A224931 -id:A224931 - cp's OEIS frontend

A340858 a(n) is the number of integer trapezoids (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

0, 0, 0, 1, 1, 1, 2, 5, 6, 3, 4, 9, 9, 7, 10, 22, 10, 10, 9, 22, 18, 14, 14, 46, 26, 21, 35, 38, 18, 31, 20, 66, 45, 22, 43, 57, 25, 25, 48, 82, 27, 46, 35, 70, 69, 43, 34, 136, 63, 57, 72, 90, 46, 76, 80, 143, 91, 42, 46, 149, 54, 47, 115, 204, 105

Offset: 1

Herbert Kociemba, Jan 24 2021

nonn

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
Without loss of generality we assume for the parallel sides c < a and for the diagonals f <= e. e and f are uniquely determined by e = sqrt((c(a^2-b^2) + a(d^2-c^2))/(a-c)) and f = sqrt((c(a^2-d^2) + a(b^2-c^2))/(a-c)).
The smallest possible trapezoid has side lengths a=4, c=3, b=d=2 and diagonals e=f=4. The smallest possible trapezoid which is not isosceles has side lengths a=8, b=9, c=3, d=11 and diagonals e=13 and f=9.

			a(7)=2 because there are two possible trapezoids: a=5, c=3, b=d=7, e=f=8 and a=7, c=4, b=d=6, e=f=8.

Cf. A224931 for parallelograms, A340859 and A340860 for isosceles and non-isosceles trapezoids.

n=65;list={};
For[a=1,a<=n,a++,
For[c=1,cse,Break[]];If[sf<=0,Continue[]];
e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
If[IntegerQ[e]&&IntegerQ[f]&&a+d>f&&d+f>a&&f+a>d&&e+b>a&&b+a>e&&a+e>b,AppendTo[list,{a,b,c,d,e,f}]]]]]]
Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&]//Length,{n,1,65}]

A340859 a(n) is the number of isosceles integer trapezoids (up to congruence) with integer side lengths a,c,b=d with n=Max(a,b,c) and integer diagonals e=f.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 6, 3, 3, 9, 6, 5, 10, 20, 9, 10, 8, 21, 18, 10, 10, 37, 21, 12, 24, 31, 14, 26, 17, 55, 32, 20, 36, 54, 22, 20, 39, 74, 24, 40, 26, 58, 59, 24, 26, 113, 47, 41, 54, 69, 33, 51, 61, 111, 65, 35, 39, 124, 38, 39, 88, 145, 79

Offset: 1

Herbert Kociemba, Jan 24 2021

nonn

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.

Without loss of generality we assume b=d and for the parallel sides c < a. e and f are uniquely determined by e = f = sqrt((c(a^2-b^2) + a(b^2-c^2))/(a-c)). The smallest possible isosceles trapezoid has side lengths a=4, c=3, b=d=2 and diagonals e=f=4.

			a(7)=2 because there are two possible trapezoids: a=5, c=3, b=d=7, e=f=8 and a=7, c=4, b=d=6, e=f=8.

Cf. A224931 for parallelograms, A340858 for general trapezoids and A340860 for non-isosceles trapezoids.

n=65;list={};
For[a=1,a<=n,a++,
For[c=1,cse,Break[]];If[sf<=0,Continue[]];
e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
If[IntegerQ[e]&&IntegerQ[f]&&a+d>f&&d+f>a&&f+a>d&&e+b>a&&b+a>e&&a+e>b,AppendTo[list,{a,b,c,d,e,f}]]]]]]
Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&&#[[2]]==#[[4]]&]//Length,{n,1,65}]

A340860 a(n) is the number of non-isosceles integer trapezoids (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 2, 1, 0, 1, 1, 0, 4, 4, 9, 5, 9, 11, 7, 4, 5, 3, 11, 13, 2, 7, 3, 3, 5, 9, 8, 3, 6, 9, 12, 10, 19, 8, 23, 16, 16, 18, 21, 13, 25, 19, 32, 26, 7, 7, 25, 16, 8, 27, 59, 26

Offset: 1

Herbert Kociemba, Jan 24 2021

nonn

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
Without loss of generality we assume for the parallel sides c < a and for the diagonals f < e. e and f are uniquely determined by e = sqrt((c(a^2-b^2) + a(d^2-c^2))/(a-c)) and f = sqrt((c(a^2-d^2) + a(b^2-c^2))/(a-c)).
The smallest possible trapezoid which is not isosceles has side lengths a=8, b=9, c=3, d=11 and diagonals e=13 and f=9.

			a(34)=2 because up to congruence there are exactly two trapezoids which are not isosceles:
a=32, b=26, c=22, d=34 and e=54, f=18;
a=34, b=11, c=32, d=12 and e=40, f=29.

