A082393 and A081232

[Don Reble]

Seqfans:

    %I A082393
    %S A082393 4,180,8,1820,12,320,9100
    %N A082393 Let p = n-th prime of the form 4k+1, take smallest solution
	(x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1;
	sequence gives value of y.
    %Y A082393 Values of x are in A081232. Cf. A082394, A081233, A081234.
	Equals A002349(p).
    %A A082393 Cino Hilliard (hillcino368(AT)hotmail.com), Apr 14 2003
    %E A082393 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com),
	Apr 15 2003
    %E A082393 Should be rechecked, I have received conflicting versions of
	this sequence. - njas, Apr 18, 2003

I wonder, in the continued fraction for square-root(p), does the
fundamental solution to x^2 - p*y^2 = 1 always appear among the
convergents (for each p)? If so, A082393 goes:

        4 180 8 1820 12 320 9100 226153980 267000 53000 6377352 20
        15140424455100 113296 519712 2113761020 3726964292220 190060
        183567298683461940 448036604040 28 386460 70255304
        649641205044600 32 820 9562401173878027020 134951575480 719780
        1819380158564160 13967198980 112422913565764752 9081060
        540608704 2712540 166660 42094239791738433660 40
        1238789998647218582160 189073995951839020880499780706260
        5025068784834899736 3388393513402120 326662411570389853632
        55067617520620 13882400040814700 1444066532654320
        159395869721270110077187138775196900 1180 702635588524014320 48
        29629176560 1586878942101888360258625080
        18741545784831997880308784340 135413180018248
        103066257550962737720 410896226494013260
        638728478116949861246791167518480580 183696788896587421699032600
        52 10485980 25003993164540 7213860 136299971388 46400
        19320788325040337217824455505160 129748968980 43276943002540
        13235458622462202510640 314136625452886403879740 16665383182260
        7377009103065498851032020 4502963741200 3933131148
        761624136944072910800 433896111669844912840
        15701968936415353889062192632 34845956052079180
        488830275367615376 3481871275306470280 456624468 18360
        9713562669309460 6219237759214762827187409503019432615976684540
        2172772383489805842331071281808456 145754861076386671221320
        5243626382156167474921782674260
        22722526912283010072320240710785462723519145740 19639620
        26678547908792 22362336179089374248695940740
        2629972211566463612149241455626172190220 1680
        618516849717028248841100388607776 8724388547388468773672620
        42328623757159480 3578536376412875571438541467679723838177648640
        286231896207756 43761444552128 26346740
        66755254524262858038831090292970460

And here's A081232 (under the same assumption):

        9 649 33 9801 73 2049 66249 1766319049 2281249 500001 62809633
        201 158070671986249 1204353 6083073 25801741449 46698728731849
        2499849 2469645423824185801 6224323426849 393 5848201 1072400673
        10085143557001249 513 13449 159150073798980475849 2262200630049
        12320649 32188120829134849 248678907849 2063810353129713793
        169648201 10157115393 52387849 3287049 838721786045180184649 801
        25052977273092427986049 3879474045914926879468217167061449
        104564907854286695713 71798771299708449 6983244756398928218113
        1182351890184201 313201220822405001 32961431500035201
        3707453360023867028800645599667005001 27849 16760473211643448449
        1153 721517598849 38902815462492318420311478049
        464018873584078278910994299849 3363593612801313
        2609429220845977814049 10499986568677299849
        16421658242965910275055840472270471049
        4765506835465395993032041249 1353 277631049 665782673992201
        195307849 3750107388553 1280001
        535781868388881310859702308423201 3607394696649 1221759532448649
        376455160998025676163201 9000987377460935993101449
        479835713751049 215454135724113414336120649 131822292741249
        116476476553 22606256615916825861249 13224937103288377430049
        480644425002415999597113107233 1068924905989944201
        15090531843660371073 108832847723078562849 14418057673 583201
        309159979019115849
        198723867690977573219668252231077415636351801801
        69833593956354694952807071952492833 4720747718766511296768801
        170800615664635332564330517881801
        742925865816843150858935268959512942700219559049 649296649
        883619927128353 744702728637239515399291224201
        87897747594260774254246835664214545379849 56449
        21002267655460685387089654909788673 299819549856198391714823049
        1462023556679567649
        124015777380821763117401108566632174583648616449
        9968927282677513 1526641059203073 923640201
        2347849777171254083668416002496336649

--
Don Reble       djr at nk.ca



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