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 1-6 of 6 results.

A144930 Numbers k arising in A144929.

Original entry on oeis.org

1, 109, 11989, 1318681, 145042921, 15953402629, 1754729246269, 193004263686961, 21228714276319441, 2334965566131451549, 256824983560183350949, 28248413226054037152841, 3107068629882383903461561, 341749300873836175343618869, 37589316027492096903894614029
Offset: 1

Views

Author

Richard Choulet, Sep 25 2008

Keywords

Comments

Numbers n such that there exists x in N : (x+1)^3 - x^3 = 7*n^2. - Richard Choulet, Oct 16 2008

Examples

			a(1) = 1 because 2^3-1^3 = 7*1. - _Richard Choulet_, Oct 16 2008

References

  • E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - N. J. A. Sloane, Mar 03 2022

Links

Crossrefs

Cf. A144729, A144727.

Programs

  • Mathematica
    LinearRecurrence[{110,-1},{1,109},20] (* Harvey P. Dale, Oct 05 2016 *)
  • PARI
    Vec(-x*(x-1)/(x^2-110*x+1) + O(x^20)) \\ Colin Barker, Jul 14 2016

Formula

a(n+2) = 110*a(n+1)-a(n). - Richard Choulet, Oct 16 2008
G.f.: -x*(x-1) / (x^2-110*x+1). - Colin Barker, Oct 17 2014

Extensions

More terms from Colin Barker, Oct 17 2014

A144927 Numbers n such that there exists an integer k with (n+7)^3-n^3=k^2.

Original entry on oeis.org

7, 1162, 128191, 14100226, 1550897047, 170584575322, 18762752388751, 2063732178187666, 226991776848254887, 24967031721129850282, 2746146497547435276511, 302051147698496750566306, 33222880100337095127017527, 3654214759889381967221362042
Offset: 1

Views

Author

Richard Choulet, Sep 25 2008

Keywords

Examples

			a(1)=7 because 14^3-7^3=49^2.

Links

Crossrefs

Cf. A144928, A144930, A144929.

Programs

  • Mathematica
    Last /@ Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && (n + 7)^3 - n^3 == k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 14}] (* or *)
Rest@ CoefficientList[Series[7 x (-1 - 55 x + 2 x^2)/((x - 1) (x^2 - 110 x + 1)), {x, 0, 14}], x] (* Michael De Vlieger, Jul 14 2016 *)
LinearRecurrence[{111,-111,1},{7,1162,128191},20] (* Harvey P. Dale, Jul 05 2024 *)
  • PARI
    Vec(7*x*(-1-55*x+2*x^2)/((x-1)*(x^2-110*x+1)) + O(x^20)) \\ Colin Barker, Jul 14 2016

Formula

a(n+2) = 110*a(n+1) - a(n) + 378.
G.f.: 7*x*(-1-55*x+2*x^2) / ( (x-1)*(x^2-110*x+1) ). - R. J. Mathar, Nov 27 2011
a(n) = 7*A144929(n). - R. J. Mathar, Nov 27 2011

A350979 a(0)=1, a(1)=652, thereafter a(n) = 254*a(n-1)-a(n-2)+378.

Original entry on oeis.org

1, 652, 165985, 42159916, 10708453057, 2719904916940, 690845140450081, 175471945769404012, 44569183380288169345, 11320397106647425609996, 2875336295905065816770017, 730324098762780070033974700, 185499445749450232722812804161, 47116128896261596331524418282572, 11967311240204696017974479430969505
Offset: 0

Views

Author

N. J. A. Sloane, Mar 06 2022

Keywords

Comments

Arises in studying the equation x^3 - 7*y^2 = 1.

References

  • P.-F. Teilhet, Query 2228, L'Intermédiaire des Mathématiciens, 11 (1904), 44-45.

Links

Crossrefs

Cf. A144929, A350980, A350981, A350982.

