A059169 Number of partitions of n into 3 parts which form the sides of a nondegenerate isosceles triangle.

0, 0, 1, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 14, 15, 14, 15, 15, 16, 15, 16, 16, 17, 16, 17, 17, 18, 17, 18, 18, 19, 18, 19, 19, 20, 19, 20, 20

Offset: 1

Author

Floor van Lamoen, Jan 13 2001

Keywords

• nonn
• easy

Comments

Also number of 0's in n-th row of triangle in A071026. - Hans Havermann, May 26 2002

Exponent of 2 in factorization of A030436(n-1) and A026655(n-1). First differences of A001971. - Ralf Stephan, Mar 21 2004

Conjecture: this is 0 followed by A026922. - R. J. Mathar, Oct 05 2008 [See the g.f. given there by Michael Somos and the one given below for the proof. - Wolfdieter Lang, May 10 2017]

a(n+1) is for n >= 0 the number of integers k in the left-sided open interval ((n+1)/4, floor(n/2)]. This is needed for the number of zeros of Chebyshev S polynomials in the open interval (-sqrt(2), sqrt(2)) given in A285869. - Wolfdieter Lang, May 10 2017

Examples

			Consider the number 13. The following partitions give a nondegenerate triangle: 4 4 5; 3 5 5; 1 6 6; 2 5 6; 3 4 6. Since the first three partitions represent isosceles triangles, we have A059169(13) = 3.
G.f. = x^3 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 2*x^12 + ...
		

Links

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

Crossrefs

Essentially the same as A008624.

Cf. A178804.

Programs

Haskell

a059169 n = a059169_list !! (n-1)
a059169_list = map abs $ zipWith (-) (tail a178804_list) a178804_list
-- Reinhard Zumkeller, Nov 15 2014

Magma

[Floor((n-1)/2) - Floor(n/4): n in [1..80]]; // G. C. Greubel, Mar 08 2018

Maple

a[1] := 0: a[2] := 0: a[3] := 1: a[4] := 0: a[5] := 1: for n from 6 to 300 do a[n] := a[n-1] + a[n-4] - a[n-5]: end do: seq(a[n], n=1..82);
a := n -> A005044(n) - A005044(n-6): A005044 := n-> floor((1/48)*(n^2 + 3*n + 21 + (-1)^(n-1)*3*n)): seq(a(n), n = 1..82); # Johannes W. Meijer, Oct 10 2013

Mathematica

CoefficientList[Series[x^2 (1 - x + x^2)/(1 - x - x^4 + x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{1,0,0,1,-1},{0,0,1,0,1},100] (* Harvey P. Dale, Feb 09 2015 *)
a[ n_] := Quotient[ n - 1, 2] - Quotient[ n, 4]; (* Michael Somos, May 05 2015 *)

PARI

{a(n) = (n - 1) \ 2 - (n \ 4)}; /* Michael Somos, Oct 14 2008 */

PARI

{a(n) = if( n<1, -a(3 - n), polcoeff( x^3 * (1 - x + x^2) / (1 - x - x^4 + x^5) + x * O(x^n), n))}; /* Michael Somos, Oct 14 2008 */

Formula

a(2*n + 2) = a(2*n - 1) = A004526(n).

a(n) = A005044(n) - A005044(n-6).

From Vladeta Jovovic, Dec 29 2001: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5).

G.f.: x^3*(1 - x + x^2)/(1 - x - x^4 + x^5). (End)

The g.f. can also be written as x^3 * (1 + x^3) / ((1 - x^2) * (1 - x^4)). - Michael Somos, May 05 2015

Euler transform of length 6 sequence [0, 1, 1, 1, 0, -1]. - Michael Somos, Oct 14 2008

a(n) = -a(3 - n) for all n in Z. - Michael Somos. Oct 14 2008

a(n) = abs(floor((n-1)*(-1)^n/4)). - Wesley Ivan Hurt, Oct 22 2013

a(n) = abs(A178804(n+1) - A178804(n)). - Reinhard Zumkeller, Nov 15 2014

a(n) = floor(n/2) - floor(n/4) - (1 if n even). - David Pasino, Jun 17 2016

E.g.f.: (4 - sin(x) - cos(x) + x*sinh(x) + (x - 3)*cosh(x))/4. - Ilya Gutkovskiy, Jun 21 2016

a(n) = floor((n-1)/2) - floor(n/4), n >= 0 (from the preceding a(n) formula). - Wolfdieter Lang, May 08 2017

a(n) = (2*n - 3 - 2*cos(n*Pi/2) - 3*cos(n*Pi) - 2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 01 2017

a(n) = Sum_{i=1..floor((n-1)/2)} (n-i-1) mod 2. - Wesley Ivan Hurt, Nov 17 2017

Extensions

More terms from Sascha Kurz, Mar 25 2002