Search: id:A000096 Results 1-1 of 1 results found. %I A000096 M1356 N0522 %S A000096 0,2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230, %T A000096 252,275,299,324,350,377,405,434,464,495,527,560,594,629,665,702,740, %U A000096 779,819,860,902,945,989,1034,1080,1127,1175,1224,1274,1325,1377,1430 %N A000096 n(n+3)/2. %C A000096 For n >= 1, a(n) = maximal number of pieces that can be obtained by cutting an annulus with n cuts. - Robert G. Wilson v (rgwv(AT)rgwv.com) %C A000096 n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr) %C A000096 n(n-3)/2 (n >= 4) is the degree of the third-smallest irreducible presentation of the symmetric group S_n (cf. James and Kerber, Appendix 1). %C A000096 a(n) is also the multiplicity of the eigenvalue (-2) of the triangle graph Delta(n+1). (See p. 19 in Biggs). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Nov 25 2001 %C A000096 For n>3 a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 18 2002 %C A000096 Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild (wild(AT)music.mcgill.ca), May 07 2004. %C A000096 Coefficient of x^2 in (1+x+2x^2)^n. - Michael Somos May 26 2004 %C A000096 A curve of order n is generally determined by n(n+3)/2 points. This function is semiprime for n = 3, 4, 7, 10, 11, 14, 19, 23, 26, 31, 34, 38, 43, ... - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 25 2005 %C A000096 a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomnio cannot be formed by connecting any other n-polyominoes except for the n-monomino and the n-monomino is not prime. E.g. for n=1, the 1-monomino is the line of length 1 and the only "prime" 1-polyominoes are the lines of length 2 and 3. This refers to "free" n-dimensional polyominoes, i.e. that can be rotated along any axis. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005 %C A000096 Solutions to the quadratic equation q(m, r) = (-3 +/- sqrt(9 + 8(m - r))) / 2, where m - r is included in a(n). Let t(m) = the triangle number (A000217) less than some number k and r = k - t(m). If k is neither prime nor a power of two and m - r is included in A000096, then m - q(m, r) will produce a value that shares a divisor with k. - Andrew Plewe, Jun 18 2005 %C A000096 Sum[4/(k*(k+1)*(k-1)),{k,2,n+1}] = ((n+3)*n)/((n+2)*(n+1)). Numerator[Sum[4/ (k*(k+1)*(k-1)),{k,2,n+1}] = (n+3)*n/2 - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006 %C A000096 a(n) = A126890(n,1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006 %C A000096 Number of rooted trees with n+3 nodes of valence 1, no nodes of valence 2 and exactly two other nodes. I.e. number of planted trees with n+2 leaves and exactly two branch points. - Theo Johnson-Freyd (theojf(AT)berkeley.edu), Jun 10 2007 %C A000096 If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-2)-subsets of X intersecting Y. > - Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007 %C A000096 For n>=1, a(n) is the number of distinct shuffles of the identity permutation on n+1 letters with the identity permutation on 2 letters (12). [From Camillia Smith (cammie(AT)math.harvard.edu), Oct 04 2008] %C A000096 A002262(a(n)) = n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2009] %C A000096 Theorem 2, p. 3, of Yashar Memarian, states "let G be a 4-regular minimal graph on the plane with n attaching points. Then G has at most (n/ 2)C2 + n vertices if n is even, else 0. This is sharp. For each n, there is a minimal 4-regular graph which achieves this bound." Since xC2 = (1/2)*(x^2) - (1/2)x, the expression (n/2)C2 + n simplifies to (1/8)*(x^2) + (3/4)*x which is n(n+3)/2 for n an even value of x. Hence I'd say: "let G be a 4-regular minimal graph on the plane with n attaching points. Then G has at most A000096(n) = n(n+3)/2 vertices if n is even, else 0. This is sharp." [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009] %D A000096 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797. %D A000096 D. Applegate, R. Bixby, V. Chvatal and W. Cook : On the solution of traveling salesman problem. In : Int. Congress of mathematics (Berlin 1998), Documenta Math., Extra Volume ICM 1998, Vol. III, pp. 645-656. %D A000096 Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. %D A000096 Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Memoires de l'Academie des Sciences de Berlin, 4, 219-233, 1750 Reprinted in Opera Omnia, Series I, Vol. 26. pp. 33-45. %D A000096 S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205. %D A000096 G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Maths. and its Appls., Vol. 16, Addison-Wesley, 1981, Reading, MA, U.S.A. %D A000096 D. