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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)
9
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

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, 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]

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, 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

MATHEMATICA

a=2; s=3; lst={a, s}; Do[a+=n; s+=a; AppendTo[lst, s], {n, 2, 6!, 1}]; lst-1 (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)

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: A011891 A184533 A178532 * A000135 A267698 A065220

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

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Last modified December 10 21:05 EST 2016. Contains 279011 sequences.