cp's OEIS Frontend

A110837 Number of ways to fold a strip of n stamps taking account of order and direction of folds.

%I A110837 #23 Jan 10 2023 16:12:37
%S A110837 1,2,8,36,176,864,4304,21448,107168,535488,2677088,13383712,66916832,
%T A110837 334575552,1672869152,8364302864,41821471424,209107142784,
%U A110837 1045535499584,5227676426944,26138381063744,130691899964544,653459494468544,3267297445575296,16336487201109056
%N A110837 Number of ways to fold a strip of n stamps taking account of order and direction of folds.
%H A110837 Seiichi Manyama, <a href="/A110837/b110837.txt">Table of n, a(n) for n = 1..1000</a>
%H A110837 <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a>
%F A110837 a(n) = 2 * Sum_{0<k<n} max{a(k), a(n-k)} starting with a(1)=1.
%F A110837 a(n) ~ 0.054816154756...*5^n.
%e A110837 a(3) = 8 since with an initial strip of three stamps there are two possible folding positions for the initial fold, each of which could be folded up or down, so there are four possible initial folds, each leaving one possible folding position which can be folded up or down, making eight possible folding patterns.
%p A110837 a:= proc(n) option remember; `if`(n=1, 1,
%p A110837       2*add(max(a(k), a(n-k)), k=1..n-1))
%p A110837     end:
%p A110837 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jan 08 2023
%t A110837 a[n_] := a[n] = If[n==1, 1, 2*Sum[Max[a[k], a[n-k]], {k, 1, n-1}]];
%t A110837 Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Jan 10 2023, after _Alois P. Heinz_ *)
%Y A110837 Cf. A001011, A075496.
%K A110837 nonn
%O A110837 1,2
%A A110837 _Henry Bottomley_, Sep 16 2005