A076024 a(n) = (2^n + 4)*(2^n - 1)/6.

0, 1, 4, 14, 50, 186, 714, 2794, 11050, 43946, 175274, 700074, 2798250, 11188906, 44747434, 178973354, 715860650, 2863377066, 11453377194, 45813246634, 183252462250, 733008800426, 2932033104554, 11728128223914, 46912504507050, 187650001250986, 750599971449514

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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.

Developed with J. R. Gallivan.

Binomial transform of A007051, with leading zero.

Second binomial transform of A078008(n-1) + 0^n/2. - Paul Barry, Apr 27 2004

Examples

			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]
		

References

• 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)

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Folding
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A007582.

Programs

GAP

List([0..30], n-> (2^n +4)*(2^n -1)/6) # G. C. Greubel, May 04 2019

Magma

[(2^n +4)*(2^n -1)/6 : n in [0..30]]; // Wesley Ivan Hurt, Jun 12 2014

Maple

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

Mathematica

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

PARI

a(n) = 1<<(2*n-1)\3 + 1<<(n-1); \\ Kevin Ryde, Nov 26 2022 [replacing previous incorrect code]

Sage

[(2^n +4)*(2^n -1)/6 for n in (0..30)] # G. C. Greubel, May 04 2019

Formula

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

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

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

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

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

a(n) = Sum_{i < n} a(i) + A073724(n-1). - Ivan N. Ianakiev, Jun 12 2014