Magic Series
 Magic series for magic cubes and hypercubes

He suggested to use n for the dimension and m for the order of a hypercube.
 There are cubes of order 5 where all straight lines are magic. You may call them strictly magic cubes. Such cubes have got 75 monagonals (= 25 rows + 25 columns + 25 pilars), 30 diagonals (= short diagonals) and 4 triagonals (= long diagonals). Each of these 109 lines is magic. The 5 numbers in each line sum up to 315. If the result of a sum of 5 distinct positive integers, each not greater than 125, equals 315 then this sum is called a magic cube series of order 5.
 For magic cube series of order m we demand:
- Number of terms in the sum: m
- Maximal value of an integer: m3
- Value of the sum: (m3 + 1)·m/2

 This concept may be used for hypercubes of any dimension n and order m:
- Number of terms in the sum: m
- Maximal value of an integer: mn
- Value of the sum: (mn + 1)·m/2

Using the same algorithm as for magic squares series, I got the following numbers of magic series (2005-02-12).

Number of magic series for hypercubes of dimension n and order m

 Dimension Orderm = 1 Orderm = 2 Orderm = 3 Orderm = 4 Orderm = 5 Orderm = 6 Order m = 7 n = 1 1 1 1 1 1 1 1 n = 2 1 2 8 86 1394 32134 957332 n = 3 1 4 85 6786 1142341 338832214 156623626331 n = 4 1 8 800 457906 751910094 2766058729144 19279934097075600 n = 5 1 16 7321 29695346 474791829591 21682691008200124 (*) n = 6 1 32 66248 1906778226 (*) (*) (*) n = 7 1 64 597325 122134408306 (*) (*) (*) Formula 1 2n-1 Nn(3) Nn(4) (*) (*) (*)

 Order-3 formula: Nn(3) = (3n - 1)2 / 8 + a with a = 0 for even n   and a = 1/2 for odd n

 Order-4 formula: Nn(4) = (2u3 - 9u2 + 18u + 16) / 72 with u = 4n (= max. number in hypercube of dimension n)

Both formula were found on February 16th, 2005.
Nn(4) was found experimentally, whereas Nn(3) was proved (not elegantly but rather complicated).
(*) In summer 2006 I determined formulae for the orders 5 to 11: Next page

