

A000096


a(n) = n*(n+3)/2.
(Formerly M1356 N0522)


250



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
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OFFSET

0,2


COMMENTS

For n >= 1, a(n) is the maximal number of pieces that can be obtained by cutting an annulus with n cuts. See illustration.  Robert G. Wilson v
n(n3)/2 (n >= 3) is the number of diagonals of an ngon.  Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
n(n3)/2 (n >= 4) is the degree of the thirdsmallest 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(n3) = dimension of the traveling salesman polytope T(n).  Benoit Cloitre, Aug 18 2002
Coefficient of x^2 in (1 + x + 2*x^2)^n.  Michael Somos, May 26 2004
a(n) is the number of "prime" ndimensional polyominoes. A "prime" npolyomino cannot be formed by connecting any other npolyominoes except for the nmonomino and the nmonomino is not prime. E.g., for n=1, the 1monomino is the line of length 1 and the only "prime" 1polyominoes are the lines of length 2 and 3. This refers to "free" ndimensional 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 triangular 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 S. Plewe, Jun 18 2005
Sum_{k=2..n+1} 4/(k*(k+1)*(k1)) = ((n+3)*n)/((n+2)*(n+1)). Numerator(Sum_{k=2..n+1} 4/(k*(k+1)*(k1)) = (n+3)*n/2.  Alexander Adamchuk, Apr 11 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 JohnsonFreyd (theojf(AT)berkeley.edu), Jun 10 2007
If X is an nset and Y a fixed 2subset of X then a(n2) is equal to the number of (n2)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
If s(n) is a sequence defined as s(1) = x, s(n) = kn + s(n1) + p for n > 1, then s(n) = a(n1)*k + (n1)*p + x.  Gary Detlefs, Mar 04 2010
a(n) = m such that the (m+1)th triangular number minus the mth triangular number is the (n+1)th triangular number: (m+1)(m+2)/2  m(m+1)/2 = (n+1)(n+2)/2.  Zak Seidov, Jan 22 2012
For n >= 1, number of different values that Sum_{k=1..n} c(k)*k can take where the c(k) are 0 or 1.  Joerg Arndt, Jun 24 2012
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(n2). 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.  Jose Abutal, Nov 19 2013
If k = a(i1) or k = a(i+1) and n = k + a(i), then C(n, k1), C(n, k), C(n, k+1) are three consecutive binomial coefficients in arithmetic progression and these are all the solutions. There are no four consecutive binomial coefficients in arithmetic progression.  Michael Somos, Nov 11 2015
a(n1) is also the number of independent components of a symmetric traceless tensor of rank 2 and dimension n >= 1.  Wolfdieter Lang, Dec 10 2015
Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rhoth derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See the Wojnar et al. link] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=Drho. These polynomials are of the form h_D(N)= ((1)^D/(D1)!)*(DN)*N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D2chi). Then a(n) are the negated coefficients of the next to the highest order term in the polynomials N^chi*g_D(N), starting at D=3.  Gregory Gerard Wojnar, Jul 19 2017
For n >= 2, a(n) is the number of summations required to solve the linear regression of n variables (n1 independent variables and 1 dependent variable).  Felipe PedrazaOropeza, Dec 07 2017
For n >= 2, a(n) is the number of sums required to solve the linear regression of n variables: 5 for two variables (sums of X, Y, X^2, Y^2, X*Y), 9 for 3 variables (sums of X1, X2, Y1, X1^2, X1*X2, X1*Y, X2^2, X2*Y, Y^2), and so on.  Felipe PedrazaOropeza, Jan 11 2018
a(n) is the area of a triangle with vertices at (n, n+1), ((n+1)*(n+2)/2, (n+2)*(n+3)/2), ((n+2)^2, (n+3)^2)).  J. M. Bergot, Jan 25 2018
Number of terms less than 10^k: 1, 4, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, ...  Muniru A Asiru, Jan 25 2018
a(n) is also the number of irredundant sets in the (n+1)path complement graph for n > 2.  Eric W. Weisstein, Apr 11 2018
a(n) is also the largest number k such that the largest Dyck path of the symmetric representation of sigma(k) has exactly n peaks, n >= 1. (Cf. A237593.)  Omar E. Pol, Sep 04 2018
For n > 0, a(n) is the number of facets of associahedra. Cf. A033282 and A126216 and their refinements A111785 and A133437 for related combinatorial and analytic constructs. See p. 40 of Hanson and Sha for a relation to projective spaces and string theory.  Tom Copeland, Jan 03 2021
For n > 0, a(n) is the number of bipartite graphs with 2n or 2n+1 edges, no isolated vertices, and a stable set of cardinality 2.  Christian Barrientos, Jun 13 2022
For n >= 2, a(n2) is the number of permutations in S_n which are the product of two different transpositions of adjacent points.  Zbigniew Wojciechowski, Mar 31 2023


REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.
G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Maths. and its Appls., Vol. 16, AddisonWesley, 1981, Reading, MA, U.S.A.
D. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.
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

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].
Maria J. Rodriguez, Black holes in all, The KIAS Newsletter, Vol.3, pp.2934, Korea Institute for Advanced Study, Dec. 2010.


FORMULA

G.f.: A(x) = x*(2x)/(1x)^3. a(n) = binomial(n+1, n1) + binomial(n, n1).
Connection with triangular numbers: a(n) = A000217(n+1)  1.
a(n) = a(n1) + n + 1.  Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
a(n) = 2*t(n)  t(n1) 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(n) = 3*a(n1)  3*a(n2) + a(n3).  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,i1]=1, A[i,j]=Catalan(ji), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n1)=coeff(charpoly(A,x),x^(n2)).  Milan Janjic, Jul 08 2010
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
G.f.: U(0) where U(k) = 1  1/((1x)^2  x*(1x)^4/(x*(1x)^2  1/U(k+1))); (continued fraction, 3step).  Sergei N. Gladkovskii, Sep 27 2012
a(n1) = (1/n!)*Sum_{j=0..n} binomial(n,j)*(1)^(nj)*j^n*(j1).  Vladimir Kruchinin, Jun 06 2013
a(n+1) = A127672(4+n, n), n >= 0, where A127672 gives the coefficients of the Chebyshev C polynomials. See the AbramowitzStegun reference.  Wolfdieter Lang, Dec 10 2015
Dirichlet g.f.: (zeta(s2) + 3*zeta(s1))/2.  Ilya Gutkovskiy, Jun 30 2016
Sum_{n>=1} (1)^(n+1)/a(n) = 4*log(2)/3  5/9.  Amiram Eldar, Jan 10 2021
Product_{n>=1} (1 + 1/a(n)) = 3.
Product_{n>=1} (1  1/a(n)) = 3*cos(sqrt(17)*Pi/2)/(4*Pi). (End)


EXAMPLE

G.f. = 2*x + 5*x^2 + 9*x^3 + 14*x^4 + 20*x^5 + 27*x^6 + 35*x^7 + 44*x^8 + 54*x^9 + ...


MAPLE



MATHEMATICA

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


PROG

(PARI) first(n) = Vec(x*(2x)/(1x)^3 + O(x^n), n) \\ Iain Fox, Dec 12 2017
(Haskell)
a000096 n = n * (n + 3) `div` 2
a000096_list = [x  x < [0..], a023531 x == 1]
(Magma) [n*(n+3)/2: n in [0..60]]; or [n: n in [0..2300]  IsSquare(8*n+9)]; // JuriStepan Gerasimov, Apr 05 2016
(GAP) a := List([0..1000], n > n*(n+3)/2); # Muniru A Asiru, Jan 25 2018


CROSSREFS

Cf. numbers of the form n*(n*kk+4)/2 listed in A226488.
Similar sequences are listed in A316466.


KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



