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%I #234 Sep 08 2022 08:44:52
%S 1,4,8,13,19,26,34,43,53,64,76,89,103,118,134,151,169,188,208,229,251,
%T 274,298,323,349,376,404,433,463,494,526,559,593,628,664,701,739,778,
%U 818,859,901,944,988,1033,1079,1126,1174,1223,1273,1324,1376,1429,1483
%N a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.
%C 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).
%C 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
%C 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
%C 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
%C 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
%C a(n) is not divisible by 3, 5, 7, or 11. - _Vladimir Shevelev_, Feb 03 2014
%C 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
%C Also, numbers m such that 8*m+17 is a square. - _Bruno Berselli_, Sep 16 2015
%C _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
%C 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
%C 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
%C From _Klaus Purath_, Dec 07 2020: (Start)
%C 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.
%C 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.
%C 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)
%C 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
%D A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).
%D G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.
%D Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.
%H Vincenzo Librandi, <a href="/A034856/b034856.txt">Table of n, a(n) for n = 1..10000</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=471">Encyclopedia of Combinatorial Structures 471</a>.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic">Two Enumerative Functions</a>.
%H W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http://www.fq.math.ca/Scanned/35-4/klostermeyer.pdf">A Pascal rhombus</a>, Fibonacci Quarterly, Vol. 35, No. 4 (1997), pp. 318-328.
%H Stavros Konstantinidis, António Machiavelo, Nelma Moreira, and Rogério Reis, <a href="https://doi.org/10.1007/s00236-021-00399-6">On the size of partial derivatives and the word membership problem</a>, Acta Informatica Vol. 58, 357-375.
%H G. C. Moisil, <a href="https://doi.org/10.2307/2268145">Recherches sur les logiques non-chrysippiennes</a>, Ann. Sci. Univ. Jassy, 26 (1940), 431-466.
%H László Németh, <a href="https://arxiv.org/abs/1905.13475">Tetrahedron trinomial coefficient transform</a>, arXiv:1905.13475 [math.CO], 2019.
%H D. D. Olesky, B. L. Shader and P. van den Driessche, <a href="https://doi.org/10.37236/1967">Permanents of Hessenberg (0,1)-matrices</a>, Electronic Journal of Combinatorics, 12 (2005), #R70.
%H J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev., Vol. 4, No. 5 (1960), pp. 473-478.
%H Renzo Sprugnoli, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Sprugnoli/sprugnoli6.html">Alternating Weighted Sums of Inverses of Binomial Coefficients</a>, J. Integer Sequences, 15 (2012), #12.6.3. - From _N. J. A. Sloane_, Nov 29 2012
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>
%F G.f.: A(x) = x*(1 + x - x^2)/(1 - x)^3.
%F a(n) = binomial(n+2, 2) - 2. - _Paul Barry_, Feb 27 2003
%F 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
%F Row sums of triangle A131818. - _Gary W. Adamson_, Jul 27 2007
%F Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - _Gary W. Adamson_, Jul 27 2007
%F Row sums of triangle A134225. - _Gary W. Adamson_, Oct 14 2007
%F a(n) = A000217(n+1) - 2. - _Omar E. Pol_, Apr 23 2008
%F From _Jaroslav Krizek_, Sep 05 2009: (Start)
%F a(n) = a(n-1) + n + 1 for n >= 1.
%F a(n) = n*(n-1)/2 + 2*n - 1.
%F a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). (End)
%F a(n) = Hyper2F1([-2, n-1], [1], -1). - _Peter Luschny_, Aug 02 2014
%F a(n) = floor[1/(-1 + Sum_{m >= n+1} 1/S2(m,n+1))], where S2 is A008277. - _Richard R. Forberg_, Jan 17 2015
%F a(n) = A101881(2*(n-1)). - _Reinhard Zumkeller_, Feb 20 2015
%F a(n) = A253909(n+3) - A000217(n+3). - _David Neil McGrath_, May 23 2015
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - _David Neil McGrath_, May 23 2015
%F 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
%F From _Klaus Purath_, Dec 07 2020: (Start)
%F a(n) = A024206(n) + A024206(n+1).
%F a(2*n-1) = -A168244(n+1).
%F a(2*n) = A091823(n). (End)
%F Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - _Amiram Eldar_, Jan 06 2021
%F a(n) + a(n+1) = A028347(n+2). - _R. J. Mathar_, Mar 13 2021
%F a(n) = A000290(n) - A161680(n-1). - _Omar E. Pol_, Mar 26 2021
%F E.g.f.: 1 + exp(x)*(x^2 + 4*x - 2)/2. - _Stefano Spezia_, Jun 05 2021
%F a(n) = A024916(n) - A244049(n). - _Omar E. Pol_, Aug 01 2021
%F a(n) = A000290(n) - A000217(n-2). - _Omar E. Pol_, Aug 05 2021
%e From _Bruno Berselli_, Mar 09 2015: (Start)
%e By the definition (first formula):
%e ----------------------------------------------------------------------
%e 1 4 8 13 19 26
%e ----------------------------------------------------------------------
%e X
%e X X X
%e X X X X X X
%e X X X X X X X X X X
%e X X X X X X X X X X X X X X X
%e X X X X X X X X X X X X X X X X X X X X X
%e X X X X X X X X X X X X X X X
%e ----------------------------------------------------------------------
%e (End)
%e From _Klaus Purath_, Dec 07 2020: (Start)
%e 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.
%e (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.
%e (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.
%e (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)
%e From _Omar E. Pol_, Aug 08 2021: (Start)
%e Illustration of initial terms: _ _
%e . _ _ |_|_|_
%e . _ _ |_|_|_ |_|_|_|_
%e . _ _ |_|_|_ |_|_|_|_ |_|_|_|_|_
%e . _ _ |_|_|_ |_|_|_|_ |_|_|_|_|_ |_|_|_|_|_|_
%e . _ |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
%e . |_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
%e .
%e . 1 4 8 13 19 26
%e ------------------------------------------------------------------------ (End)
%p a := n -> hypergeom([-2, n-1], [1], -1);
%p seq(simplify(a(n)), n=1..53); # _Peter Luschny_, Aug 02 2014
%t 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 *)
%t Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)
%t Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* _Michael De Vlieger_, Aug 29 2016 *)
%o (Magma) [Binomial(n + 1, 2) + n - 1: n in [1..60]]; // _Vincenzo Librandi_, May 21 2011
%o (Maxima) A034856(n) := block(
%o n-1+(n+1)*n/2
%o )$ /* _R. J. Mathar_, Mar 19 2012 */
%o (PARI) A034856(n)=(n+3)*n\2-1 \\ _M. F. Hasler_, Jan 21 2015
%o (Haskell)
%o a034856 = subtract 1 . a000096 -- _Reinhard Zumkeller_, Feb 20 2015
%Y Subsequence of A165157.
%Y Triangular numbers (A000217) minus two. a(n) = T(3, n-2), array T as in A049600.
%Y Third diagonal of triangle in A059317.
%Y Cf. A000096, A027379, A101881, A113452, A113453, A113454, A113455, A130296, A131818, A134225.
%Y Cf. A000124, A000217, A000290, A005408, A005843, A008277, A161680, A253909, A267633.
%Y Cf. A024206, A028347, A038889, A038890, A091823, A124199, A168244.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Zerinvary Lajos_, May 12 2006
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