Pseudo Magic Rings
(derivative note: definition below)

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In MagicSequences.html I derived that all magic squares of order 3 can be seen as:

a1 + a2 = a3 a1 + a3 = a4
a1	a2	-a3
-a4	0	a4
a3	-a2	-a1

subtracting the two basic relations within an order 3 magic square:
a₁ + a₂ = a₃
a₁ + a₃ = a₄
gives after some little rearangement:
a₂ + a₄ = 2 a₃
interestingly this relation between a cornerpoint and its knightjump connected numbers is independent of the central number x₀ and a needed relation between the numbers in any order 3 magic square.

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The Pseudo Magic Ring
(definition)

In Generalized Object notation these rings ought to be written as:
³IN-Magic Rings {....}
with {...} generating function and condition statement,
below I however use the shorthand ³R_dif

Pseudo Magic Ring 3Rdif (definition)
			Sd = f + c
a	b	c	Sh = a + b + c
d		e	Sch = d + e
f	g	h	Sh = f + g + h
Sv = a + d + f	Scv = b + g	Sv = c + e + h	Sd = a + h
satisfying conditions: 2a = e + g 2c = d + g 2f = b + e 2h = b + d dif = Sv - Sh = Sch - Sd = Sd - Scv = [(b + g) - (d + e)] / 2

once the numbers b,d,e and g are given the diff equalities are automatically satisfied
current samples for the pseudo magic ring is the ring of any order 3 magic square (in which case the mentioned difference S_v - S_h = 0 and henche ³R₀)
in order for the relation 2a = e + g to be satisfied the relation either e <= a <= g or g <= a <= e need to be obeyed, which means that one of the central ring values needs to be the smallest and another one the largest. Let's choose e to be the smallest number of the 8 numbers, working out the possibilities gives:

possibilities for inequality sequence (choosing e to be the smallest number) (also assuming non-equal numbers)
ring type	relations	simplification	remark
I	e < a < g g < c < d e < f < b b < h < d	e < a < g < c < d e < f < b < h < d	type 1 square c < b type 2 square c > b
II	e < a < g g < c < d e < f < b d < h < b	e < a < g < c < d < h < b e < f < b	ring with dif < 0
II	e < a < g d < c < g e < f < b b < h < d	e < a < g e < f < b < h < d < c < g	horizontal mirror of above
III	e < a < g d < c < g e < f < b d < h < b		neither type II (d < c and d < h)

in order for a ring to be part of a magic square the dif needs be 0 (ie the ring be an ³R₀).
As derived in MagicSequences.html the smallest and biggest number in a magic square must be on opposite sites of the central number, which means the ring must be of type I, a type II or III ring can't be part of a magic square.
two types of magic squares are known to exist (conditions listed in above table). These conditions subdivides ring type I into type Ia (c < b) and type Ib (c > b). Further conditions posed on the numbers subdivides the other ring types as well.
Below two ring type I are listed and one type II, using only square numbers.

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The Pseudo Magic Ring with square numbers

According to Chebrakov's result mentioned in:
The Prime Puzzles and Problems Connection puzzle #79
a magic square consisting of square numbers:
³IN-Magic Squares {f(i)=m_i² (m_i in IN)}
can't exists

In order to get a magic square of order 3 we need four semi-pythagorean triplets (SPT's) (a,b,c) satisfying
a² + b² = 2 c²
intertwined in such a manner that they form a pseudo magic ring
of course when (a,b,c) is an SPT so is n(a,b,c) = (na,nb,nc) which give meaning to the term "primitive SPT"
of course only ³R₀ can be made into a real magic square with a central number
Chebrakov's result means that a pseudo magic ring of only square numbers can never be made to give an ³R₀
Using only primitive SPT's I found a ³R_201.120, a ³R_10.273.824 and a ³R_-18.267.216
(the first two of type I and the third of type II)

All primitive SPT's of numbers below 10.000 are in prim_spts.zip the three found rings are also listed in that file, currently I only used those to form rings. Some future uploads might hold non primitive spt rings as well.

