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%I M1594 #85 Jan 13 2023 18:08:00
%S 1,2,6,13,24,40,62,91,128,174,230,297,376,468,574,695,832,986,1158,
%T 1349,1560,1792,2046,2323,2624,2950,3302,3681,4088,4524,4990,5487,
%U 6016,6578,7174,7805,8472,9176,9918,10699,11520,12382,13286,14233,15224
%N 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).
%C Both the bagel and the torus are solid (apart from the hole in the middle, of course)! - _N. J. A. Sloane_, Oct 03 2012
%D 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.)
%D Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, pp. 373-374 and Plate 27.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A003600/b003600.txt">Table of n, a(n) for n = 0..1000</a>
%H George Hart, <a href="http://momath.org/home/math-monday-slice-a-bagel-into-13-pieces-with-three-cuts/">Slice a Bagel into 13 Pieces with Three Cuts</a>
%H Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H Clifford A. Pickover, <a href="/A003600/a003600.png">Illustration of a(3)=13</a> [Plate 27 from Computers and the Imagination, used with permission]
%H N. J. A. Sloane, <a href="/A003600/a003600_1.png">Illustration for a(2)=6 and a(3)=13</a> [Based on part of Fig. 62 in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions, colored and annotated]
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusCutting.html">Torus Cutting.</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = binomial(n+2, n-1) + binomial(n, n-1).
%F 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
%F Binomial transform of [1, 1, 3, 0, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Nov 08 2007
%F G.f.: (1 - 2*x + 4*x^2 - 3*x^3 + x^4) / (1 - x)^4. - _Colin Barker_, Jun 28 2012
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 29 2012
%F a(n) = A108561(n+4,3) for n > 0. - _Reinhard Zumkeller_, Jun 10 2005
%F a(n) = A000292(n+1) - A000124(n) for n > 0. - _Torlach Rush_, Aug 04 2018
%F 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
%t 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 *)
%t LinearRecurrence[{4,-6,4,-1},{1,2,6,13,24},50] (* _Harvey P. Dale_, Oct 22 2016 *)
%o (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
%o (PARI) a(n)=if(n,n*(n^2+3*n+8)/6,1) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000124 (slicing a pancake), A000125 (a cake).
%Y Cf. A004148.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Mira Bernstein_
%E More terms from _James A. Sellers_, Aug 22 2000
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