%I A034856
%S A034856 1,4,8,13,19,26,34,43,53,64,76,89,103,118,134,151,169,188,208,229,251,
%T A034856 274,298,323,349,376,404,433,463,494,526,559,593,628,664,701,739,778,
%U A034856 818,859,901,944,988,1033,1079,1126,1174,1223,1273,1324,1376,1429,1483
%N A034856 C(n + 1, 2) + n - 1.
%C A034856 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 A034856 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-subests of X intersecting 
               both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Oct 03 2007
%C A034856 Subsequence of A165157. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Sep 05 2009]
%D A034856 D. D. Olesky, B. L. Shader and P. van den Driessche, Permanents of Hessenberg 
               (0,1)-matrices, Electronic Journal of Combinatorics, 12 (2005) #R70.
%D A034856 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. 
               Dev. 4 (1960), 473-478.
%H A034856 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A034856 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A034856 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=471">
               Encyclopedia of Combinatorial Structures 471</a>
%H A034856 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http:/
               /www.math.wvu.edu/~mays/Papers/apd2.ps">A Pascal rhombus</a>, Fibonacci 
               Quarterly, 35 (1997), 318-328.
%F A034856 G.f.: A(x) = x*(1+x-x^2)/(1-x)^3.
%F A034856 With offset 0, this is C(n+3, 2)-2 = (n^2+5n+2)/2 - Paul Barry (pbarry(AT)wit.ie), 
               Feb 27 2003
%F A034856 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 (pbarry(AT)wit.ie), Jul 
               01 2003
%F A034856 Row sums of triangle A131818 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 27 2007
%F A034856 Binomial transform of (1, 3, 1, 0, 0, 0,...). Also equals A130296 * [1,
               2,3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007
%F A034856 Row sums of triangle A134225 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 14 2007
%F A034856 a(n) = A000217(n+1) - 2. - Omar E. Pol (info(AT)polprimos.com), Apr 23 
               2008
%F A034856 a(n) = ((n(n+3)+2)/2)-2. - Omar E. Pol (info(AT)polprimos.com), May 18 
               2008
%F A034856 If F(1)=4, F(2)=8, F(n)=2*F(n-1)-F(n-2)+1; [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Jan 22 2009]
%F A034856 a(1) = 1, a(n) = a(n-1) + n + 1 for n >= 1. a(n) = 1/2 * n * (n-1) + 
               (2n - 1). a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). 
               [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 05 2009]
%F A034856 a(n)=n+a(n-1)+1 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 07 2009]
%e A034856 F(3)=28-4+1=13; F(4)=19; F(5)=26 and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Jan 22 2009]
%e A034856 For n=2, a(2)=2+1+1=4; n=3, a(3)=3+4+1=8; n=4, a(4)=4+8+1=13 [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
%p A034856 a:=array(1...53): a[1]:=1: print(1,a[1]); for i from 2 to 53 do a[i]:= 
               a[i-1]+(binomial(i+1,i)):print(i,a[i]); od: - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 27 2007
%p A034856 with (combinat):a:=n->sum(fibonacci(2,i), i=0..n):seq(a(n)-2, n=2..54); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2008
%p A034856 a:=n->sum(k, k=2..n):seq(a(n)/2+sum(k, k=3..n)/2, n=2..54); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2008
%t A034856 i=2; s=-1; lst={}; Do[s+=n+i; If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 
               1}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 29 
               2008]
%t A034856 s = 1; lst = {s}; Do[s += n; AppendTo[lst, s], {n, 3, 54, 1}]; lst [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
%Y A034856 Triangular numbers (A000217) minus two. a(n)=T(3, n-2), array T as in 
               A049600.
%Y A034856 Cf. A000096, A027379. Third diagonal of triangle in A059317.
%Y A034856 Cf. A113452-A113455.
%Y A034856 Cf. A130296, A131818.
%Y A034856 Cf. A134225.
%Y A034856 Sequence in context: A004081 A127264 A130236 this_sequence A064609 A056738 
               A143978
%Y A034856 Adjacent sequences: A034853 A034854 A034855 this_sequence A034857 A034858 
               A034859
%K A034856 nonn
%O A034856 1,2
%A A034856 N. J. A. Sloane (njas(AT)research.att.com).
%E A034856 More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 
               2006