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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

%I M1119 N0427

%S 1,2,4,8,16,31,57,99,163,256,386,562,794,1093,1471,1941,2517,3214,

%T 4048,5036,6196,7547,9109,10903,12951,15276,17902,20854,24158,27841,

%U 31931,36457,41449,46938,52956,59536,66712,74519,82993,92171,102091,112792,124314,136698

%N 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.

%C a(n) is the sum of the first five terms in the n-th row of Pascal's triangle. - _Geoffrey Critzer_, Jan 18 2009

%C {a(k): 1 <= k <= 5} = divisors of 16. - _Reinhard Zumkeller_, Jun 17 2009

%C Equals binomial transform of [1, 1, 1, 1, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Mar 02 2010

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

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

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

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

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

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

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

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

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

%D Ross Honsberger; Mathematical Gems I, Chap. 9.

%D Ross Honsberger; Mathematical Morsels, Chap. 3.

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

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

%D 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.

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

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

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

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

%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).

%H T. D. Noe, <a href="/A000127/b000127.txt">Table of n, a(n) for n = 1..1000</a>

%H Alan Calvitti, <a href="/A000127/a000127.jpg">Illustration of initial terms</a>

%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 M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Griffiths2/griffiths17.html">Remodified Bessel Functions via Coincidences and Near Coincidences</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.

%H R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

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

%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 Math Forum, <a href="http://mathforum.org/library/drmath/view/55262.html">Regions of a circle Cut by Chords to n points</a>.

%H R. J. Mathar, <a href="/A247158/a247158.pdf">The number of binary nxm matrices with at most k 1's in each row or column</a>, (2014) Table 4 column 1.

%H Leo Moser and W. Bruce Ross, <a href="http://www.jstor.org/stable/3219224">Mathematical Miscellany, On the Danger of Induction</a>, Mathematics Magazine, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 109-114.

%H M. Noy, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-short-solution-of-a-problem-in-combinatorial-geometry">A Short Solution of a Problem in Combinatorial Geometry</a>, Mathematics Magazine, pp. 52-3 69(1) 1996 MAA

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Patrick Popescu-Pampu, <a href="http://images.math.cnrs.fr/Demarrage-trompeur.html">Démarrage trompeur</a>, Images des Mathématiques, CNRS, 2017 (in French).

%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 H. P. Robinson, <a href="/A002664/a002664.pdf">Letter to N. J. A. Sloane, Mar 21 1985</a>

%H Grant Sanderson, <a href="https://www.youtube.com/watch?v=K8P8uFahAgc">Circle division solution</a> (2015)

%H K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/198507Sol.htm">A Blase of Glory</a> [Broken link?]

%H K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/199909sol.htm">Moser's Problem</a> [Broken link?]

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers</a>

%H R. Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)

%F 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.

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

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

%F G.f.: (1 - 3*x + 4*x^2 - 2*x^3 + x^4)/(1-x)^5. (for offset 0) - _Simon Plouffe_ in his 1992 dissertation

%F E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24)*exp(x) (for offset 0). [Typos corrected by _Juan M. Marquez_, Jan 24 2011]

%F 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

%F a(n) = A000124(A000217(n-1)) - n*A000217(n-2) - A034827(n), n > 1. - _Melvin Peralta_, Feb 15 2016

%F a(n) = A223718(-n). - _Michael Somos_, Dec 23 2017

%e 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

%e G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 31*x^6 + 57*x^7 + 99*x^8 + 163*x^9 + ...

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

%p 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

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

%t 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 *)

%t Table[(n^4 - 6 n^3 + 23 n^2 - 18 n + 24) / 24, {n, 100}] (* _Vincenzo Librandi_, Feb 16 2015 *)

%t a[ n_] := Binomial[n, 4] + Binomial[n, 2] + 1; (* _Michael Somos_, Dec 23 2017 *)

%o (Haskell)

%o a000127 = sum . take 5 . a007318_row -- _Reinhard Zumkeller_, Nov 24 2012

%o (MAGMA) [(n^4-6*n^3+23*n^2-18*n+24)/24: n in [1..50]]; // _Vincenzo Librandi_, Feb 16 2015

%o (PARI) a(n)=(n^4-6*n^3+23*n^2-18*n+24)/24 \\ _Charles R Greathouse IV_, Mar 22 2016

%o (PARI) {a(n) = binomial(n, 4) + binomial(n, 2) + 1}; /* _Michael Somos_, Dec 23 2017 */

%Y 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, A223718.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E Formula corrected and additional references from torsten.sillke(AT)lhsystems.com

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

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Last modified November 14 06:49 EST 2018. Contains 317162 sequences. (Running on oeis4.)