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A000096 n*(n+3)/2.
(Formerly M1356 N0522)
138
0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430, 1484, 1539, 1595, 1652, 1710, 1769 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n >= 1, a(n) = maximal number of pieces that can be obtained by cutting an annulus with n cuts. - Robert G. Wilson v

n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)

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).

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

For n>3 a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre, Aug 18 2002

Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild, May 07 2004

Coefficient of x^2 in (1 + x + 2*x^2)^n. - Michael Somos, May 26 2004

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, Mar 25 2005

a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomino 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

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

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, Apr 11 2006

a(n) = A126890(n,1) for n>0. - Reinhard Zumkeller, Dec 30 2006

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

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 Janjic, Jul 30 2007

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). - Camillia Smith Barnes, Oct 04 2008

A002262(a(n)) = n. - Reinhard Zumkeller, May 20 2009

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." - Jonathan Vos Post, Oct 14 2009

If s(n) is a sequence defined as s(1) = x, s(n)=kn+s(n-1)+p..n>1 then s(n)=a(n-1)*k+(n-1)*p +x. - Gary Detlefs, Mar 04 2010

The number of degrees of freedom in solving Einstein's equations (coupled nonlinear partial differential equations) carried by the unknown metric g_(mu nu) for black holes/black rings of different topologies in n dimensions (see Rodriguez, p.32). - Jonathan Vos Post, Jan 30 2011

No primes except a(1) = 2 and a(2) = 5. - Reinhard Zumkeller, Jul 18 2011

a(n) = m such that (m+1)-th triangle number minus m-th triangle number is (n+1)-th triangle number: (m+1)(m+2)/2-m(m+1)/2 = (n+1)(n+2)/2. - Moshe Levin, Jan 22 2012

For n>=1, number of different values that sum( k=1..n, c(k)*k ) can take were the c(k) are 0 or 1. - Joerg Arndt, Jun 24 2012

A023532(a(n)) = 0. - Reinhard Zumkeller, Dec 04 2012

Sum(n>0, 1/a(n)) = 11/9. - Enrique Pérez Herrero, Nov 26 2013

Comment from Jose Abutal, Nov 19 2013. On an n X n chessboard (n >= 2), the number of possible checkmate positions in the case of king and rook versus a lone king is 0, 16, 40, 72, 112, 160, 216, 280, 352,. ..., which is 8*a(n-2).  For a 4 X 4 board the number is 40. The number of positions possible was counted including all mirror images and rotations for all four sides of the board.

REFERENCES

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. 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.

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.

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.

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. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.

Maria J. Rodriguez, "Black holes in all", The KIAS Newsletter, Vol.3, pp.29-34, Korea Institute for Advanced Study, Dec. 2010.

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

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000

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].

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers.

S. P. Humphries, Home page

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1018

Milan Janjic, Two Enumerative Functions

A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37.

Yashar Memarian, On the Maximum Number of Vertices of Minimal Embedded Graphs, Oct 13, 2009. Jonathan Vos Post, Oct 14 2009

E. Pérez Herrero, Binomial Matrix (I), Psychedelic Geometry Blogspot 09/22/09 [Enrique Pérez Herrero, Sep 22 2009]

P. Moree, Convoluted convolved Fibonacci numbers

P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

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.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Sandifer, E. How Euler Did It

Eric Weisstein's World of Mathematics, Cramer-Euler Paradox.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: A(x) = x*(2-x)/(1-x)^3. a(n) = binomial(n+1, n-1)+binomial(n, n-1).

a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005

a(n) = 2*t(n)-t(n-1) where t() are the triangular numbers, e.g., a(5)=2*t(5)-t(4)=2*15-10=20. - Jon Perry, Jul 23 2003

a(-3-n) = a(n). - Michael Somos, May 26 2004

2*a(n) = A008778(n) - A105163(n). - Creighton Dement, Apr 15 2005

a(n) = C(3+n, 2) - C(3+n, 1). - Zerinvary Lajos, Dec 09 2005

a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk, May 20 2006

a(n) = 3a(n-1)-3a(n-2)+a(n-3). - Paul Curtz, Jan 02 2008

Starting (2, 5, 9, 14,...) = binomial transform of (2, 3, 1, 0, 0, 0,...). - Gary W. Adamson, Jul 03 2008

For n >= 0, a(n+2) = b(n+1) - b(n), where b(n) is the sequence A005586. - K.V.Iyer, Apr 27 2009

Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n-1)=coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jul 08 2010

a(n) = sum((k+1)!/k!, k=1..n). - Gary Detlefs, Aug 03 2010

a(n) = n(n+1)/2+n = A000217(n) + n. - Moshe Levin, Jan 22 2012

E.g.f.: F(x) = 1/2*x*exp(x)*(x+4) satisfies the differential equation F''(x)-2*F'(x)+F(x) = exp(x). - Peter Bala, Mar 14, 2012

a(n) = binomial(n+3, 2) - (n+3). - Robert G. Wilson v, Mar 15 2012

a(n) = A181971(n+1, 2) for n > 0. - Reinhard Zumkeller, Jul 09 2012

a(n) = A214292(n+2, 1). - Reinhard Zumkeller, Jul 12 2012

G.f.: -U(0)  where U(k)= 1 - 1/((1-x)^2 - x*(1-x)^4/(x*(1-x)^2 - 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 27 2012

a(n) = A014132(n,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012

a(n-1) = 1/n!*sum(j=0..n, binomial(n,j)*(-1)^(n-j)*j^n*(j-1)). - Vladimir Kruchinin, Jun 06 2013

a(n) = 2n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

a(n) = sum_{i=2..n+1} i. - Wesley Ivan Hurt, Jun 28 2013

a(n) = sum_{i=1..n} (n - i + 2). - Wesley Ivan Hurt, Mar 31 2014

MAPLE

A000096 := n->n*(n+3)/2; seq(A000096(n), n=0..50);

A000096 :=z*(-2+z)/(z-1)**3; # [Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

Table[n*(n+3)/2, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)

LinearRecurrence[{3, -3, 1}, {0, 2, 5}, 60] (* Harvey P. Dale, Apr 30 2013 *)

PROG

(PARI) {a(n) = n * (n+3)/2} /* Michael Somos, May 26 2004 */

(Haskell)

import Data.List (elemIndices)

a000096 n = n * (n + 3) `div` 2

a000096_list = elemIndices 0 a023532_list

-- Reinhard Zumkeller, Dec 04 2012

CROSSREFS

Triangular numbers (A000217) minus one.

Cf. A000217, A034856, A000124, A005581-A005584.

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

A column of triangle A014473.

First skew subdiagonal of A033282.

Cf. A067550.

Complement of A007401.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - Bruno Berselli, Jun 10 2013

A diagonal of A079508.

Sequence in context: A212342 A080956 A132337 * A134189 A109470 A112873

Adjacent sequences:  A000093 A000094 A000095 * A000097 A000098 A000099

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, May 04 2000

STATUS

approved

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Last modified September 20 16:08 EDT 2014. Contains 247015 sequences.