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Searching for Semi Pythagorean Triplets I found 127 such "primitive" triplets with numbers under 1000. running through this data and searching for a ring I found the following
(primes shown in red ('1' is not considered to be a prime(!?))

pseudo magic ring with squares of numbers < 1000 only one using primitive SPT's
29	239	481	listing as SPT's
679		1
169	41	509
with square numbers (3R201.120)			type I
			259.922
841	57.121	231.361	289.323
461.041		1	461.042
28.561	1.881	259.081	289.323
490.443	58.802	490.443	259.922

Interestingly this ring of square numbers has equal summing horizontal triplets also equal summing vertical triplets and equal summing diagonals, further the difference between the horizontal and vertical triplets is equal to the difference between the diagonal sum and both central sums
(this ring was the basis for the ring definition)

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below 10.000 there are 1258 primitive SPT's forming but two more (primitive SPT) rings:

pseudo magic ring with squares of numbers < 10.000 only one using primitive SPT's
25	2513	3665	listing as SPT's
5183		17
1777	31	4073
with square numbers (3R10.273.824)			type I
			16.589.954
625	6.315.169	13.432.225	19.748.019
26.863.489		289	26.863.778
3.157.729	961	16.589.329	19.748.019
30.021.843	6.316.130	30.021.843	16.589.954

and

pseudo magic ring with squares of numbers < 10.000 only one using primitive SPT's
3365	8897	5305	listing as SPT's
6647		3247
6697	3479	7853
with square numbers (3R-18.267.216)			type II
			72.992.834
11.323.225	79.156.609	28.143.025	118.622.859
44.182.609		10.543.009	54.725.618
44.849.809	12.103.441	61.669.609	118.622.859
100.355.643	91.260.050	100.355.643	72.992.834

notice the last has a negative dif, the first two rings are of type I and the last of type II.
(The three rings listed are the only possible three (if my program is correct) with primitive SPT's of numbers < 10.000, not known whether such a ring of type III exists)
Chebrakov's result must be based on some non-existence theorema I can't figure just yet!

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Two SPT's (a,b,c) and (d,e,f) can be combined four ways with lcm(a,d), lcm(a,e),lcm(b,d) and lcm(b,e) which makes triplets [lcm(a,d) (a,b,c) / a] = [lcm(a,d),lcm(a,b)b/a,lcm(a,d)c/a] etc. in this way four SPT's can always be joined together. In order to make a ring however the end of this chain of five numbers must be equal. This limits the possibilities An itersative procedure using the known SPT's gave me several possibilities for these "compound SPT-rings" (several I'll upload in some future date, currently the least diff I have is 6.216):

pseudo magic ring with squares of numbers < 10.000 compound SPT ring
SPT decomposition
17			7
1	7	5	7	17	13	17	31	25	31	49	41
giving the ring
25			119			41			listing as SPT's
49						17
85			31			91
with square numbers (3R-6.216)									type II
									8.906
625			14.161			1.681			16.467
2.401						289			2.690
7.225			961			8.281			16.467
10.251			15.122			10.251			8.906

which has a prime number count of 11 in it's SPT decomposition

pseudo magic ring with squares of numbers < 10.000 compound SPT rings
no 1: using spts (0,1,3,6)
31	7	1 7 5			217 1519 1085			(3R922.056)
	1	7 17 13			217 527 403
1	49	17 31 25			833 1519 1225
	1	31 49 41			527 833 697
no 2: using spts (0,1,3,6)
factors		spt's			triplet			(3R6.216)			8.906
1	17	1 7 5			25	119	41	625	14.161	1.681	16.467
	7	7 17 13			49		17	2.401		289	2.690
1	1	17 31 25			85	31	91	7.225	961	8.281	16.467
	1	31 49 41						10.251	15.122	10.251	8.906
no 3: using spts (0,1,44,838)
391	7	1 7 5			2.737 19.159 13.685			(3R195.827.640)
	1	7 17 13			2.737 6.647 5.083
1	49	71 391 281			3.479 19.159 13.769
	1	3.479 6.647 5.305			3.479 6.647 5.305
no 4: using spts (0,1,44,838)
71	17	1 7 5			1.207 8.449 6.035			(3R-51.003.960)
	7	7 17 13			3.479 8.449 6.461
1	17	71 391 281			1.207 6.647 4.777
	1	3.479 6.647 5.305			3.479 6.647 5.305