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A000125 Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1.
(Formerly M1100 N0419)
58
1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Note that a(n) = a(n-1) + A000124(n-1). This has the following geometrical interpretation: Define a number of planes in space to be in general arrangement when (1) no two planes are parallel, (2) there are no two parallel intersection lines, (3) there is no point common to four or more planes. Suppose there are already n-1 planes in general arrangement, thus defining the maximal number of regions in space obtainable by n-1 planes and now one more plane is added in general arrangement. Then it will cut each of the n-1 planes and acquire intersection lines which are in general arrangement. (See the comments on A000124 for general arrangement with lines.) These lines on the new plane define the maximal number of regions in 2-space definable by n-1 straight lines, hence this is A000124(n-1). Each of this regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n)=a(n-1)+A000124(n-1). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

More generally, we have: A000027(n) = binomial(n,0) + binomial(n,1) (the natural numbers), A000124(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) (the Lazy Caterer's sequence), a(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) + binomial(n,3) (Cake Numbers). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number of 3-subsets of X which have no exactly one element in common with Y. - Milan Janjic, Dec 28 2007

a(n) is the number of compositions (ordered partitions) of n+1 into four or fewer parts or equivalently the sum of the first four terms in the n-th row of Pascal's triangle. [From Geoffrey Critzer, Jan 23 2009]

{a(k): 0 <= k < 4} = divisors of 8. [From Reinhard Zumkeller, Jun 17 2009]

a(n) is also the maximum number of different values obtained by summing n consecutive positive integers with all possible 2^n sign combinations. This maximum is first reached when summing the interval [n, 2n-1]. - Olivier Gérard, Mar 22 2010

a(n) contains only 5 perfect squares > 1: 4, 64, 576, 676000, and 75203584. The incidences of > 0 are given by A047694. - Frank M Jackson, Mar 15 2013

REFERENCES

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_3.

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 27.

A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014, http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.

H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.

D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.

S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.

D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. H. Stickels, Mindstretching Puzzles. Sterling, NY, 1994 p. 85.

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.

A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First published: San Francisco: Holden-Day, Inc., 1964)

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Eric Weisstein's World of Mathematics, Cylinder Cutting

Eric Weisstein's World of Mathematics, Space Division by Planes

R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller, Jun 17 2009]

Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = (n+1)*(n^2-n+6)/6 = (n^3 + 5*n + 6) / 6.

G.f.: (1-2*x+2x^2)/(1-x)^4; - [Simon Plouffe in his 1992 dissertation.]

E.g.f.: (1+x+x^2/2+x^3/6)*exp(x).

a(n) = binomial(n,3)+binomial(n,2)+binomial(n,1)+binomial(n,0). [Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006]

Paraphrasing the previous comment: the sequence is the binomial transform of [1,1,1,1,0,0,0,...]. - Gary W. Adamson, Oct 23 2007

EXAMPLE

a(4)=15 because there are 15 compositions of 5 into four or fewer parts. a(6)=42 because the sum of the first four terms in the 6th row of Pascal's triangle is 1+6+15+20=42 [From Geoffrey Critzer, Jan 23 2009]

For n=5, (1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 35) and their opposite are the 26 different sums obtained by summing 5,6,7,8,9 with any sign combination. - Olivier Gérard, Mar 22 2010

MAPLE

A000125 := n->(n+1)*(n^2-n+6)/6;

MATHEMATICA

a=1; lst={a}; k=1; e=0; Do[a+=k; AppendTo[lst, a]; e++; k+=e, {n, 0, 6!}]; lst [From Vladimir Joseph Stephan Orlovsky, Dec 15 2008]

Table[(n^3+5n+6)/6, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 2, 4, 8}, 50] (* Harvey P. Dale, Jan 19 2013 *)

PROG

(PARI) a(n)=(n^2+5)*n/6+1 \\ Charles R Greathouse IV, Jun 15 2011

CROSSREFS

Cf. A000124, A003600. Bisections give A100503, A100504.

Row sums of A077028.

A005408, A000124, A016813, A086514, A058331, A002522, A161701 - A161705, A000127, A161706 - A161708, A080856, A161710 - A161713, A161715, A006261. [Reinhard Zumkeller, Jun 17 2009]

Cf. A063865 [Olivier Gérard, Mar 22 2010]

Cf. A051601. [Bruno Berselli, Aug 02 2013]

Sequence in context: A159243 A089140 A204555 * A129961 A133551 A114226

Adjacent sequences:  A000122 A000123 A000124 * A000126 A000127 A000128

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Feb 22 2000

STATUS

approved

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Last modified August 27 22:12 EDT 2014. Contains 246149 sequences.