cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A220978 a(n) = 3^(2*n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6*n+3) + 1.

Original entry on oeis.org

1, 19, 217, 2107, 19441, 176419, 1592137, 14342347, 129120481, 1162202419, 10460176057, 94142647387, 847287015121, 7625592702019, 68630363015977, 617673353237227, 5559060437415361, 50031544711579219, 450283904728735897, 4052555149532191867
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A198410(n+2): 3^(6*n+3) + 1 = (3^(2*n+1) + 1) * a(n) * A198410(n+2).

Links

Crossrefs

Cf. A092440, A085601, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[3^(2n+1) - 3^(n+1) + 1, {n, 0, 30}]
LinearRecurrence[{13,-39,27},{1,19,217},30] (* Harvey P. Dale, Mar 17 2013 *)
  • PARI
    Vec((1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)) + O(x^30)) \\ Michel Marcus, Feb 12 2015

Formula

a(n) = 13*a(n-1) - 39*a(n-2) + 27*a(n-3).
G.f.: (1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)).

A220979 a(n) = 5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1: the left Aurifeuillian factor of 5^(10n+5) - 1.

Original entry on oeis.org

11, 12851, 9384251, 6054921251, 3808599606251, 2383422998031251, 1490020755615156251, 931310653778075781251, 582075119020843503906251, 363797694444713592519531251, 227373652160169124603222656251, 142108544241637027263641113281251
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220980.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220980-A220990.

Programs

  • Mathematica
    Table[5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1, {n, 0, 30}]
  • PARI
    a(n)=5^(4*n+2)-5^(3*n+2)+3*5^(2*n+1)-5^(n+1)+1 \\ Charles R Greathouse IV, Sep 28 2015

Formula

Aurifeuillian factorization: 5^(10n+5) - 1 = (5^(2n+1) - 1) * a(n) * A220980(n).
G.f.: -(4296875*x^4+2662500*x^3+464450*x^2+4260*x+11) / ((x-1)*(5*x-1)*(25*x-1)*(125*x-1)*(625*x-1)). - Colin Barker, Jan 03 2013

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

113, 34925927, 4651514210561, 556919483179733591, 65684998500756890925713, 7730533744900130305342957127, 909535949164303794596648514307361, 107006774488854204226839526889653524791, 12589253114717671385404089651370543317211313
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220984.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * a(n) * A220984(n).
G.f.: -(184010736563880737*x^6 +268740854387875086*x^5 +14564007567924591*x^4 +73553506117028*x^3 +123792021759*x^2 +19415886*x +113) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A220984 The right Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

911, 46489241, 4845303761663, 560176314330212777, 65739735996793498937711, 7731453717973685046293120441, 909551411151743369070229385367263, 107007034358477098527617255914118283977, 12589257482346423369016062830670344414194511
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220983.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * A220983(n) * a(n).
G.f.: -(1483484787696419039*x^6 -1087259214306211086*x^5 +71725962948861585*x^4 -562870083909028*x^3 +609660625665*x^2 -78551886*x +911) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A220980 a(n) = 5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1: the right Aurifeuillian factor of 5^(10n+5) - 1.

Original entry on oeis.org

71, 19151, 10165751, 6152578751, 3820806643751, 2384948876968751, 1490211490478593751, 931334495635986718751, 582078099253082277343751, 363798066973743438730468751, 227373698726297855377246093751, 142108550062403118610382324218751
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220979.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1, {n, 0, 20}]

Formula

Aurifeuillian factorization: 5^(10n+5) - 1 = (5^(2n+1) - 1) * A220979(n) * a(n).
G.f.: -(27734375*x^4-22687500*x^3+2417450*x^2-36300*x+71) / ((x-1)*(5*x-1)*(25*x-1)*(125*x-1)*(625*x-1)). [Colin Barker, Jan 03 2013]

A220981 a(n) = 6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1: the left Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

13, 39493, 58809673, 78002205553, 101481622729633, 131604778271166913, 170578072060319947393, 221073129991920857571073, 286511629376393032228157953, 371319255900007820952456748033, 481229795439713382306649129101313
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220982.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1, {n, 0, 20}]
LinearRecurrence[{1555,-345210,12427560,-72550080,60466176},{13,39493,58809673,78002205553,101481622729633},20] (* Harvey P. Dale, Oct 01 2021 *)

Formula

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * a(n) * A220982(n).
G.f.: -(21835008*x^4+24984288*x^3+1885788*x^2+19278*x+13) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

A220982 a(n) = 6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1: the right Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

97, 55117, 62169337, 78727802257, 101638351073377, 131638631590149697, 170585384377200633217, 221074709452366968135937, 286511970539849391404729857, 371319329591314394530363646977, 481229811357035602199451623479297
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220981.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1, {n, 0, 20}]

Formula

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * A220981(n) * a(n).
G.f.: -(162922752*x^4-124050528*x^3+9947772*x^2-95718*x+97) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

A220987 The left Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

58367, 3812903020530517, 107454987376543082369146967, 2808133028073215608147547774721982717, 72885505321551844061773948114862247606146502767, 1890579685660625069233746109183146734516524279847333062117
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220988.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[z = 11^n; 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1, {n, 0, 10}]

Formula

a(n) = 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1 with z = 11^n.
Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * a(n) * A220988(n).

A220988 The right Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

407353, 4572972882642803, 109245858982819139102535553, 2812355783638980226466572392952970603, 72895462357781065526518523423275265184080402953, 1890603163831201090586603020695655490130990020251181357603
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220987.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[z = 11^n; 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1, {n, 0, 10}]

Formula

a(n) = 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1 with z = 11^n.
Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * A220987(n) * a(n).

A220989 a(n) = 12^(2n+1) - 6 * 12^n + 1: the left Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

7, 1657, 247969, 35821441, 5159655937, 743006877697, 106993187463169, 15407021359595521, 2218611104160546817, 319479999339664244737, 46005119908998197280769, 6624737266944778960896001, 953962166440636632998608897
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220990.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[12^(2n+1) - 6 * 12^n + 1, {n, 0, 20}]

Formula

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * a(n) * A220990(n).
G.f.: -(1008*x^2+558*x+7) / ((x-1)*(12*x-1)*(144*x-1)). [Colin Barker, Jan 03 2013]
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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