Number of magic series for hypercubes of dimension 3
(= Number of magic cube series)
```   N3(01) = 1
N3(02) = 4
N3(03) = 85
N3(04) = 6786
N3(05) = 1142341
N3(06) = 338832214
N3(07) = 156623626331
N3(08) = 104510988949316
N3(09) = 95268144607230087
N3(10) = 113890197280403493542
N3(11) = 173010424861377562731014
N3(12) = 325702485631908523866475222
N3(13) = 744530535411231865659859497116
N3(14) = 2032002010218114238831528413724970
N3(15) = 6527910014315748187894308484050347987
N3(16) = 24387539775140714873044335703400231155136
N3(17) = 104848714411898862472408574437293947290702029
N3(18) = 514044750180549050442567090101496413473966382586
N3(19) = 2851028862119546169537090377842408485779881321936932
N3(20) = 17761622824968263447480846131961889702954933930143640222
N3(21) = 123507908155489415259714919787990912099853019354696330382798
N3(22) = 953171345162459056079884898190780075022378458290023885264280280
N3(23) = 8122418609655059149917607124169919389170847292056282240224982459626
N3(24) = 76071078142263489144271535348244319093125085424867933793021114142686578
N3(25) = 779713664953507723219864299512253552154552656079012342466306984743947563650
N3(26) = 8712657247151011533667805994365535615436854780746585820112459443349827000311924
N3(27) = 105760216311436165183093905738108357978036899914057350653021726339429929822378954480
N3(28) = 1390042617972072415527107819834364205977961640723184756497410212317708290067795924300716
N3(29) = 19722122087194379309353177795515784954079133716100546130420206136600289567836012674671474866
N3(30) = 301217770585949430821677726637765162566699227115283269125110601447255945201684345965851046031534
N3(31) = 4939426566399261462756076081272647860686256065099788060372415751105529825052176969165454985576861687
N3(32) = 8675333889314245080115100105795164944345175600520457112625490684274074817000120194445203249053716751
8874
N3(33) = 1628260383813869207998922142039476022036287910014547576454513639913242413797043349921234206115039249
029401559
N3(34) = 3258867359557494742040012044191360271780776833323501077374328036372135471813127804372905190369023718
3571482153844
N3(35) = 6941429955900658689354188844561486663268998734833537572536847347647772016708628487119214577730510848
14223437931760829
N3(36) = 1570568428641886366948015262874480318427961799771800848140536925691269867264395917355337998996209592
1083958005905871514746
N3(37) = 3768114474329823826299675817021952584305940597993146395470748371760538024101231176029559084496053034
90089692439321097280968348
N3(38) = 9570346906458367638793398255351711173255772809159197130577281447291472048909951274939589720919764528
599171987491881597483497978056
N3(39) = 2569120821555720707700542468344195880287696558391464567715117453885076390575874772987559433933441916
33967279223746800013114750890957995
N3(40) = 7278605055911668334821326725993341386246010067035229244115656361166847064174969690779723955339042061
538446483745723106534260336145395931754
N3(41) = 2173230779213848713536695719675543557861326443189198808084324465855241186783613395813072181858966735
18037723537283407054157185125937526884924547
N3(42) = 6829298411921338006682867206461792760412418988361589425869202105922011693612718080319580419149794264
506973600454773215852929617378344144218139843824
N3(43) = 2255829345518394586285834583584524474629406046712981866372906733588621009189447252437345689625845578
89452984034625257951893900947585591524987253212582406
N3(44) = 7822947984413726559094779131725299808529220978220624834706456876701893988167444602361210917017816391
244014741950061011673276914420498487689389377679974192876
N3(45) = 2844904642704956862508680330434354929244711824913316419924930601010110970725088345988089565096916364
94755258460051892130689339658828802896364370951497532824714454
N3(46) = 1083727861587641336710995329753513129605879455959676060092481969704257882604072398074152992702161914
9097277308360159892159164917901117506631287598885629149441267071878
N3(47) = 4319881088507975829734230068822030067860139202912025731928318359364488659389416303018655828090856628
49226845895553833223861380722128636160327965907150014469783239765091690
N3(48) = 1800057232871625132978552441397348100338861819821751135488500215455500164316531542021202857166758771
0051235878239481057938863848598002175495581689104574656199947129450455741708
N3(49) = 7833333191430341278522787494348181579391407349629970428943064607841593488692410810837146963661921532
47802155657625782308681281586213051813161391603678207726006497112196985676654724
N3(50) = 3556748105603989234823119461374755695798529979798571501566339610225751331578675310920357327105610726
2293749404075323047302623168999163662007179828837967313311350537310758255886452496692

N3(100)= 1471350942827991124132253858550690876482115191673038840314670850799568188033396757689961539703310655
4380072163888531628289538043515224898124360560826499542710495764658104038928328140558424407620051866
5039086551579035293327528828557245522946934356819877526113136664678366937360806224281040278522557304
3081638034089207192136464852931347472144009117175208822482012115577686696977857267865190895031912398
192433455025101020359249399781677698

N3(150)= 1015058984725358739400257699818044218023590596485541978713833627223210811651338257677296530889713903
0056093263627800617386845914271944516628615329076983794334069831210768258772763395118597863672331257
8081021439380085010695658833415355043182608965341965982231493494519258236664210511758828477548219439
0359878244829930520391840397393259870046877174070410904514819520247123687477971530444393867641202786
6233914139092615808379867838855469633644757645585704994563927304798131046810102121199940111511015758
0540246955004876030039159587357436238442923397797383805993529121997826170159936169994278743238138031
8284031879995332764948393543955193488655364806381579931914169711745131792809099172521836335784741529
7714356686

N3(200)= 6406245515885694440138075060451214459650333203982281354698031613068195252070201317013328004209162979
8147607141028776082434192614145168526114479513723186013887408111014287453788219428114789661865946880
3458876399192479599647187147576537407906306856719146722519601563362195782957743365530014342436292986
7467830231111424425893316256767952066759575185445549422917928633818512778473246945148088978933440273
6708056507696856453193293973221525059511328266987297938341065433394089023673929360114548642469594638
7787064303242954037675534628462083078616193751998912705492276758132620502274957914655065984726726692
5808560261999649572546512749094943457513047606293554574800040591300324848141533352749661591563184577
2519068419390512594327344076351949372879228808236366803784197802581313165018575960665119890856996186
6186078912160892998908246882678458914582195423246398826274992491065690940477925067198639656527401462
89715573235066384525755138302440920995385958790567136979639900848174778076112927531019709760630608

