OFFSET
0,3
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
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).
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
EXAMPLE
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]
MAPLE
MATHEMATICA
Table[(2^n+4)*(2^n-1)/6, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 12 2014 *)
PROG
(PARI) a(n) = 1<<(2*n-1)\3 + 1<<(n-1); \\ Kevin Ryde, Nov 26 2022 [replacing previous incorrect code]
(Magma) [(2^n +4)*(2^n -1)/6 : n in [0..30]]; // Wesley Ivan Hurt, Jun 12 2014
(Sage) [(2^n +4)*(2^n -1)/6 for n in (0..30)] # G. C. Greubel, May 04 2019
(GAP) List([0..30], n-> (2^n +4)*(2^n -1)/6) # G. C. Greubel, May 04 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Britney C. Gallivan (ogallivan(AT)verizon.net), Sep 30 2002
STATUS
approved