A211539 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w = 2n - 2x + y.

0, 0, 2, 3, 7, 9, 15, 18, 26, 30, 40, 45, 57, 63, 77, 84, 100, 108, 126, 135, 155, 165, 187, 198, 222, 234, 260, 273, 301, 315, 345, 360, 392, 408, 442, 459, 495, 513, 551, 570, 610, 630, 672, 693, 737, 759, 805, 828, 876, 900, 950, 975, 1027, 1053

Offset: 0

Author

Clark Kimberling, Apr 15 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

a(n) = sum of natural numbers in interval (floor((n+1)/2),n]. - Jaroslav Krizek, Mar 05 2014

For n > 0, 2*a(n-1) is the sum of the largest parts of the partitions of 2n into two distinct even parts. - Wesley Ivan Hurt, Dec 19 2017

From Paul Curtz, Oct 23 2018: (Start)

Consider the 51 first nonnegative numbers in the following boustrophedon distribution:

35--36--37--38--39--40--41--42--43--44--45

34--33--32--31--30--29--28--27--26--46

12--13--14--15--16--17--18--25--47

11--10---9---8---7--19--24--48

1---2---3---6--20--23--49

0---4---5--21--22--50

a(n+1) is the union of the main vertical (0,2, 9,15, 30,40, ... ) and of the shifted main antidiagonal (3,7, 18,26, 45,57, ... ). (End)

Sum of the shortest side lengths of all integer-sided triangles with perimeter 3(n+1) whose sides lengths are in arithmetic progression (For example, when n=4 there are two triangles with perimeter 3(4+1) = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(4) = 3+4 = 7). - Wesley Ivan Hurt, Nov 01 2020

Examples

			G.f. = 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 15*x^6 + 18*x^7 + ... - _Michael Somos_, Nov 14 2018
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A211422.

Cf. A000217, A001318, A005449, A045943, A045944, A065423, A211538.

Programs

GAP

a:=[0];; for n in [2..55] do if n mod 2 = 0 then Add(a,a[n-1]+n); else Add(a,a[n-1]+(n-1)/2); fi; od; Concatenation([0],a); # Muniru A Asiru, Oct 26 2018

Magma

I:=[0,0,2,3,7]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Mar 12 2014

Maple

a:=n->add(k,k=floor((n+1)/2)+1..n): seq(a(n),n=0..55); # Muniru A Asiru, Oct 26 2018

Mathematica

t[n_] := t[n] = Flatten[Table[2 w + 2 x - y - 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211539 *)
FindLinearRecurrence[t]
CoefficientList[Series[(x^3 + 2 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 12 2014 *)

PARI

a(n)=(1/16)*(6*n^2+2*n-3+(2*n+3)*(-1)^n) \\ Ralf Stephan, Mar 10 2014

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

From Jaroslav Krizek, Mar 05 2014: (Start)

a(n) = T(n+1) - T(floor((n+1)/2)) - (n+1), where T(k) = A000217(k).

a(n) = Sum_{k=floor((n+1)/2)+1..n} k.

a(n) = a(n-1) + n for even n; a(n) = a(n-1) + (n-1)/2 for odd n. (End)

From Ralf Stephan, Mar 10 2014: (Start)

a(n) = (1/16) * (6n^2 + 2n - 3 + (2n+3)*(-1)^n ).

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

From Paul Curtz, Oct 22 2018: (Start)

a(2n) = A005449(n), a(2n+1) = A045943(n).

a(2n) + a(2n+1) = A045944(n).

a(3n) = 3*(0, 1, 5, 10, 19, 28, 42, ...).

a(n+1) = a(n) + A065423(n+2).

a(-n) = A211538(n+2). (End)

E.g.f.: (3*x*(1 + x)*cosh(x) + (-3 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Nov 02 2020

a(n) = A001318(n+1) - (n+1). - Davide Rotondo, Apr 07 2024