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A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1. 88
1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, 151, 169, 188, 208, 229, 251, 274, 298, 323, 349, 376, 404, 433, 463, 494, 526, 559, 593, 628, 664, 701, 739, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1174, 1223, 1273, 1324, 1376, 1429, 1483 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of 1's in the n X n lower Hessenberg (0,1)-matrix (i.e., the matrix having 1's on or below the superdiagonal and 0's above the superdiagonal).

If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007

Number of binary operations which have to be added to Moisil's algebras to obtain algebraic counterparts of n-valued Łukasiewicz logics. See the Wójcicki and Malinowski book, page 31. - Artur Jasinski, Feb 25 2010

Also (n + 1)!(-1)^(n + 1) times the determinant of the n X n matrix given by m(i,j) = i/(i+1) if i=j and otherwise 1. For example, (5+1)! * ((-1)^(5+1)) * Det[{{1/2, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1}, {1, 1, 3/4, 1, 1}, {1, 1, 1, 4/5, 1}, {1, 1, 1, 1, 5/6}}] = 19 = a(5), and (6+1)! * ((-1)^(6+1)) * Det[{{1/2, 1, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1, 1}, {1, 1, 3/4, 1, 1, 1}, {1, 1, 1, 4/5, 1, 1}, {1, 1, 1, 1, 5/6, 1}, {1, 1, 1, 1, 1, 6/7}}] = 26 = a(6). - John M. Campbell, May 20 2011

2*a(n-1) = n*(n+1) - 4, n>=0, with a(-1) = -2 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 17 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013

a(n) is not divisible by 3, 5, 7, or 11. - Vladimir Shevelev, Feb 03 2014

With a(0) = 1 and a(1) = 2, a(n-1) is the number of distinct values of 1 +- 2 +- 3 +- ... +- n, for n > 0. - Derek Orr, Mar 11 2015

Also, numbers m such that 8*m+17 is a square. - Bruno Berselli, Sep 16 2015

Omar E. Pol's formula from Apr 23 2008 can be interpreted as the position of an element located on the third diagonal of an triangular array (read by rows) provided n > 1. - Enrique Pérez Herrero, Aug 29 2016

a(n) is the sum of the numerator and denominator of the fraction that is the sum of 2/(n-1) + 2/n; all fractions are reduced and n > 2. - J. M. Bergot, Jun 14 2017

a(n) is also the number of maximal irredundant sets in the (n+2)-path complement graph for n > 1. - Eric W. Weisstein, Apr 12 2018

From Klaus Purath, Dec 07 2020: (Start)

a(n) is not divisible by primes listed in A038890. The prime factors are given in A038889 and the prime terms of the sequence are listed in A124199.

Each odd prime factor p divides exactly 2 out of any p consecutive terms with the exception of 17, which appears only once in such an interval of terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -3 (mod p), see examples.

If A is a sequence satisfying the recurrence t(n) = 5*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 4 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i for i>0. (End)

Mark each point on a 4^n grid with the number of points that are visible from the point; for n > 1, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 23 2021

REFERENCES

A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).

G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.

Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 471.

Milan Janjic, Two Enumerative Functions.

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, Vol. 35, No. 4 (1997), pp. 318-328.

G. C. Moisil, Recherches sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 26 (1940), 431-466.

László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

D. D. Olesky, B. L. Shader and P. van den Driessche, Permanents of Hessenberg (0,1)-matrices, Electronic Journal of Combinatorics, 12 (2005), #R70.

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev., Vol. 4, No. 5 (1960), pp. 473-478.

Renzo Sprugnoli, Alternating Weighted Sums of Inverses of Binomial Coefficients, J. Integer Sequences, 15 (2012), #12.6.3. - From N. J. A. Sloane, Nov 29 2012

Eric Weisstein's World of Mathematics, Maximal Irredundant Set.

Eric Weisstein's World of Mathematics, Path Complement Graph.

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

Index entries for sequences related to Łukasiewicz

FORMULA

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

a(n) = binomial(n+2, 2) - 2. - Paul Barry, Feb 27 2003

With offset 5, this is binomial(n, 0) - 2*binomial(n, 1) + binomial(n, 2), the binomial transform of (1, -2, 1, 0, 0, 0, ...). - Paul Barry, Jul 01 2003

Row sums of triangle A131818. - Gary W. Adamson, Jul 27 2007

Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007

Row sums of triangle A134225. - Gary W. Adamson, Oct 14 2007

a(n) = A000217(n+1) - 2. - Omar E. Pol, Apr 23 2008

From Jaroslav Krizek, Sep 05 2009: (Start)

a(n) = a(n-1) + n + 1 for n >= 1.

a(n) = n*(n-1)/2 + 2*n - 1.