```
Terms 17 to 30 were added from 2007-01-04 to 2007-02-03 (calculated with Gerbicz's algorithm)
Later 50 terms were calculated by a common PC (3 GHz, 1 GB RAM) within two days.
All numbers for orders > 50 were calculated in 2013 with the method of Dirk Kinnaes.
Number of magic series for hypercubes of dimension 4
```   N4(01) = 1
N4(02) = 8
N4(03) = 800
N4(04) = 457906
N4(05) = 751910094
N4(06) = 2766058729144
N4(07) = 19279934097075600
N4(08) = 228594811277217786320
N4(09) = 4264819655248960858342656
N4(10) = 118132052487666384802213007240
N4(11) = 4643768453772212643643327799069020
N4(12) = 249910611212426874400489215388031291754
N4(13) = 17879943059627473884359465048565343208306660
N4(14) = 1659694058087666324322349501564041153231721074328
N4(15) = 195811840766424991031393098828021058542782512606563208
N4(16) = 28851267146351983233823608598248115643943930325659092454940
N4(17) = 5228863104684345825174562208556576509465064597808542674228301378
N4(18) = 1150288132762271809213881969579045654302729771403333915087698909956912
N4(19) = 303593807909897913226526653065876795977788518791081663327003953776116575660
N4(20) = 95141304887696226735982367947818131861125408699521534114572649089416273825549048

N4(30) = 2098638659424343929327316561091573489925102531006550724812628752340187131974624333952158505430765286
485571149136065994945813284537113976552

N4(40) = 2225463558912637234807260241554796228893871604172108460624694732515941880265526830600236805068402592
8025093660720301317419254244693748705538190621768225321200399930786354775578799833314723279181723540
40

N4(50) = 6377807557182457458433629410686306418947568052088945549980823082854078022580506297909206313147886341
6952960493065091011155185114641571623893990954342073670719051174001062857472820939973951748152653735
77344974657558034059241082347764831515486555758377889243675145798552

N4(60) = 9414526840971139887814151681637325317695253710633867479367536638554935700222892778817086281615458462
6741769457389126663904027273882542090488411057692537059371585744661978880571374628136652670775780750
7861785479430462704589609523364725232206355974587170826818774684367642411906077843912127546708443036
2309821361962877715567855198967463430

N4(70) = 2430132585092435367573576883250941569683824759504088489054715861626277755775378566116327097323815015
3083355815728961104790804033242906665754237437194987911406080194400167883121175788284160082106663486
6929696068318173122028716599376608220277259038441702314925034449533244248051504442135633141121822732
7415812744360182984881784177385725130670505655908557061732895117770523452883068260563423201274884052
977470322

```
Terms 10 to 20 were added from 2007-01-06 to 2007-01-28 (calculated with Gerbicz's algorithm)
All numbers for m > 20 were calculated in 2013 with Dirk Kinnaes' program.