0,2

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 R. Janjic (agnus(AT)blic.net), Dec 28 2007

a(n) is the number of compositions (ordered partiitions) of n+1 into four or fewer parts or equivalently the sum of the first four terms in the nth row of Pascal's triangle. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 23 2009]

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

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

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)

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

Index entries for sequences related to linear recurrences with constant coefficients

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

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

G.f.: (1-2x+2x^2)/(1-x)^4; - Paul Barry (pbarry(AT)wit.ie), Jun 21 2005

C(n,3)+C(n,1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 24 2006

a(n) = sum of (n+1)-th row terms of A077028. Also, binomial transform of [1,1,1,1,0,0,0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 23 2007

a(n)=C(n+3,n)+n,n>=1 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2009]

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 (critzer.geoffrey(AT)usd443.org), Jan 23 2009]

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

seq(binomial(n, 3)+binomial(n, 2)+binomial(n, 1)+binomial(n, 0), n=0..29); - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

A000125:=(1-2*z+2*z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]

a=1; lst={a}; k=1; e=0; Do[a+=k; AppendTo[lst, a]; e++; k+=e, {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008]

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

Cf. A077028.

A005408, A000124, A016813, A086514, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Sequence in context: A082562 A159243 A089140 this_sequence A129961 A133551 A114226

Adjacent sequences: A000122 A000123 A000124 this_sequence A000126 A000127 A000128

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 22 2000