Magic Series
Multimagic series
In multimagic series additionally the sums of certain powers of the numbers have to equal constant values.
There are bimagic, trimagic, tetramagic, ... series for squares, cubes and hypercubes of any dimension.

Example on magic squares of order 4 :
Magic constant: S1 = (1+2+3+...+16)/4 = 34
Bimagic constant: S2 = (12+22+32+...+162)/4 = 374
Trimagic constant: S3 = (13+23+33+...+163)/4 = 4624
The sequence  2, 8, 9, 15  is a trimagic series because it fulfills three conditions
  (1)   2  + 8  + 9  + 15  = 34 = S1
  (2)   22 + 82 + 92 + 152 = 374 = S2
  (3)   23 + 83 + 93 + 153 = 4624 = S3

The algorithm used for normal magic series does not work for multimagic series.

Bimagic series

Say Bi(m) is the number of bimagic series of order m and dimension 2 (i.e. series for bimagic squares).

Achille Rilly (France) calculated Bi(8) correctly in 1906 and later Bi(7) before he died in 1909.
Bi(9) and Bi(10) were found by Christian Boyer (France) in May 2002.
I calculated Bi(11) in August 2002 with backtracking methods.
Bi(12) was reported in March 2004 by Fredrik Jansson (Finland). His computer worked about 50 days on this task, using backtracking methods.
In 2005 Lorenz Schlangen from Solingen, Germany introduced a new method. He splits the numbers of the series into 3 sets and saves the bimagic sums of each set in a table. With this split sum method Bi(12) can be determined in less than a minute (!) - what a progress.
Lorenz Schlangen computed Bi(13), Bi(14) and Bi(15).
In September 2005 I wrote a gb32-program based on Lorenz's ideas. This program could determine Bi(16) within 16 hours.
The calculation of Bi(17) took about 4 weeks, using the split sum method.
In the next years further enumerations were done by Micheal Quist (USA), Lee Morgenstern (USA) and Dirk Kinnaes (Belgium).
All about multimagic series is described at the famous website www.multimagie.com (external link) of Christian Boyer.
   NEW:    Number of bimagic series from order 31 to 34 calculated by Lee Morgenstern in October 2020.
   Counting method of Lee Morgenstern (PDF)

Order Number of
bimagic series 		First computed
in ... by ... from ...
3 0 (?) 1892 Michel Frolov France
4 2 (?) 1892 Michel Frolov France
5 8 ? ? ?
6 98 ? ? ?
7 1844 before 1909 Achille Rilly France
8 38039 1906 Achille Rilly France
9 949738 May 2002 Christian Boyer France
10 24643236 May 2002 Christian Boyer France
11 947689757 August 2002 Walter Trump Germany
12 45828982764 March 2004 Fredrik Jansson Finland
13 2151748695931 July 2005 Lorenz Schlangen Germany
14 123821075526032 July 2005 Lorenz Schlangen Germany
15 8131094055190149 August 2005 Lorenz Schlangen Germany
16 573957471153552576 September 2005 Walter Trump Germany
17 44010987379157415768 October 2005 Walter Trump Germany
18 3655486139293429450720 August 2008 Michael Quist USA
19 333633403912637510806972 August 2008 Michael Quist USA
20 32862657239386515532593520 August 2008 Michael Quist USA
21 3431453899306300581868236386 May 2013 Quist / Morgenstern USA
22 384125946998166710305616659402 May 2013 Quist / Morgenstern USA
23 45801639842337348432002857205878 May 2013 Quist / Morgenstern USA
24 5773407884408951768360741341457291 December 2014 Lee Morgenstern USA
25 768467875608077797720790265285636797 December 2014 Lee Morgenstern USA
26 107710220763567919574782844625088827344 December 2014 Lee Morgenstern USA
27 15880475347526962316266889239839811859979 December 2014 Lee Morgenstern USA
28 2455456581123976779162274548131606029110869 December 2014 Lee Morgenstern USA
29 397017067970073855910942668942599683652058914 August 2015 Dirk Kinnaes Belgium
30 67063309991205148544594890672812817873237628826 August 2015 Dirk Kinnaes Belgium
31 11812446561362208327466140766008607985470443496036 October 2020 Lee Morgenstern USA
32 2165477600372220609578974274970618517151706364579790 October 2020 Lee Morgenstern USA
33 412599409233456658488874049396540090344928282077003902 October 2020 Lee Morgenstern USA
34  81594548527266498796028942169759259784230336662571184344 October 2020 Lee Morgenstern USA

A bimagic series of order 16 consists of 4k even and (16-4k) odd numbers. Thus the problem can be partitioned into 3 tasks.
Note that each series with 4 even numbers has a complementary series with 12 even numbers (replace all numbers i by 257-i).
  a)   Bi(16, 0 even) = Bi(16, 16 even) =   44 103 659 723 445
  b)   Bi(16, 4 even) = Bi(16, 12 even) =   57 725 723 619 430 429
  c)   Bi(16,  8 even) =   458 417 816 595 244 828
Bi(16,  4 even) =   57 725 723 619 430 429
Bi(16,  0 even) =   44 103 659 723 445
 --------------------------------
Sum of all Bi(16,4k even) = Bi(16) =   573 957 471 153 552 576

 Bi(03) = 0
 Bi(04) = 2
 Bi(05) = 8
 Bi(06) = 98
 Bi(07) = 1844
 Bi(08) = 38039
 Bi(09) = 949738
 Bi(10) = 24643236
 Bi(11) = 947689757
 Bi(12) = 45828982764
 Bi(13) = 2151748695931
 Bi(14) = 123821075526032
 Bi(15) = 8131094055190149
 Bi(16) = 573957471153552576
 Bi(17) = 44010987379157415768
 Bi(18) = 3655486139293429450720
 Bi(19) = 333633403912637510806972
 Bi(20) = 32862657239386515532593520
 Bi(21) = 3431453899306300581868236386
 Bi(22) = 384125946998166710305616659402
 Bi(23) = 45801639842337348432002857205878
 Bi(24) = 5773407884408951768360741341457291
 Bi(25) = 768467875608077797720790265285636797
 Bi(26) = 107710220763567919574782844625088827344
 Bi(27) = 15880475347526962316266889239839811859979
 Bi(28) = 2455456581123976779162274548131606029110869
 Bi(29) = 397017067970073855910942668942599683652058914
 Bi(30) = 67063309991205148544594890672812817873237628826
 Bi(31) = 11812446561362208327466140766008607985470443496036
 Bi(32) = 2165477600372220609578974274970618517151706364579790
 Bi(33) = 412599409233456658488874049396540090344928282077003902
 Bi(34) = 81594548527266498796028942169759259784230336662571184344

Trimagic series

Order Trimagic series
3 0
4 2
5 2
6 0
7 0
8 121
9 126
10 0
11 31187
12 2226896
13 17265701
14 0
15 69303997733
16 1683487116508
17 112205432382966
During the 20th century the numbers of trimagic series of the orders 8, 9 and 11 were published incorrectly in magazines and books.
Christian Boyer was the first to calculate these numbers accurately from 2001 (order 8) to May 2002 (orders 9 and 11).
In 2002 Walter Trump determined all trimagic series of order 12 in order to construct a trimagic order-12 square, the smallest possible trimagic square.
In 2004 Fredrik Jansson from Finland calculated the number of trimagic order-13 series.
Michael Quist computed the number of trimagic series in May 2008.
In December 2014 Lee Morgenstern was able to enumerate the trimagic series of orders 16 and 17.

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Walter Trump, Nürnberg, Germany, (c) 2005-02-08 (last modified: 2020-10-28)