%I A000127 M1119 N0427
%S A000127 1,2,4,8,16,31,57,99,163,256,386,562,794,1093,1471,1941,2517,3214,
%T A000127 4048,5036,6196,7547,9109,10903,12951,15276,17902,20854,24158,27841,
%U A000127 31931,36457,41449,46938,52956,59536,66712,74519,82993,92171,102091
%N 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.
%C A000127 a(n) is the sum of the first five terms in the nth row of Pascal's triangle. 
               [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 18 2009]
%C A000127 {a(k): 1 <= k <= 5} = divisors of 16. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 17 2009]
%D A000127 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventues in 
               Applied Mathematics, Princeton Univ. Press, 1999. See p. 28.
%D A000127 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D A000127 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 
               1996, Chap. 3.
%D A000127 J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des 
               Nombres, Problem 33 pp. 18; 128 Ellipses Paris 2004.
%D A000127 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 A000127 M. Gardner, Mathematical Circus, pp. 177; 180-1 Alfred A. Knopf NY 1979
%D A000127 M. Gardner, The Colossal Book of Mathematics, 2001, p. 561.
%D A000127 James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
%D A000127 M. de Guzman, Aventures Mathematiques, Prob. B pp. 115-120 PPUR Lausanne 
               1990
%D A000127 Ross Honsberger; Mathematical Gems I, Chap. 9.
%D A000127 Ross Honsberger; Mathematical Morsels, Chap. 3.
%D A000127 Jeux Mathematiques et Logiques, Vol. 3 pp. 12; 51 Prob. 14 FFJM-SERMAP 
               Paris 1988
%D A000127 J. N. Kapur, Reflections of a Mathematician, Chap.36, pp. 337-343, Arya 
               Book Depot, New Delhi 1996.
%D A000127 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 
               292-298.
%D A000127 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 A000127 I. Niven, Mathematics of Choice, pp. 158; 195 Prob. 40 NML 15 MAA 1965
%D A000127 M. Noy, "A Short Solution of a Problem in Combinatorial Geometry", Mathematics 
               Magazine, pp. 52-3 69(1) 1996 MAA
%D A000127 C. S. Ogilvy, Tomorrow's Math, pp. 144-6 OUP 1972
%D A000127 Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, 
               Prometheus Books, NY, 2007, page 81-87.
%D A000127 D. J. Price, Some unusual series occurring in n-dimensional geometry, 
               Math. Gaz., 30 (1946), 149-150.
%D A000127 A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 
               213.
%D A000127 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000127 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000127 T. D. Noe, <a href="b000127.txt">Table of n, a(n) for n=1..1000</a>
%H A000127 Alan Calvitti, <a href="a000127.jpg">Illustration of initial terms</a>
%H A000127 Math Forum, <a href="http://mathforum.org/library/drmath/view/55262.html">
               Regions of a circle Cut by Chords to n points</a>.
%H A000127 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000127 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000127 K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/198507Sol.htm">
               A Blase of Glory</a>
%H A000127 K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/199909sol.htm">
               Moser's Problem</a>
%H A000127 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CircleDivisionbyChords.html">Link to a section of The World of Mathematics.</
               a>
%H A000127 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StrongLawofSmallNumbers.html">Strong Law of Small Numbers</a>
%H A000127 R. Zumkeller, <a href="a161700.txt">Enumerations of Divisors</a> [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
%H A000127 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000127 C(n-1, 4)+C(n-1, 3)+ ... +C(n-1, 0) = C(n, 4)+C(n, 2)+1 = C(n, 4)+C(n-1, 
               2)+n.
%F A000127 a(n) = Sum_{0 <= k <= 2} C(n, 2k) - Joel Sanderi (sanderi(AT)itstud.chalmers.se), 
               Sep 08 2004
%F A000127 (n^4-6n^3+23n^2-18n+24)/24.
%F A000127 G.f.:1-3x+4x^2-2x^3+x^4/(1-x)^5 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jan 18 2009]
%e A000127 a(7)=99 because the first five terms in the 7th row of Pascal's triangle 
               are 1+7+21+35+35=99 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jan 18 2009]
%p A000127 A000127 := n->1/24*n^4-1/4*n^3+23/24*n^2-3/4*n+1;
%p A000127 A000127 := n->(n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24;
%p A000127 A000127:=-(1-3*z+4*z**2-2*z**3+z**4)/(z-1)**5; [S. Plouffe in his 1992 
               dissertation.]
%p A000127 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 (zerinvarylajos(AT)yahoo.com), Mar 08 2008
%p A000127 seq(sum(binomial(n,m), m=1..4)+1,n=0..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 17 2008
%t A000127 f[n_] := Sum[Binomial[n, i], {i, 0, 4}]; Table[f@n, {n, 0, 40}] (* Robert 
               G. Wilson v (rgwv(AT)rgwv.com), Jun 29 2007 *)
%Y A000127 A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, 
               A161702, A161703, A161704, A161706, A161707, A161708, A161710, A080856, 
               A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
%Y A000127 Sequence in context: A054016 A051039 A056183 this_sequence A133552 A000128 
               A106399
%Y A000127 Adjacent sequences: A000124 A000125 A000126 this_sequence A000128 A000129 
               A000130
%K A000127 nonn,easy,nice
%O A000127 1,2
%A A000127 N. J. A. Sloane (njas(AT)research.att.com).
%E A000127 Formula corrected and additional references from TORSTEN.SILLKE(AT)LHSYSTEMS.COM.
%E A000127 Additional correction from Jonas Paulson (jonasso(AT)sdf.lonestar.org), 
               Oct 30 2003