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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) 70

%I M1100 N0419 #260 Nov 19 2022 10:01:35

%S 1,2,4,8,15,26,42,64,93,130,176,232,299,378,470,576,697,834,988,1160,

%T 1351,1562,1794,2048,2325,2626,2952,3304,3683,4090,4526,4992,5489,

%U 6018,6580,7176,7807,8474,9178,9920,10701,11522,12384,13288,14235,15226

%N Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.

%C 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

%C (1) no two planes are parallel,

%C (2) there are no two parallel intersection lines,

%C (3) there is no point common to four or more planes.

%C 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

%C 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

%C 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 do not have exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007

%C 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. - _Geoffrey Critzer_, Jan 23 2009

%C {a(k): 0 <= k < 4} = divisors of 8. - _Reinhard Zumkeller_, Jun 17 2009

%C 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

%C a(n) contains only 5 perfect squares > 1: 4, 64, 576, 67600, and 75203584. The incidences of > 0 are given by A047694. - _Frank M Jackson_, Mar 15 2013

%C Given n tiles with two values - an A value and a B value - a player may pick either the A value or the B value. The particular tiles are [n, 0], [n-1, 1], ..., [2, n-2] and [1, n-1]. The sequence is the number of different final A:B counts. For example, with n=4, we can have final total [5, 3] = [4, _] + [_, 1] + [_, 2] + [1, _] = [_, 0] + [3, _] + [2, _] + [_, 3], so a(4) = 2^4 - 1 = 15. The largest and smallest final A+B counts are given by A077043 and A002620 respectively. - _Jon Perry_, Oct 24 2014

%C For n>=3, a(n) is also the number of maximal cliques in the (n+1)-triangular graph (the 4-triangular graph has a(3)=8 maximal cliques). - _Andrew Howroyd_, Jul 19 2017

%C a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*. Coincidentally, A000124 counts binary words of the form 0*1*0*. See Alexandersson and Nabawanda for proof. - _Per W. Alexandersson_, May 15 2021

%C For n > 0, let the n-dimensional cube, {0,1}^n be provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 3. Example: n = 4. Let x = (0,0,0,0) be in {0,1}^4.

%C d(x,y) = 0: y in {(0,0,0,0)}.

%C d(x,y) = 1: y in {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}.

%C d(x,y) = 2: y in {(1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1)}.

%C d(x,y) = 3: y in {(1,1,1,0), (1,1,0,1), (1,0,1,1), (0,1,1,1)}.

%C All these y are at a distance <= 3 from (0,0,0,0), so a(4) = 15. (See Peter C. Heinig's formula). - _Yosu Yurramendi_, Dec 14 2021

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

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

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

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

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

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

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

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

%D 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)

%H T. D. Noe, <a href="/A000125/b000125.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Alexandersson and O. Nabawanda, <a href="https://arxiv.org/abs/2104.04220">Peaks are preserved under run-sorting</a>, arXiv:2104.04220 [math.CO], 2021.

%H A. M. Baxter and L. K. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf">Ascent sequences avoiding pairs of patterns</a>, 2014.

%H M. L. Cornelius, <a href="/A006261/a006261_1.pdf">Variations on a geometric progression</a>, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)

%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.

%H R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>

%H Zachary Hoelscher, <a href="https://arxiv.org/abs/2102.07083">Semicomplete Arithmetic Sequences, Division of Hypercubes, and the Pell Constant</a>, arXiv:2102.07083 [math.NT], 2021. Mentions this sequence.

%H Marie Lejeune, <a href="https://hdl.handle.net/2268/259266">On the k-binomial equivalence of finite words and k-binomial complexity of infinite words</a>, Ph. D. Thesis, Université de Liège (Belgium, 2021).

%H D. A. Lind, <a href="http://www.fq.math.ca/Scanned/3-4/lind.pdf">On a class of nonlinear binomial sums</a>, Fib. Quart., 3 (1965), 292-298.

