cp's OEIS Frontend

A133257 The number of edges on a piece of paper that has been folded n times (see comments for more precise definition).

%I A133257 #17 May 06 2024 12:38:39
%S A133257 4,7,11,17,25,37,53,77,109,157,221,317,445,637,893,1277,1789,2557,
%T A133257 3581,5117,7165,10237,14333,20477,28669,40957,57341,81917,114685,
%U A133257 163837,229373,327677,458749,655357,917501,1310717,1835005,2621437,3670013,5242877,7340029
%N A133257 The number of edges on a piece of paper that has been folded n times (see comments for more precise definition).
%C A133257 The angle each (straight) fold line makes with the long dimension of the original rectangle alternates in degrees as 90, 0, 90, 0, and so on. Each fold doubles the number of rectangles and halves the area of each rectangle.
%C A133257 The first five or six terms in the sequence can be verified experimentally with a standard piece of paper.
%H A133257 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).
%F A133257 a(n) = 2*a(n-2) + 3 for n >= 2.
%F A133257 a(n) = 2^((n-2)/2)*((7+5*sqrt(2))+(7-5*sqrt(2))*(-1)^n)-3.
%F A133257 a(n) = a(n-1)+2*a(n-2)-2*a(n-3). G.f.: -(4*x^2-3*x-4) / ((x-1)*(2*x^2-1)). - _Colin Barker_, Jul 07 2014
%e A133257 When n = 0, the piece of paper hasn't been folded yet and has 4 edges. Thus a(0) = 4.
%e A133257 When n = 1, we have folded the piece of paper once. The fold splits 2 of the original edges in half, resulting in 6 edges, and it creates one new edge at the fold itself, for 7 edges in total. Thus a(1) = 7.
%o A133257 (PARI) Vec(-(4*x^2-3*x-4)/((x-1)*(2*x^2-1)) + O(x^100)) \\ _Colin Barker_, Jul 07 2014
%Y A133257 Cf. A014577, A014707.
%K A133257 nonn,easy
%O A133257 0,1
%A A133257 Harold M. Frost, III (halfrost(AT)charter.net), Dec 19 2007
%E A133257 Edited, extended, and formulas by _Nathaniel Johnston_, Nov 11 2012