1,2

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-subests of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Oct 03 2007

Subsequence of A165157. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 05 2009]

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.

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 471

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.

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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Jul 01 2003

Row sums of triangle A131818 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007

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

Row sums of triangle A134225 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 14 2007

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

a(n) = ((n(n+3)+2)/2)-2. - Omar E. Pol (info(AT)polprimos.com), May 18 2008

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]

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]

a(n)=n+a(n-1)+1 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]

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]

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]

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

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

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

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]

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]

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.

Cf. A134225.

Sequence in context: A004081 A127264 A130236 this_sequence A064609 A056738 A143978

Adjacent sequences: A034853 A034854 A034855 this_sequence A034857 A034858 A034859

nonn,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006