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A059169
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Number of partitions of n into 3 parts which form the sides of a nondegenerate isosceles triangle.
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23
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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
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OFFSET
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1,7
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COMMENTS
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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
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LINKS
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FORMULA
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a(2*n + 2) = a(2*n - 1) = A004526(n).
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) = 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
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EXAMPLE
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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 + ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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 */
(Haskell)
a059169 n = a059169_list !! (n-1)
a059169_list = map abs $ zipWith (-) (tail a178804_list) a178804_list
(Magma) [Floor((n-1)/2) - Floor(n/4): n in [1..80]]; // G. C. Greubel, Mar 08 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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