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A076024
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a(n) = (2^n + 4)*(2^n - 1)/6.
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4
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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)
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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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]
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MAPLE
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A076024:=n->(2^n + 4)*(2^n - 1)/6; seq(A076024(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014
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MATHEMATICA
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Table[(2^n+4)*(2^n-1)/6, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 12 2014 *)
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PROG
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(PARI) th(n)=if(n<1, y, th(n-1)*(th(n-1)+1)/2)
for(n=2, 30, print1(numerator(polcoeff(th(n), 2^n-3))", "))
(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
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CROSSREFS
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Cf. A007582.
Sequence in context: A087945 A051924 A272687 * A062807 A117421 A034743
Adjacent sequences: A076021 A076022 A076023 * A076025 A076026 A076027
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KEYWORD
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easy,nonn
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AUTHOR
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Britney C. Gallivan (ogallivan(AT)verizon.net), Sep 30 2002
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STATUS
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approved
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