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4. %D A000096 A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37. %D A000096 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000096 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A000096 Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000 %H A000096 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. %H A000096 S. P. Humphries, Home page %H A000096 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1018 %H A000096 Milan Janjic, Two Enumerative Functions %H A000096 Yashar Memarian, On the Maximum Number of Vertices of Minimal Embedded Graphs, Oct 13, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009] %H A000096 Barbarel Tres Mil, Binomial Matrix (I), Psychedelic Geometry Blogspot 09/22/09 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 22 2009] %H A000096 P. Moree, Convoluted convolved Fibonacci numbers %H A000096 P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2. %H A000096 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. %H A000096 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000096 C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. %H A000096 Sandifer, E. How Euler Did It %H A000096 Eric Weisstein's World of Mathematics, Cramer-Euler Paradox. %H A000096 Index entries for two-way infinite sequences %H A000096 Index entries for sequences related to linear recurrences with constant coefficients %F A000096 G.f.: A(x) = x*(2-x)/(1-x)^3. a(n)=binomial(n+1, n-1)+binomial(n, n-1). %F A000096 a(n)=2*t(n)-t(n-1), e.g. a(5)=2*t(5)-t(4)=2*15-10=20. - Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003 %F A000096 a(-3-n)=a(n). - Michael Somos May 26 2004 %F A000096 a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005 %F A000096 2*a(n) = A008778(n) - A105163(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 15 2005 %F A000096 a(n) = C(3+n, 2)-C(3+n, 1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 09 2005 %F A000096 a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006 %F A000096 a(n)=3a(n-1)-3a(n-2)+a(n-3). - Paul Curtz (bpcrtz(AT)free.fr), Jan 02 2008 %F A000096 Starting (2, 5, 9, 14,...) = binomial transform of (2, 3, 1, 0, 0, 0, ...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2008 %F A000096 For n >= 0, a(n+2) = b(n+1) - b(n), where b(n) is the sequence A005586. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Apr 27 2009] %F A000096 a(n)=binomial(n+2,n)-1=binomial(n+2,2)-1 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 22 2009] %p A000096 A000096 := n->n*(n+3)/2; %p A000096 [seq(binomial(n,2)-n,n=3..55)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006 %p A000096 seq((GAMMA(n+3)/GAMMA(n+1)/2)-1,n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2007 %p A000096 seq(sum(mul(gcd(k+2,j),j=0..n), k=0..n), n=-1..51); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007 %p A000096 seq(add((k), k=2..n), n=1..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 14 2007 %p A000096 A000096:=z*(-2+z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.] %p A000096 a:=n->sum(numer (k/(k+3)), k=2..n): seq(a(n), n=1..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2008 %p A000096 a:=n->sum(2+sum(1, k=1..n), k=2..n)/2:seq(a(n), n=1...43); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008] %t A000096 lst={};Do[AppendTo[lst, n*(n+3)/2], {n, 0, 5!}];lst ...and/or... s=0; lst={s};Do[s+=n+1;AppendTo[lst, s], {n, 1, 5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008] %o A000096 (PARI) a(n)=n*(n+3)/2 %Y A000096 Triangular numbers (A000217) minus one. Cf. A000217, A034856, A000124, A005581-A005584. %Y A000096 Occurs as a diagonal in A074079/A074080, i.e.: A074079(n+3, n) = A000096(n-1) for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2. - Antti Karttunen, Aug 20, 2002. %Y A000096 A column of triangle A014473. %Y A000096 Cf. A067550. %Y A000096 Sequence in context: A075543 A132315 A132336 this_sequence A080956 A132337 A134189 %Y A000096 Adjacent sequences: A000093 A000094 A000095 this_sequence A000097 A000098 A000099 %K A000096 nonn,easy,nice %O A000096 0,2 %A A000096 N. J. A. Sloane (njas(AT)research.att.com). %E A000096 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000 Search completed in 0.002 seconds