A100702 Number of layers of dough separated by butter in successive foldings of croissant dough.

1, 3, 7, 19, 55, 163, 487, 1459, 4375, 13123, 39367, 118099, 354295, 1062883, 3188647, 9565939, 28697815, 86093443, 258280327, 774840979, 2324522935, 6973568803, 20920706407, 62762119219, 188286357655, 564859072963

Offset: 0

Author

Daniel Wolf (djwolf1(AT)axelero.hu), Dec 09 2004

Keywords

• easy
• nonn

Comments

At each trebling of layers following the first, two sets of layers, not separated from their neighbors by butter, are combined. Traditional patisserie stops at 55 layers, but forgetful chefs have been known to make additional folds to 163 layers.

This sequence also describes the number of moves of the k-th disk solving (non-optimally) the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (see the "CROSSREFS" in A183120). For other Magnetic Tower of Hanoi related sequences cf. A183111-A183125.

Same as A052919 except first term is 1, not 2. - Omar E. Pol, Feb 20 2011

References

• J. Child and M. Beck, Mastering the Art of French Cooking, Vol. 2

Links

• Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
• Index entries for linear recurrences with constant coefficients, signature (4, -3).

Crossrefs

Cf. A052919.

Programs

Mathematica

Join[{1}, LinearRecurrence[{4, -3}, {3, 7}, 25]] (* Jean-François Alcover, Jul 28 2018 *)

PARI

a(n)=([0,1; -3,4]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Jan 28 2018

Formula

For n > 1, a(n) = 3*a(n-1) - 2.

From R. J. Mathar, Jun 30 2009: (Start)

a(n) = 1 + 2*3^(n-1), n > 0.

a(n) = 4*a(n-1) - 3*a(n-2), n > 2.

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