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A336974
Sum of the largest two side lengths of all distinct integer-sided triangles with perimeter n.
0
0, 0, 2, 0, 4, 4, 11, 6, 21, 15, 35, 27, 52, 43, 83, 62, 109, 97, 152, 125, 201, 172, 258, 225, 321, 286, 410, 353, 489, 448, 597, 531, 714, 645, 843, 768, 981, 903, 1157, 1047, 1318, 1231, 1520, 1398, 1734, 1608, 1964, 1830, 2206, 2068, 2498, 2318, 2770, 2620, 3095, 2900, 3435, 3235
OFFSET
1,3
FORMULA
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (n - k).
Conjectures from Colin Barker, Aug 10 2020: (Start)
G.f.: x^3*(2 + 2*x + 4*x^2 + 4*x^3 + 7*x^4 + 5*x^5 + 5*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
a(3) = 2; There is one integer-sided triangle with perimeter 3, [1,1,1]. The sum of the largest two side lengths is 1 + 1 = 2.
a(7) = 11; There are two distinct integer-sided triangles with perimeter 7, [1,3,3] and [2,2,3]. The sum of the largest two side lengths of these triangles is 3 + 3 + 2 + 3 = 11.
MATHEMATICA
Table[Sum[Sum[(n - k)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]
CROSSREFS
Cf. A005044.
Sequence in context: A021493 A195395 A296805 * A084247 A300307 A286606
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 09 2020
STATUS
approved