A282617 -id:A282617

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A282615

Number of self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

+10
10

0, 1, 1, 3, 4, 9, 20, 35, 102, 160, 736, 930, 5972, 6766, 59017, 61814, 671651 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	COMMENTS	`An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).` `A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.` `\| separable \| inseparable \| either \|` `-------------------+-----------+-------------+---------+` `self-conjugate \| A282615 \| A279197 \| A282616 \|` `non-self-conjugate \| A282618 \| A282617 \| A282619 \|` `either \| A279199 \| A202705 \| A104429 \|`
	LINKS	`Table of n, a(n) for n=1..17.`
	FORMULA	`a(n) = A282616(n) - A279197(n).` `a(n) = A279199(n) - A282618(n).`
	EXAMPLE	`For n = 4 the a(4) = 3 solutions are:` `(10,12,11),(7,9,8),(4,6,5),(1,3,2),` `(10,12,11),(5,9,7),(4,8,6),(1,3,2), and` `(8,12,10),(7,11,9),(2,6,4),(1,5,3).`
	CROSSREFS	`Cf. A104429, A202705, A279197, A279199, A282616, A282617, A282618, A282619.` `All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.`
	KEYWORD	`nonn,more`
	AUTHOR	`Peter Kagey, Feb 19 2017`
	EXTENSIONS	`a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017` `a(17) from Fausto A. C. Cariboni, Mar 22 2017`
	STATUS	`approved`

A282616

Number of self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

+10
5

1, 2, 3, 5, 15, 20, 75, 93, 588, 602, 4954, 4854, 51068, 48779, 597554, 567644, 8039742 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.` `\| separable \| inseparable \| either \|` `-------------------+-----------+-------------+---------+` `self-conjugate \| A282615 \| A279197 \| A282616 \|` `non-self-conjugate \| A282618 \| A282617 \| A282619 \|` `either \| A279199 \| A202705 \| A104429 \|`
	LINKS	`Table of n, a(n) for n=1..17.`
	FORMULA	`a(n) = A282615(n) + A279197(n).` `a(n) = A104429(n) - A282619(n).`
	EXAMPLE	`For n = 3 the a(3) = 3 solutions are:` `(7,9,8),(4,6,5),(1,3,2),` `(3,9,6),(2,8,5),(1,7,4), and` `(6,8,7),(2,4,3),(1,9,5).`
	CROSSREFS	`Cf. A104429, A202705, A279197, A279199, A282615, A282617, A282618, A282619.`
	KEYWORD	`nonn,more`
	AUTHOR	`Peter Kagey, Feb 19 2017`
	EXTENSIONS	`a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017` `a(17) from Fausto A. C. Cariboni, Mar 22 2017`
	STATUS	`approved`

A282618

Number of non-self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

+10
5

0, 0, 2, 6, 26, 108, 492, 2562, 14790, 98874, 720614, 5908394, 52572682, 516141316, 5422012074, 61889630476, 749456000504 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).` `A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.` `\| separable \| inseparable \| either \|` `-------------------+-----------+-------------+---------+` `self-conjugate \| A282615 \| A279197 \| A282616 \|` `non-self-conjugate \| A282618 \| A282617 \| A282619 \|` `either \| A279199 \| A202705 \| A104429 \|`
	LINKS	`Table of n, a(n) for n=1..17.`
	FORMULA	`a(n) = A282619(n) - A282617(n).` `a(n) = A279199(n) - A282615(n).`
	EXAMPLE	`For n = 3 the a(3) = 2 solutions are:` `(5,9,7),(4,8,6),(1,3,2), and` `(7,9,8),(2,6,4),(1,5,3).`
	CROSSREFS	`Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282617, A282619.`
	KEYWORD	`nonn,more`
	AUTHOR	`Peter Kagey, Feb 19 2017`
	EXTENSIONS	`a(10)-a(16) from Fausto A. C. Cariboni, Feb 27 2017` `a(17) from Fausto A. C. Cariboni, Mar 22 2017`
	STATUS	`approved`

A282619

Number of non-self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

+10
5

0, 0, 2, 10, 40, 212, 1086, 6550, 43978, 326462, 2704096, 24307174, 240782702, 2546166908, 29250772016, 355838290758, 4658858733922 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.` `\| separable \| inseparable \| either \|` `-------------------+-----------+-------------+---------+` `self-conjugate \| A282615 \| A279197 \| A282616 \|` `non-self-conjugate \| A282618 \| A282617 \| A282619 \|` `either \| A279199 \| A202705 \| A104429 \|`
	LINKS	`Table of n, a(n) for n=1..17.`
	FORMULA	`a(n) = A282617(n) + A282618(n).` `a(n) = A104429(n) - A282616(n).`
	EXAMPLE	`For n = 3 the a(3) = 3 solutions are` `(5,9,7),(4,8,6),(1,3,2),` `(7,9,8),(2,6,4),(1,5,3).`
	CROSSREFS	`Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282617, A282618.`
	KEYWORD	`nonn,more`
	AUTHOR	`Peter Kagey, Feb 19 2017`
	EXTENSIONS	`a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017` `a(17) from Fausto A. C. Cariboni, Mar 22 2017`
	STATUS	`approved`

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Last modified August 29 21:34 EDT 2024. Contains 375518 sequences. (Running on oeis4.)