The Number of Singular Vector Tuples of a general, [n, n, n], tensor . By Shalosh B. Ekhad Theorem: Let a(n) be the number of singular vector tuples of a, 3, -dimensional tensor whose dimensions are all, n The first, 70, terms of that sequence are [1, 6, 37, 240, 1621, 11256, 79717, 572928, 4164841, 30553116, 225817021, 1679454816, 12556853401, 94313192616, 711189994357, 5381592930816, 40848410792017, 310909645663332, 2372280474687277, 18141232682656320, 139010366280363601, 1067160872528170536, 8206301850166625797, 63203453697218605440, 487480961825645988721, 3764885111909676266856, 29112650384540312124397, 225377607037294291831008, 1746648730442554074446041, 13549903631370307542264936, 105214484529950220234997141, 817709228332592281063815168, 6360391464220968384566019745, 49511928791684034124645730676, 385707501237368849206626328717, 3006846684379174364465426255616, 23455977069620091841962275705089, 183092319857177553953377636176456, 1430037723632571918813195121704997, 11175656244135989347042802862284928, 87384630486487451749353666860367841, 683632453962320044584305182576200288, 5350886598153793514481537897337176493, 41902005901674406571033924405839308768, 328277426887267454292261947436072998521, 2572978040988090019983161573900671390056, 20174938455027696505833506309037744673717, 158256711967850325867904932175164613805568, 1241878121565109721305598151979300621187041, 9748914697534511954263668793857706936885416, 76557533012847845401553482298831558586858781, 601406624861928916400901634219448551509045216, 4725977743774598911951207466985442875802004041, 37149518272412304070221079042862933387873130216, 292110984170941385701186299397195372784867345221, 2297582777856431871486896289498367921392694659456, 18076679242941480643112276737449814240450399614961, 142261109059091890572028251448021584831777493283976, 1119875738825722199969846898012482760095108120469421, 8817913547145642408782946965788762982054339519598816, 69449739062865201034991808469623104398486763332361881, 547117715028973597496117057639950244728453193275378536, 4311149329690424172211655089600627818487132234309850581, 33978514528762393389117490950553726381113420890825711616, 267862418462871260646089896898133632801592296425890350401, 2112090051496950649407979640746003519860385829377421719956, 16657262165394749047223410973405780093084150560578941606221, 131396189046352933535318043187955359702871390804208939532416, 1036686394280916864442473886582846818530956264890966401359201, 8180787959749319986573210152825554401372254551057541041233416] Theorem 1: The sequence a(n) satisfies the following linear recurrence equat\ ion with polynomial coefficients 4 3 2 2 72 (n + 2) (245 n + 3094 n + 14447 n + 29474 n + 22100) (n + 1) a(n) - 6 5 4 3 2 (n + 2) (21805 n + 330981 n + 2012733 n + 6230951 n + 10263446 n 7 6 5 + 8425060 n + 2639760) a(n + 1) + (-13230 n - 249641 n - 1998705 n 4 3 2 - 8785333 n - 22847777 n - 35069178 n - 29331496 n - 10279296) a(n + 2) 7 6 5 4 3 + (21560 n + 413637 n + 3343917 n + 14735333 n + 38132651 n 2 6 + 57777574 n + 47273504 n + 16026528) a(n + 3) - (n + 4) (4410 n 5 4 3 2 + 70147 n + 452903 n + 1516515 n + 2769127 n + 2601986 n + 975888) a(n + 4) + (n + 5) (n + 4) (n + 3) 4 3 2 (245 n + 2114 n + 6635 n + 8882 n + 4224) a(n + 5) = 0 subject to the initial conditions a(1) = 1, a(2) = 6, a(3) = 37, a(4) = 240, a(5) = 1621 and in Maple notation 72*(n+2)*(245*n^4+3094*n^3+14447*n^2+29474*n+22100)*(n+1)^2*a(n)-(n+2)*(21805*n ^6+330981*n^5+2012733*n^4+6230951*n^3+10263446*n^2+8425060*n+2639760)*a(n+1)+(-\ 13230*n^7-249641*n^6-1998705*n^5-8785333*n^4-22847777*n^3-35069178*n^2-29331496 *n-10279296)*a(n+2)+(21560*n^7+413637*n^6+3343917*n^5+14735333*n^4+38132651*n^3 +57777574*n^2+47273504*n+16026528)*a(n+3)-(n+4)*(4410*n^6+70147*n^5+452903*n^4+ 1516515*n^3+2769127*n^2+2601986*n+975888)*a(n+4)+(n+5)*(n+4)*(n+3)*(245*n^4+ 2114*n^3+6635*n^2+8882*n+4224)*a(n+5) = 0 Just for fun, the, 3000, -th term of this sequence, i.e. the number of singular vector tuples of a , 3, -dimensional tensor, all whose dimensions are equal to, 3000, is equal to 2277896955015916911500519305067646829216367186714766881181764764642381659301499\ 5617017688070318100443927019987340376196845937510408264738762424804282411584793\ 0834953758304364879411732126206501288612773043060615219967222058488964945656647\ 1361064392117700003777872570122242003153745330008348862468735412878404722463047\ 0030641430975894845585298564982183977697785656890933641229025777705204419245433\ 1410060754418711178109872077820512455422422607946219412524236263037516354798155\ 5252762727697249874673029909120033406432642447302368302207559710405983342102280\ 0682207358578694805796359390715159372773231926082577320965437146489867061389333\ 0197111764243414987642045082108126422591953226570717188296454977578922087092233\ 5238712070372965843401134578778825685667140789130159723126102296654159887602390\ 8069988996534813690838372156705963369848571111096346719405300055672015760995056\ 4985887120697771953153867364404025166657221647419301757055185422070040657929689\ 1881510108100118567436424019918822912164680182560196183144267497084793127360305\ 1738275226804716653380203954776319218977853315510547894854089244626601680720459\ 1140374703930019469049176878486694788500909985424757128011940955174357360724218\ 9588891193026980573347164332372314649522940753241997488268894849975413689599063\ 2132097848404777966812597232369434129756515938761247628762660457039156782060729\ 9038817612038132369124828838190552454089455465420736293150781708186341644696672\ 7523275724321220993030382883366158728806907754568025337674232468319689490252015\ 4850260731201851433178834921040718521889933955388915472536363904756900200478967\ 0778853492399539225996517026165229864097195230702178364607186314284041577777439\ 2821853308284123035608562558114055521257384711858483090372304657286711635034683\ 0461115966518189739018466385235458798944278503043853659712751157837524156285989\ 3885144678501601614629600750260002359077280192913014276113398212526068984653288\ 8966708273639476525395570537001853754572772560137747575243541459620269186674286\ 2480346262656337617335724714238668289628220888570489829233403557721929383607472\ 1334766746960978672560600633596447495421745861805793192243275512460461303573153\ 3426932568531998521146297953770060249903049863012142482691407063391648333277383\ 8785396256870104841124885346965212335395739665305446858209644047985829632689608\ 4228331935059573334986245828485657701251305388403176681735358835156793578076852\ 6003946412519869268803724252322559369395474463932779261331900333012769079344591\ 8640318940859034098698270006293550203601715357653843410172531405135964557293025\ 6419612236498779437644683095673942379465811223854509368688321547084073459074985\ 1443917742444099734413550511403103914909824519678355512947442011335061513035129\ 05565849083542440320 Theorem 2: The asymptotic formula for a(n), to order, 6, is given by 1/2 n / 13 1477 93707 8343061 2866730137 1204239422533\ 2 3 8 |1 - --- + ----- - ----- + ------- - ---------- + -------------|/(3 | 3 n 2 3 4 5 6 | \ 27 n 81 n 243 n 2187 n 19683 n / Pi n) and in Maple notation 2/3*3^(1/2)/Pi*8^n/n*(1-13/3/n+1477/27/n^2-93707/81/n^3+8343061/243/n^4-\ 2866730137/2187/n^5+1204239422533/19683/n^6) This ends this paper that took, 11.660, seconds to generate. Have a good day.