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

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.

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.

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)

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

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

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).

(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

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.]

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

Cf. A077028.

Sequence in context: A026474 A082562 A089140 this_sequence A129961 A133551 A114226

Adjacent sequences: A000122 A000123 A000124 this_sequence A000126 A000127 A000128

nonn,easy,nice

njas

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