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A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.
(Formerly M1119 N0427)
47
1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841, 31931, 36457, 41449, 46938, 52956, 59536, 66712, 74519, 82993, 92171, 102091 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the sum of the first five terms in the n-th row of Pascal's triangle. [Geoffrey Critzer, Jan 18 2009]

{a(k): 1 <= k <= 5} = divisors of 16. [Reinhard Zumkeller, Jun 17 2009]

Equals binomial transform of [1, 1, 1, 1, 1, 0, 0, 0,...] [Gary W. Adamson, Mar 02 2010]

REFERENCES

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 28.

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

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, Chap. 3.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 33 pp. 18; 128 Ellipses Paris 2004.

A. Deledicq and D. Missenard, A La Recherche des Regions Perdues, Math. & Malices, No. 22 Summer 1995 issue pp. 22-3 ACL-Editions Paris.

M. Gardner, Mathematical Circus, pp. 177; 180-1 Alfred A. Knopf NY 1979

M. Gardner, The Colossal Book of Mathematics, 2001, p. 561.

James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).

M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.

M. de Guzman, Aventures Mathematiques, Prob. B pp. 115-120 PPUR Lausanne 1990

Ross Honsberger; Mathematical Gems I, Chap. 9.

Ross Honsberger; Mathematical Morsels, Chap. 3.

Jeux Mathematiques et Logiques, Vol. 3 pp. 12; 51 Prob. 14 FFJM-SERMAP Paris 1988

J. N. Kapur, Reflections of a Mathematician, Chap.36, pp. 337-343, Arya Book Depot, New Delhi 1996.

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

C. D. Miller, V. E. Heeren, J. Hornsby, M. L. Morrow and J. Van Newenhizen, Mathematical Ideas, Tenth Edition, Pearson, Addison-Wesley, Boston, 2003, Cptr 1, 'The Art of Problem Solving, page 6.

I. Niven, Mathematics of Choice, pp. 158; 195 Prob. 40 NML 15 MAA 1965

M. Noy, "A Short Solution of a Problem in Combinatorial Geometry", Mathematics Magazine, pp. 52-3 69(1) 1996 MAA

C. S. Ogilvy, Tomorrow's Math, pp. 144-6 OUP 1972

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 81-87.

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

A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Alan Calvitti, Illustration of initial terms

Math Forum, Regions of a circle Cut by Chords to n points.

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.

K. Uhland, A Blase of Glory [Broken link?]

K. Uhland, Moser's Problem [Broken link?]

Eric Weisstein's World of Mathematics, Circle Division by Chords

Eric Weisstein's World of Mathematics, Strong Law of Small Numbers

R. Zumkeller, Enumerations of Divisors

Index entries for sequences related to linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

FORMULA

a(n) = C(n-1, 4)+C(n-1, 3)+ ... +C(n-1, 0) = A055795(n) +1 = C(n, 4)+C(n-1, 2)+n.

a(n) = Sum_{0 <= k <= 2} C(n, 2k). - Joel Sanderi (sanderi(AT)itstud.chalmers.se), Sep 08 2004

a(n) = (n^4-6n^3+23n^2-18n+24)/24.

G.f.: (1-3x+4x^2-2x^3+x^4)/(1-x)^5. [Simon Plouffe in his 1992 dissertation.]

E.g.f. : (1+x+x^2/2+x^3/6+x^4/24)exp(x). [Typos corrected by Juan M. Marquez, Jan 24 2011]

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n>4 [Harvey P. Dale, Aug 24 2011]

EXAMPLE

a(7)=99 because the first five terms in the 7th row of Pascal's triangle are 1+7+21+35+35=99 [Geoffrey Critzer, Jan 18 2009]

MAPLE

A000127 := n->(n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24;

with (combstruct):ZL:=[S, {S=Sequence(U, card<r), U=Set(Z, card>=1)}, unlabeled]: seq(count(subs(r=6, ZL), size=m), m=0..41); - Zerinvary Lajos, Mar 08 2008

MATHEMATICA

f[n_] := Sum[Binomial[n, i], {i, 0, 4}]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Jun 29 2007 *)

Total/@Table[Binomial[n-1, k], {n, 50}, {k, 0, 4}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {1, 2, 4, 8, 16}, 50] (* Harvey P. Dale, Aug 24 2011 *)

PROG

(Haskell)

a000127 = sum . take 5 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

CROSSREFS

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261, A007318, A008859 - A008863, A219531.

Sequence in context: A054016 A051039 A056183 * A133552 A174439 A000128

Adjacent sequences:  A000124 A000125 A000126 * A000128 A000129 A000130

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Formula corrected and additional references from TORSTEN.SILLKE(AT)LHSYSTEMS.COM.

Additional correction from Jonas Paulson (jonasso(AT)sdf.lonestar.org), Oct 30 2003

STATUS

approved

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Last modified October 31 15:41 EDT 2014. Contains 248868 sequences.