = 1 and a(2) = 2. Subsequent terms are generated like this: if a(m) is the last term available -- say a(2) -- then a(m+1) = a(m) * a(m-1), a(m+2) = a(m) * a(m-1) * a(m-2), ..., a(2*m-1) = a(m) * a(m-1) * a(m-2) * ... * a(2) * a(1), a(2*m) = a(2*m-1) * a(2*m-2), and so on.

1, 2, 2, 4, 4, 16, 32, 64, 64, 4096, 131072, 2097152, 8388608, 33554432, 67108864, 134217728, 134217728, 18014398509481984, 1208925819614629174706176, 40564819207303340847894502572032, 340282366920938463463374607431768211456

Offset: 1

Author

Amarnath Murthy, Aug 01 2005

Keywords

• easy
• nonn

Comments

By choosing appropriate values for a(1) and a(2), many such sequences can be generated.

Examples

			a(3) = a(2)*a(1) = 2. [Now a(3) is the last term available.]
a(4) = a(3)*a(2) = 4.
a(5) = a(3)*a(2)*a(1) = 4. [Now a(5) is the last term available.]
a(6) = a(5)*a(4) = 16.
a(7) = a(5)*a(4)*a(3) = 32.
a(8) = a(5)*a(4)*a(3)*a(2) = 64.
a(9) = a(5)*a(4)*a(3)*a(2)*a(1) = 64. [Now a(9) is the last term available.]
a(10) = a(9)*a(8) = 4096.
a(11) = a(9)*a(8)*a(7) = 131072.
...
a(17) = a(9)*a(8)*...*a(1) = 134217728. [Now a(17) is the last term available.]
a(18) = a(17)*a(16) = 18014398509481984.
[Example extended by _Petros Hadjicostas_, Nov 13 2019]
		

Crossrefs

Cf. A000051 (index of "available" terms as described above), A050049 (an additive version of this sequence), A329474 (log[2] of this sequence).

Programs

Maple

a := proc(n) option remember;
`if`(n < 3, [1, 2][n], a(n - 1) * a(2^ceil(log[2](n - 1)) + 2 - n));
end proc;
seq(a(n), n = 1..25); # Petros Hadjicostas, Nov 13 2019

Formula

From Petros Hadjicostas, Nov 13 2019: (Start)

a(n) = a(n-1) * a(m) for n >= 3, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 2.

a(A000051(n)) = a(2^n + 1) = a(2^n) for n >= 1.

a(A000051(n) + 1) = a(2^n + 2) = a(2^n + 1) * a(2^n) = a(2^n)^2 for n >= 1.

log[2](a(n)) = A329474(n) for n >= 1. (End)

Extensions

More terms from Joshua Zucker, May 10 2006