A003600 Maximal number of pieces obtained by slicing a torus (or a bagel) with n cuts: (n^3 + 3*n^2 + 8*n)/6 (n > 0).
1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, 297, 376, 468, 574, 695, 832, 986, 1158, 1349, 1560, 1792, 2046, 2323, 2624, 2950, 3302, 3681, 4088, 4524, 4990, 5487, 6016, 6578, 7174, 7805, 8472, 9176, 9918, 10699, 11520, 12382, 13286, 14233, 15224
Offset: 0
References
- M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions. Simon and Schuster, NY, 1961. See Chapter 13. (See pages 113-116 in the English edition published by Pelican Books in 1966.)
- Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, pp. 373-374 and Plate 27.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- George Hart, Slice a Bagel into 13 Pieces with Three Cuts
- Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
- Clifford A. Pickover, Illustration of a(3)=13 [Plate 27 from Computers and the Imagination, used with permission]
- N. J. A. Sloane, Illustration for a(2)=6 and a(3)=13 [Based on part of Fig. 62 in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions, colored and annotated]
- N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 1.
- Eric Weisstein's World of Mathematics, Torus Cutting.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
I:=[1, 2, 6, 13, 24]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 29 2012
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Mathematica
CoefficientList[Series[(1-2*x+4*x^2-3*x^3+x^4)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 29 2012 *) LinearRecurrence[{4,-6,4,-1},{1,2,6,13,24},50] (* Harvey P. Dale, Oct 22 2016 *) -
PARI
a(n)=if(n,n*(n^2+3*n+8)/6,1) \\ Charles R Greathouse IV, Oct 07 2015
Formula
a(n) = binomial(n+2, n-1) + binomial(n, n-1).
a(n) = coefficient of z^3 in the series expansion of G^n (n>0), where G=[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. of A004148 (secondary structures of RNA molecules). - Emeric Deutsch, Jan 11 2004
Binomial transform of [1, 1, 3, 0, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 08 2007
G.f.: (1 - 2*x + 4*x^2 - 3*x^3 + x^4) / (1 - x)^4. - Colin Barker, Jun 28 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 29 2012
a(n) = A108561(n+4,3) for n > 0. - Reinhard Zumkeller, Jun 10 2005
a(n) = A000125(n+1) - 2, as one can see by thinking of the donut hole as a slit in a cake, i.e. an (n+1)st cut in the cake that doesn't quite reach the edges of the cake and so leaves two pieces unseparated. - Glen Whitney, Mar 31 2019
E.g.f.: 1 + exp(x)*x*(12 + 6*x + x^2)/6. - Stefano Spezia, Apr 19 2025
Extensions
More terms from James Sellers, Aug 22 2000
Comments