%H Svante Linusson, <a href="http://dx.doi.org/10.1007/s004930050055">The number of M-sequences and f-vectors</a>, Combinatorica, vol 19 no 2 (1999) 255-266.

%H R. J. Mathar, <a href="/A247158/a247158.pdf">The number of binary mxm matrices with at most k 1's in each row or columna</a>, (2014) Table 3 column 1.

%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H Sebastian Mizera and Sabrina Pasterski, <a href="https://arxiv.org/abs/2204.02505">Celestial Geometry</a>, arXiv:2204.02505 [hep-th], 2022.

%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H D. J. Price, <a href="http://www.jstor.org/stable/3609091">Some unusual series occurring in n-dimensional geometry</a>, Math. Gaz., 30 (1946), 149-150.

%H L. Pudwell and A. Baxter, <a href="http://faculty.valpo.edu/lpudwell/slides/pp2014_pudwell.pdf">Ascent sequences avoiding pairs of patterns</a>, 2014.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H H. P. Robinson, <a href="/A000124/a000124.pdf">Letter to N. J. A. Sloane, Aug 16 1971, with attachments</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CakeNumber.html">Cake Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubeDivisionbyPlanes.html">Cube Division by Planes</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CylinderCutting.html">Cylinder Cutting</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpaceDivisionbyPlanes.html">Space Division by Planes</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>

%H Reinhard Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</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) = (n+1)*(n^2-n+6)/6 = (n^3 + 5*n + 6) / 6.

%F G.f.: (1 - 2*x + 2x^2)/(1-x)^4. - [_Simon Plouffe_ in his 1992 dissertation.]

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

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

%F Paraphrasing the previous comment: the sequence is the binomial transform of [1,1,1,1,0,0,0,...]. - _Gary W. Adamson_, Oct 23 2007

%F From _Ilya Gutkovskiy_, Jul 18 2016: (Start)

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

%F a(n) = Sum_{k=0..n} A152947(k+1).

%F Inverse binomial transform of A134396.

%F Sum_{n>=0} a(n)/n! = 8*exp(1)/3. (End)

%F a(n) = -A283551(-n). - _Michael Somos_, Jul 07 2022

%e 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. - _Geoffrey Critzer_, Jan 23 2009

%e 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

%e G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 42*x^6 + 64*x^7 + ... - _Michael Somos_, Jul 07 2022

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

%t Table[(n^3 + 5 n + 6)/6, {n, 0, 50}] (* _Harvey P. Dale_, Jan 19 2013 *)

%t LinearRecurrence[{4, -6, 4, -1}, {1, 2, 4, 8}, 50] (* _Harvey P. Dale_, Jan 19 2013 *)

%t Table[Binomial[n, 3] + n, {n, 20}] (* _Eric W. Weisstein_, Jul 21 2017 *)

%o (PARI) a(n)=(n^2+5)*n/6+1 \\ _Charles R Greathouse IV_, Jun 15 2011

%o (PARI) Vec((1-2*x+2*x^2)/((1-x)^4) + O(x^100)) \\ _Altug Alkan_, Oct 16 2015

%o (Magma) [(n^3+5*n+6)/6: n in [0..50]]; // _Vincenzo Librandi_, Nov 08 2014

%o (Python)

%o def A000125_gen(): # generator of terms

%o a, b, c = 1, 1, 1

%o while True:

%o yield a

%o a, b, c = a+b, b+c, c+1

%o it = A000125_gen()

%o A000125_list = [next(it) for _ in range(50)] # _Cole Dykstra_, Aug 03 2022

%Y Bisections give A100503, A100504.

%Y Row sums of A077028.

%Y Cf. A000124, A003600, A005408, A016813, A086514, A058331, A002522, A161701-A161705, A000127, A161706 - A161708, A080856, A161710-A161713, A161715, A006261, A063865, A051601, A077043, A002620.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Minor typo in comments corrected by _Mauro Fiorentini_, Jan 02 2018

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