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a(n) = (2^n + 4)*(2^n - 1)/6.
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%I #37 Nov 27 2022 00:18:34

%S 0,1,4,14,50,186,714,2794,11050,43946,175274,700074,2798250,11188906,

%T 44747434,178973354,715860650,2863377066,11453377194,45813246634,

%U 183252462250,733008800426,2932033104554,11728128223914,46912504507050,187650001250986,750599971449514

%N a(n) = (2^n + 4)*(2^n - 1)/6.

%C Provides loss function for folding paper in half. It tells how much normalized paper has been lost with n folds. The sequence sets a limit on the number of times things of finite thickness can be folded in one direction.

%C Developed with J. R. Gallivan.

%C Binomial transform of A007051, with leading zero.

%C Second binomial transform of A078008(n-1) + 0^n/2. - _Paul Barry_, Apr 27 2004

%D Britney C. Gallivan, How to fold paper in half twelve times (an "impossible challenge" solved and explained), Historical Society of Pomona Valley, Pomona California, (2002)

%H Ivan Panchenko, <a href="/A076024/b076024.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Folding.html">Folding</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).

%F a(n) = Sum_{k <= n} A007582(k).

%F G.f.: x*(1-3*x)/((1-x)*(1-2*x)*(1-4*x)).

%F E.g.f.: (3*exp(2*x) + exp(4*x) - 4*exp(x))/6 = (exp(2*x)*(2*cosh(x) - sinh(x)) - 2)/3.

%F a(n) = Sum_{k=0..n} C(n, k)*(3^(k-1) + 1 - 4*0^k/3)/2.

%F a(n) = Sum_{k=0..n} C(n, k+1)*(3^k + 1).

%F a(n) = Sum_{i < n} a(i) + A073724(n-1). - _Ivan N. Ianakiev_, Jun 12 2014

%e a(12) = 2798250 means that for the 12th folding of paper in half that 2798250 times as much material has been lost to potential folding as was lost on the first fold. [corrected by _Rick L. Shepherd_, May 08 2003]

%p A076024:=n->(2^n + 4)*(2^n - 1)/6; seq(A076024(n), n=0..30); # _Wesley Ivan Hurt_, Jun 12 2014

%t Table[(2^n+4)*(2^n-1)/6, {n,0,30}] (* _Wesley Ivan Hurt_, Jun 12 2014 *)

%o (PARI) a(n) = 1<<(2*n-1)\3 + 1<<(n-1); \\ _Kevin Ryde_, Nov 26 2022 [replacing previous incorrect code]

%o (Magma) [(2^n +4)*(2^n -1)/6 : n in [0..30]]; // _Wesley Ivan Hurt_, Jun 12 2014

%o (Sage) [(2^n +4)*(2^n -1)/6 for n in (0..30)] # _G. C. Greubel_, May 04 2019

%o (GAP) List([0..30], n-> (2^n +4)*(2^n -1)/6) # _G. C. Greubel_, May 04 2019

%Y Cf. A007582.

%K easy,nonn

%O 0,3

%A Britney C. Gallivan (ogallivan(AT)verizon.net), Sep 30 2002