A007902 Number of pebbling configurations with n pebbles.
1, 1, 2, 4, 9, 20, 46, 105, 243, 561, 1301, 3014, 6995, 16227, 37668, 87426, 202961, 471150, 1093819, 2539348, 5895408, 13686805, 31775756, 73771474, 171270732, 397628399, 923150354, 2143222823, 4975795414, 11552012449, 26819637103, 62265592589, 144558421877
Offset: 1
Examples
a(4) = 4: |_|_|_|_|_ |_|_|_|_|_ |_|_|_|_|_ |_|_|_|_|_ |Q|_|_|_|_ |_|_|_|_|_ |_|_|_|_|_ |_|_|_|_|_ |_|Q|_|_|_ |Q|Q|_|_|_ |_|Q|_|_|_ |_|_|_|_|_ |_|Q|_|_|_ |_|_|Q|_|_ |Q|_|Q|_|_ |Q|Q|Q|_|_ |_|Q|_|_|_ |_|Q|_|_|_ |_|_|Q|_|_ |_|_|_|Q|_ . - _Alois P. Heinz_, Dec 20 2013
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
- F. R. K. Chung, R. L. Graham, J. A. Morrison and A. M. Odlyzko, Pebbling a chessboard, Amer. Math. Monthly 102 (1995), pp. 113-123.
- A. Khodulev, Pebble spreading, Kvant, July 1982, pp. 28-31, 55.
- Charles Knessl, On the number of reachable configurations for the chessboard pebbling problem, Mathematical and Computer Modelling, Volume 47, Issues 1-2 (2008), 127-139.
- M. Kontsevich, Problem M715, Kvant, November 1981, p. 21.
- Zvezdelina Stankova and Brady Haran, Pebbling a Chessboard, Numberphile video (2013)
Crossrefs
Cf. A007901.
Programs
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Maple
G:= proc(k, m) option remember; `if`(k<1, 0, `if`(m=0, 2*G(k-1, 0) +G(k, 1) +`if`(k=2, 1, 0), `if`(m=1, G(k-3, 0) +2*G(k-2, 1) +G(k-1, 2) +G(k-4, 1), G(k-m-2, m-1) +2*G(k-m-1, m) +G(k-m, m+1)))) end: a:= n-> `if`(n=1, 1, G(n, 0)): seq(a(n), n=1..40); # Alois P. Heinz, Dec 20 2013 -
Mathematica
G[k_, m_] := G[k, m] = If[k<1, 0, If[m == 0, 2*G[k-1, 0]+G[k, 1]+If[k == 2, 1, 0], If[m == 1, G[k-3, 0]+2*G[k-2, 1]+G[k-1, 2]+G[k-4, 1], G[k-m-2, m-1]+2*G[k-m-1, m]+G[k-m, m+1]]]]; a[n_] := If[n == 1, 1, G[n, 0]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
Formula
a(n) ~ c * d^n, where d = 2.321642199494229709895447236309905876768690729938226667582430304..., c = 0.122687073421485997619475676632990508558955463577161642002764414... (Knessl, 2006). - Vaclav Kotesovec, Sep 06 2014