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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).
(Formerly M1594)
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Both the bagel and the torus are solid (apart from the hole in the middle, of course)! - N. J. A. Sloane, Oct 03 2012
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
Kyu-Hwan Lee, 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.
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) = A000292(n+1) - A000124(n) for n > 0. - Torlach Rush, Aug 04 2018
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
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 *)
PROG
(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
(PARI) a(n)=if(n, n*(n^2+3*n+8)/6, 1) \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Cf. A000124 (slicing a pancake), A000125 (a cake).
Cf. A004148.
Sequence in context: A184533 A338991 A178532 * A283551 A362438 A000135
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Aug 22 2000
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)