

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
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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 113116 in the English edition published by Pelican Books in 1966.)
Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, pp. 373374 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
KyuHwan Lee, Sejin 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]
Eric Weisstein's World of Mathematics, Torus Cutting.
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).


FORMULA

a(n) = binomial(n+2, n1) + binomial(n, n1).
a(n) = coefficient of z^3 in the series expansion of G^n (n>0), where G=[1z+z^2sqrt(12zz^22z^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(n1)  6*a(n2) + 4*a(n3)  a(n4).  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[(12*x+4*x^23*x^3+x^4)/(1x)^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(n1)6*Self(n2)+4*Self(n3)Self(n4): 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: A011891 A184533 A178532 * A283551 A000135 A281865
Adjacent sequences: A003597 A003598 A003599 * A003601 A003602 A003603


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane, Mira Bernstein


EXTENSIONS

More terms from James A. Sellers, Aug 22 2000


STATUS

approved