a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). (End)

a(n) = Hyper2F1([-2, n-1], [1], -1). - Peter Luschny, Aug 02 2014

a(n) = floor[1/(-1 + Sum_{m >= n+1} 1/S2(m,n+1))], where S2 is A008277. - Richard R. Forberg, Jan 17 2015

a(n) = A101881(2*(n-1)). - Reinhard Zumkeller, Feb 20 2015

a(n) = A253909(n+3) - A000217(n+3). - David Neil McGrath, May 23 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - David Neil McGrath, May 23 2015

For n > 1, a(n) = 4*binomial(n-1,1) + binomial(n-2,2), comprising the third column of A267633. - Tom Copeland, Jan 25 2016

From Klaus Purath, Dec 07 2020: (Start)

a(n) = A024206(n) + A024206(n+1).

a(2*n-1) = -A168244(n+1).

a(2*n) = A091823(n). (End)

Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Jan 06 2021

a(n) + a(n+1) = A028347(n+2). - R. J. Mathar, Mar 13 2021

a(n) = A000290(n) - A161680(n-1). - Omar E. Pol, Mar 26 2021

E.g.f.: 1 + exp(x)*(x^2 + 4*x - 2)/2. - Stefano Spezia, Jun 05 2021

a(n) = A024916(n) - A244049(n). - Omar E. Pol, Aug 01 2021

a(n) = A000290(n) - A000217(n-2). - Omar E. Pol, Aug 05 2021

EXAMPLE

From Bruno Berselli, Mar 09 2015: (Start)

By the definition (first formula):

----------------------------------------------------------------------

  1       4         8           13            19              26

----------------------------------------------------------------------

                                                              X

                                              X              X X

                                X            X X            X X X

                    X          X X          X X X          X X X X

          X        X X        X X X        X X X X        X X X X X

  X      X X      X X X      X X X X      X X X X X      X X X X X X

          X        X X        X X X        X X X X        X X X X X

----------------------------------------------------------------------

(End)

From Klaus Purath, Dec 07 2020: (Start)

Assuming a(i) is divisible by p with 0 < i < p and a(k) is the next term divisible by p, then from i + k == -3 (mod p) follows that k = min(p*m - i - 3) != i for any integer m.

(1) 17|a(7) => k = min(17*m - 10) != 7 => m = 2, k = 24 == 7 (mod 17). Thus every a(17*m + 7) is divisible by 17.

(2) a(9) = 53 => k = min(53*m - 12) != 9 => m = 1, k = 41. Thus every a(53*m + 9) and a(53*m + 41) is divisible by 53.

(3) 101|a(273) => 229 == 71 (mod 101) => k = min(101*m - 74) != 71 => m = 1, k = 27. Thus every a(101*m + 27) and a(101*m + 71) is divisible by 101. (End)

From Omar E. Pol, Aug 08 2021: (Start)

Illustration of initial terms:                             _ _

.                                           _ _           |_|_|_

.                              _ _         |_|_|_         |_|_|_|_

.                   _ _       |_|_|_       |_|_|_|_       |_|_|_|_|_

.          _ _     |_|_|_     |_|_|_|_     |_|_|_|_|_     |_|_|_|_|_|_

.   _     |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|

.  |_|    |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|

.

.   1       4         8          13            19              26

------------------------------------------------------------------------ (End)

MAPLE

a := n -> hypergeom([-2, n-1], [1], -1);

seq(simplify(a(n)), n=1..53); # Peter Luschny, Aug 02 2014

MATHEMATICA

f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] (* Robert G. Wilson v, Jun 11 2010 *)

Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)

Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Aug 29 2016 *)

PROG

(MAGMA) [Binomial(n + 1, 2) + n - 1: n in [1..60]]; // Vincenzo Librandi, May 21 2011

(Maxima) A034856(n) := block(

        n-1+(n+1)*n/2

)$ /* R. J. Mathar, Mar 19 2012 */

(PARI) A034856(n)=(n+3)*n\2-1 \\ M. F. Hasler, Jan 21 2015

(Haskell)

a034856 = subtract 1 . a000096 -- Reinhard Zumkeller, Feb 20 2015

CROSSREFS

Subsequence of A165157.

Triangular numbers (A000217) minus two. a(n) = T(3, n-2), array T as in A049600.

Third diagonal of triangle in A059317.

Cf. A000096, A027379, A101881, A113452, A113453, A113454, A113455, A130296, A131818, A134225.

Cf. A000124, A000217, A000290, A005408, A005843, A008277, A161680, A253909, A267633.

Cf. A024206, A028347, A038889, A038890, A091823, A124199, A168244.

Sequence in context: A312209 A312210 A312211 * A183865 A064609 A327566

Adjacent sequences:  A034853 A034854 A034855 * A034857 A034858 A034859

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Zerinvary Lajos, May 12 2006

STATUS

approved

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Last modified December 1 14:10 EST 2021. Contains 349430 sequences. (Running on oeis4.)