A107572 -id:A107572 - cp's OEIS frontend

A107576 a(n)=perimeter of n-th triangle listed at A107572.

9, 11, 12, 13, 13, 14, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26

Offset: 1

Clark Kimberling, May 16 2005

nonn

The number of perimeters equal to n+6 is A005044(n).

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the perimeters are 9,11,12,13,13.

Cf. A107572, A107573, A107574, A107575, A107577, A005044, A070080.

A107577 a(n)=greatest integer p such that a^p + b^p > c^p, where (a,b,c) is the n-th integer-sided triangle listed at A107572.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first such triangle is (2,3,4), for which 2^1 + 3^1 > 4^1 but 2^2 + 3^2 <= 4^2, so a(1)=1.

Cf. A107572.

A107573 a(n)=least sidelength of n-th triangle listed at A107572.

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 2, 3, 4, 3, 4, 2, 3, 4, 4, 3, 4, 5, 2, 3, 4, 4, 5, 3, 4, 5, 5, 2, 3, 4, 4, 5, 5, 6, 3, 4, 5, 5, 6, 2, 3, 4, 4, 5, 5, 6, 6, 3, 4, 5, 5, 6, 6, 7, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 3, 4, 5, 5, 6, 6, 7, 7, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 3, 4, 5, 5, 6, 6, 7, 7, 7, 8, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the least sidelengths are 2,2,3,2,3

Cf. A107572, A107574, A107575, A107576, A107577, A005044, A070080.

A107574 a(n)=middle sidelength of n-th triangle listed at A107572.

Original entry on oeis.org

3, 4, 4, 5, 4, 5, 6, 5, 5, 6, 5, 7, 6, 5, 6, 7, 6, 6, 8, 7, 6, 7, 6, 8, 7, 6, 7, 9, 8, 7, 8, 6, 7, 7, 9, 8, 7, 8, 7, 10, 9, 8, 9, 7, 8, 7, 8, 10, 9, 8, 9, 7, 8, 8, 11, 10, 9, 10, 8, 9, 7, 8, 9, 8, 11, 10, 9, 10, 8, 9, 8, 9, 12, 11, 10, 11, 9, 10, 8, 9, 10, 8, 9, 9, 12, 11, 10, 11, 9, 10, 8, 9, 10, 9, 13

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the middle sidelengths are 3,4,4,5,4.

Cf. A107572, A107573, A107575, A107576, A107577, A005044, A070080.

A107575 a(n)=greatest sidelength of n-th triangle listed at A107572.

Original entry on oeis.org

4, 5, 5, 6, 6, 6, 7, 7, 6, 7, 7, 8, 8, 8, 7, 8, 8, 7, 9, 9, 9, 8, 8, 9, 9, 9, 8, 10, 10, 10, 9, 10, 9, 8, 10, 10, 10, 9, 9, 11, 11, 11, 10, 11, 10, 10, 9, 11, 11, 11, 10, 11, 10, 9, 12, 12, 12, 11, 12, 11, 12, 11, 10, 10, 12, 12, 12, 11, 12, 11, 11, 10, 13, 13, 13, 12, 13, 12, 13, 12, 11

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the greatest sidelengths are 4,5,5,6,6.

Cf. A107572, A107573, A107574, A107576, A107577, A005044, A070080.

A370408 Lexicographically earliest sequence of positive integers such that no three equal terms appear at distinct indices that are the side lengths of a triangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 4, 4, 1, 5, 5, 2, 3, 6, 6, 7, 1, 7, 4, 8, 8, 2, 3, 9, 5, 9, 10, 10, 11, 1, 4, 11, 6, 12, 12, 13, 13, 2, 7, 3, 5, 14, 14, 15, 8, 15, 16, 16, 17, 17, 1, 6, 18, 4, 9, 18, 19, 19, 10, 20, 7, 20, 21, 2, 11, 21, 3, 22, 22, 5, 8, 23, 12, 23, 24, 24, 13, 25, 25, 26, 26, 27, 27, 28, 1, 9, 28, 29, 4

Offset: 1

Neal Gersh Tolunsky, Feb 17 2024

nonn

In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z.
So if x < y and a(x) = a(y) = t then we cannot have a(z) = t for any z in the range y < z < x+y.
Another way to construct the sequence: Place 1's at the earliest permitted positions (in this case, at Fibonacci indices). Each subsequent value (2’s, 3’s, etc.) is placed at the earliest permitted indices not already occupied by a smaller value. For example, 3's could be placed in a Fibonacci pattern beginning with 7, 9 (7, 9, 16, 25, etc.), but i=7+9=16 is already occupied by the value 2, so 3 gets the next smallest position i=17. i=9+17=26 is again occupied by a 2, so we give 3 the next smallest unoccupied position i=27.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A367196, A107572 (triangle side lengths), A100480.

list={1};Do[k=1;While[lst=Join[list,{k}];!And@@(And@@(({a,b,c}=#;(-a+b+c)(a-b+c)(a+b-c))<=0&/@Subsets[Flatten[Position[lst,#]],{3}])&/@Union@lst),k++];AppendTo[list,k],{n,92}];list (* Giorgos Kalogeropoulos, Feb 20 2024 *)

from itertools import combinations as C, count, islice
def agen(): # generator of terms
    yield from [1, 1, 1]
    sides = {1: [1, 2, 3]}
    for n in count(4):
        an = next(an for an in count(1) if an not in sides or all(not all((nMichael S. Branicky, Feb 24 2024

More terms from Giorgos Kalogeropoulos, Feb 20 2024

A263772 Perimeters of integer-sided scalene triangles.

Original entry on oeis.org

9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Rick L. Shepherd, Oct 27 2015

All natural numbers larger than 8 except 10.
Equivalently, numbers n that can be partitioned into three distinct parts a, b, and c, where a + b > c, a + c > b, and b + c > a (or, without loss of generality, into (a, b, c) with a < b < c < a + b). A subsequence of A009005. The unique terms in A107576.
For k > 2, (k-1, k, k+1) gives perimeter 3k and (k-1, k+1, k+2) gives perimeter 3k + 2. For k > 3, the scalene triangle (k-1, k, k+2) has perimeter 3k + 1.

			The integer-sided scalene triangle of least perimeter has sides of lengths 2, 3, and 4, so a(1) = 2 + 3 + 4 = 9.

Cf. A005044, A009005, A107572, A107576.

vector(100, n, if(n==1, 9, n+9)) \\ Altug Alkan, Oct 28 2015

a(n) = n + 9 for n > 1.

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A107576 a(n)=perimeter of n-th triangle listed at A107572.

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A107577 a(n)=greatest integer p such that a^p + b^p > c^p, where (a,b,c) is the n-th integer-sided triangle listed at A107572.

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A107573 a(n)=least sidelength of n-th triangle listed at A107572.

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A107574 a(n)=middle sidelength of n-th triangle listed at A107572.

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A107575 a(n)=greatest sidelength of n-th triangle listed at A107572.

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A370408 Lexicographically earliest sequence of positive integers such that no three equal terms appear at distinct indices that are the side lengths of a triangle.

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A263772 Perimeters of integer-sided scalene triangles.

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