A065071 Minimum number of identical bricks of length 1 which, when stacked without mortar in the naive way, form a stack of length >=n.

1, 5, 32, 228, 1675, 12368, 91381, 675215, 4989192, 36865413, 272400601, 2012783316, 14872568832, 109894245430, 812014744423, 6000022499694, 44334502845081, 327590128640501, 2420581837980562, 17885814992891027

Offset: 1

Author

John W. Layman, Nov 08 2001

Keywords

• nonn
• easy

Comments

Note that one can do "better" in terms of projections if one groups the bricks asymmetrically into lozenges with holes. See the Ainsley and Drummond references. Ainsley considers only the case of four bricks, but achieves an overhang of (15 - 4*sqrt(2))/8, compared with 25/24 for the harmonic pile. - D. G. Rogers, Aug 31 2005

Lim_{n -> inf} a(n)/a(n-1) = exp(2). - Robert G. Wilson v, Jan 26 2017

Examples

			Obviously a(1)=1. If the center of gravity of one brick is placed at the end of a second brick, the length of the stack of 2 bricks is 1.5. If the c.g. of that stack is placed at the end of a third brick, the length of the stack is 1.75. Continuing, we get a stack of length 1.916666... for 4 bricks and a stack of length 2.0416666... for 5 bricks. Thus a(2)=5.
		

References

• N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000
• S. Ainley, Finely Balanced, Math. Gaz., 63 (1979), 272.
• R. Dickau, Harmonic numbers and the book-stacking problem
• J. E. Drummond, On stacking bricks to achieve a large overhang, Math. Gaz., 65 (1981), 40-42.
• Eric Weisstein's World of Mathematics, Book Stacking Problem

Crossrefs

Cf. harmonic numbers H(n) = A001008/A002805, A002387, A004080.

Programs

Mathematica

A002387[n_] := Floor[ Exp[n - EulerGamma] + 1/2]; a[n_] := A002387[2n - 2] + 1; a[1] = 1; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Dec 13 2011, after Charles R Greathouse IV *)
f[n_] := k /. FindRoot[HarmonicNumber[k -1] == 2n, {k, Exp[ 2n]}, WorkingPrecision -> 100] // Ceiling; Array[f, 21, 0] (* Robert G. Wilson v, Jan 26 2017 after Jean-François Alcover in A014537 *) (* note that the index is off by one *)

Formula

a(n) = A002387(2n) + 1 = A014537(n) + 1.

Extensions

More terms from Vladeta Jovovic, Nov 14 2001