A089645 Penny Flipping, or Flipping Coins: Given a stack of n coins, flip the top coin, then the stack of the top two coins, then the stack of the top three etc... starting again with the top coin after flipping all n coins. A flip of m coins reverses their order and inverts their state. This is the number of flips required to restore the stack to its original configuration.

2, 3, 9, 11, 24, 35, 28, 31, 80, 60, 121, 119, 116, 195, 75, 79, 204, 323, 228, 199, 146, 264, 529, 504, 200, 675, 540, 251, 840, 899, 186, 191, 1088, 748, 1225, 324, 740, 1140, 1521, 1079, 1680, 336, 1204, 484, 540, 460, 1692, 1151, 734, 2499

Offset

1,1

Comments

Here "original configuration" seems to mean each coin in original orientation and either original or reverse order; for the original order and either original or reverse orientation n*A002326(n) flips required, while for both original order and original orientation n*A003558(n) required. - Henry Bottomley, Jan 19 2007

Birtwistle (1973) attributes the problem to Iain Bride and John Gilder of UMIST (University of Manchester Institute of Science and Technology).

References

G. M. Birtwistle, Simula Begin, Auerbach Publishers, Philadelphia, 1973 [Uses this problem to illustrate the power of the Simula language]

Popular Computing (Calabasas, CA), Penny Flipping, Vol. 3 (No. 23, Feb 1973), pages PC23-10 to PC23-13) and Vol. 3 (No. 29, Aug 1975), pages PC29-6 to PC29-8. Gives first 32 terms.

Links

Table of n, a(n) for n=1..50.

B. B. Newman, The Flippin' Coins Problem, Mathematics Magazine, Vol. 54 (1981), pp. 51-59.

Formula

For n>1, if A002326(n)=A003558(n) then a(n)=n*A002326(n), otherwise a(n)=n*A002326(n)-1. - Henry Bottomley, Jan 19 2007

Example

For 3 coins (starting with HHH) the flips move the stack through the sequence: HHH -1-> THH -2-> THH -3-> TTH -1-> HTH -2-> HTH -3-> THT -1-> HHT -2-> TTT -3-> HHH. (-n-> indicates n coins are flipped)

Mathematica

b[n_] := MultiplicativeOrder[2, 2n+1];
c[n_] := MultiplicativeOrder[2, 2n+1, {-1, 1}];
a[1] = 2; a[n_] := If[b[n] == c[n], n*b[n], n*b[n]/2 - 1];
Array[a, 50] (* Jean-François Alcover, Oct 29 2017, after Henry Bottomley *)

Crossrefs

Cf. A002326, A003558, A056951.

Sequence in context: A057292 A098016 A372073 * A214259 A287680 A377882

Adjacent sequences: A089642 A089643 A089644 * A089646 A089647 A089648

Keyword

easy,nonn,nice

Author

Richard Forster (gbrl01(AT)yahoo.co.uk), Jan 02 2004

Status

approved