Cf. A224931 for parallelograms, A340858 for general trapezoids and A340859 for isosceles trapezoids.

n=65;list={};
For[a=1,a<=n,a++,
For[c=1,cse,Break[]];If[sf<=0,Continue[]];
e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
If[IntegerQ[e]&&IntegerQ[f]&&a+d>f&&d+f>a&&f+a>d&&e+b>a&&b+a>e&&a+e>b,AppendTo[list,{a,b,c,d,e,f}]]]]]]
Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&&#[[2]]!=#[[4]]&]//Length,{n,1,65}]

A365049 a(n) is the number of distinct parallelograms with integer sides and area n, and where at least one height is an integer.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 2, 5, 5, 4, 2, 10, 2, 4, 8, 9, 2, 9, 2, 10, 8, 4, 2, 20, 5, 4, 8, 10, 2, 16, 2, 13, 8, 4, 8, 23, 2, 4, 8, 20, 2, 16, 2, 10, 18, 4, 2, 34, 5, 9, 8, 10, 2, 16, 8, 20, 8, 4, 2, 40, 2, 4, 18, 19, 8, 16, 2, 10, 8, 16, 2, 45, 2, 4, 18, 10, 8, 16, 2, 34, 13

Offset: 1

Felix Huber, Aug 18 2023

nonn

If n is not a square, there are A000005(n)/2 rectangles with A027750(n,i)*A027750(n,j) = n, i < j. If n is a square, there are (A000005(n)-1)/2 rectangles with A027750(n,i)*A027750(n,j) = n, i < j and a square with A027750(n,(A000005(n)+1)/2)^2 = n. From these rectangles and, if present, the square, further parallelograms of equal area and integer sides can be formed. A046079(A027750(n,k)) is the number of possibilities there are for each side of the rectangle or for the side of the square.

			For area n = 9 there is one rectangle (sides of lengths: 1,9) and a square (3,3) with integer sides. From both, further parallelograms with area n = 9 and integer sides can be formed. Since (9,12,15) and (9,40,41) are the only Pythagorean triples with leg 9, from the rectangle (1,9) exactly the two further parallelograms (1,15) and (1,41) with height 9 can be formed, but no further parallelogram with height 1. Since (3,4,5) is the only Pythagorean triple with leg 3, from the square (3,3) exactly one further parallelogram (3,5) with height 3 can be formed. Therefore for area n = 9 there are a(9) = 5 distinct parallelograms with integer sides.

F. Richman, Pythagorean triples.
Wikipedia, Parallelogram.
Wikipedia, Table of divisors.

Cf. A000005, A027750, A046079, A224931, A214602.

from math import prod
from itertools import takewhile
from sympy import factorint, divisors
def A365049(n): return sum(1+(prod((e+(p&1)<<1)-1 for p, e in factorint(d).items())>>1)+(prod((e+(p&1)<<1)-1 for p, e in factorint(n//d).items())>>1 if d*dChai Wah Wu, Aug 21 2023

If n is a square, then a(n) = 1 + A046079(A027750(n, (A000005(n) + 1)/2)) + Sum_{i = 1..(A000005(n) - 1)/2} (1 + A046079(A027750(n,i)) + A046079(n/A027750(n,i)));

otherwise, a(n) = Sum_{i = 1..A000005(n)/2} (1 + A046079(A027750(n,i)) + A046079(n/A027750(n,i))).

A224933 Minimal side length b <= a = n of integer parallelograms.

Original entry on oeis.org

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

Offset: 1

Reiner Moewald, Apr 20 2013

nonn

Reiner Moewald, Table of n, a(n) for n = 1..100

Cf. A224931, A224932.

cp's OEIS Frontend

A340858 a(n) is the number of integer trapezoids (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

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A340859 a(n) is the number of isosceles integer trapezoids (up to congruence) with integer side lengths a,c,b=d with n=Max(a,b,c) and integer diagonals e=f.

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A340860 a(n) is the number of non-isosceles integer trapezoids (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

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A365049 a(n) is the number of distinct parallelograms with integer sides and area n, and where at least one height is an integer.

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A224933 Minimal side length b <= a = n of integer parallelograms.

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