A308089 Sum of the perimeters of all integer-sided triangles with perimeter n.

0, 0, 3, 0, 5, 6, 14, 8, 27, 20, 44, 36, 65, 56, 105, 80, 136, 126, 190, 160, 252, 220, 322, 288, 400, 364, 513, 448, 609, 570, 744, 672, 891, 816, 1050, 972, 1221, 1140, 1443, 1320, 1640, 1554, 1892, 1760, 2160, 2024, 2444, 2304, 2744, 2600, 3111, 2912

Offset

1,3

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Wikipedia, Integer Triangle

Index entries for linear recurrences with constant coefficients, signature (-1,0,2,4,2,-1,-5,-5,-1,2,4,2,0,-1,-1).

Formula

a(n) = n * A005044(n).

From Colin Barker, May 13 2019: (Start)

G.f.: x^3*(3 + 3*x + 5*x^2 + 5*x^3 + 8*x^4 + 6*x^5 + 6*x^6) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

a(n) = -a(n-1) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 5*a(n-7) - 5*a(n-8) - a(n-9) + 2*a(n-10) + 4*a(n-11) + 2*a(n-12) - a(n-14) - a(n-15) for n>15.

(End)

Example

From Wesley Ivan Hurt, Aug 10 2020: (Start)
a(3) = 3; There is one integer-sided triangle with perimeter 3, [1,1,1]. Since there is only one perimeter, the sum is 3.
a(7) = 14; There are two integer-sided triangles with perimeter 7, [1,3,3] and [2,2,3]. The sum of the perimeters is 7+7 = 14. (End)

Mathematica

Table[n*Sum[Sum[Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
LinearRecurrence[{-1, 0, 2, 4, 2, -1, -5, -5, -1, 2, 4, 2, 0, -1, -1}, {0, 0, 3, 0, 5, 6, 14, 8, 27, 20, 44, 36, 65, 56, 105}, 60] (* Harvey P. Dale, Apr 17 2022 *)

Prog

(PARI) concat([0, 0], Vec(x^3*(3 + 3*x + 5*x^2 + 5*x^3 + 8*x^4 + 6*x^5 + 6*x^6) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^60))) \\ Colin Barker, May 13 2019

Crossrefs

Cf. A005044.

Sequence in context: A052483 A308116 A308123 * A249859 A213724 A350082

Adjacent sequences: A308086 A308087 A308088 * A308090 A308091 A308092

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 12 2019

Status

approved