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A034856
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a(n) = binomial(n+1, 2) + n - 1 = n(n + 3)/2 - 1.
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52
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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
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OFFSET
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1,2
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COMMENTS
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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 Lukasiewicz logics. See the Wojcicki 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
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REFERENCES
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Karpenko, A. S., Lukasiewicz's Logics and Prime Numbers, 2006 (English translation). [From Artur Jasinski, Mar 01 2010]
G. C. Moisil, Recherches sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 26, 1940, 431-466. [From Artur Jasinski, Mar 01 2010]
G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972. [From Artur Jasinski, Mar 01 2010]
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. 4 (1960), 473-478.
R. 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
Wojcicki and Malinowski, eds., Lukasiewicz Sentential Calculi, Wroclaw: Ossolineum, 1977.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Guo-Niu Han, Enumeration of Standard Puzzles
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, 35 (1997), 318-328.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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G.f.: A(x) = x*(1+x-x^2)/(1-x)^3.
With offset 0, this is C(n+3, 2)-2 = (n^2+5n+2)/2 - Paul Barry, Feb 27 2003
With offset 5, this is C(n, 0)-2C(n, 1)+C(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
a(n) = (n(n+3)+2)/2-2. - Omar E. Pol, May 18 2008
a(n) = a(n-1) + n + 1 for n >= 1. a(n) = n * (n-1)/2 + 2n - 1. a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). [From Jaroslav Krizek, Sep 05 2009]
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MATHEMATICA
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f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] [From Robert G. Wilson v, Jun 11 2010]
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PROG
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(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 */
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CROSSREFS
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Subsequence of A165157. [From Jaroslav Krizek, Sep 05 2009]
Triangular numbers (A000217) minus two. a(n)=T(3, n-2), array T as in A049600.
Cf. A000096, A027379. Third diagonal of triangle in A059317.
Cf. A113452-A113455.
Cf. A130296, A131818, A134225.
Sequence in context: A206814 A130236 A198464 * A183865 A064609 A056738
Adjacent sequences: A034853 A034854 A034855 * A034857 A034858 A034859
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006
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STATUS
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approved
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