Formula

From Chai Wah Wu, Mar 07 2022: (Start)
a(n) = 255*a(n-1) - 255*a(n-2) + a(n-3) for n > 3.
G.f.: x*(20*x^2 - 397*x - 1)/((x - 1)*(x^2 - 254*x + 1)). (End)

A274971 Numbers k such that (x+1)^3 - x^3 = k*y^2 has integer solutions.

Original entry on oeis.org

1, 7, 19, 31, 37, 43, 61, 67, 79, 91, 103, 127, 139, 151, 157, 163, 169, 199, 211, 217, 223, 247, 271, 283, 307, 313, 331, 343, 349, 367, 373, 379, 397, 403, 427, 439, 463, 469, 487, 499, 511, 523, 547, 553, 571, 577, 607, 613, 619, 631, 643, 661, 679, 691
Offset: 1

Views

Author

Colin Barker, Jul 13 2016

Keywords

Examples

			7 is in the sequence because, for instance, (167^3-166^3)/7 = 11881 = 109^2.

Links

Crossrefs

Cf. A001921 (k=1), A144929 (k=7), A145124 (k=19), A145323 (k=31), A145700 (k=37), A145336 (k=43), A274972 (k=61), A145212 (k=67), A145309 (k=79), A145530 (k=91), A147530 (k=103), A145720 (k=127).
Cf. A003215 is a subsequence; A004611 contains this sequence.

Programs

  • Mathematica
    A004611=Select[Range[500],And@@(Mod[#,3]==1&)/@(First/@FactorInteger[#])&]; Select[A004611,Reduce[x^2+3== 12*#*y^2,{x,y},Integers]=!=False &] (* Ray Chandler, Jul 24 2016 *)

Extensions

More terms using solver at Alpern link by Ray Chandler, Jul 23 2016

A145693 Numbers X such that there exists Y in N with X^2=21*Y^2+7.

Original entry on oeis.org

14, 1526, 167846, 18461534, 2030600894, 223347636806, 24566209447766, 2702059691617454, 297201999868472174, 32689517925840321686, 3595549769842566913286, 395477785164756520139774, 43498960818353374648461854, 4784490212233706454810664166
Offset: 1

Views

Author

Richard Choulet, Oct 16 2008

Keywords

Examples

			a(1)=14 because the first relation is 14^2=21*3^2+7.

Links

Crossrefs

Cf. A144927, A144928, A144929, A144930.

Programs

  • Magma
    I:=[14,1526]; [n le 2 select I[n] else 110*Self(n-1)-Self(n-2): n in [1..15]]; // Vincenzo Librandi, Oct 21 2014
  • Mathematica
    CoefficientList[Series[14 (1 - x)/(x^2 - 110 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, Oct 21 2014 *)
LinearRecurrence[{110,-1},{14,1526},20] (* Harvey P. Dale, Sep 19 2024 *)
  • PARI
    Vec(-14*x*(x-1)/(x^2-110*x+1) + O(x^20)) \\ Colin Barker, Oct 21 2014

Formula

a(n+2) = 110*a(n+1)-a(n).
G.f.: -14*x*(x-1) / (x^2-110*x+1). - Colin Barker, Oct 21 2014

Extensions

Editing and more terms from Colin Barker, Oct 21 2014

A350982 a(0)=0, a(1)=49, thereafter a(n) = 14*a(n-1)-a(n-2)+42.

Original entry on oeis.org

0, 49, 728, 10185, 141904, 1976513, 27529320, 383434009, 5340546848, 74384221905, 1036038559864, 14430155616233, 200986140067440, 2799375805327969, 38990275134524168, 543064476078010425, 7563912389957621824, 105351708983328695153, 1467360013376644110360, 20437688478289688849929
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2022

Keywords

Comments

Arises in studying the equation x^3 - 7*y^2 = 1.

References

  • P.-F. Teilhet, Query 2228, L'Intermédiaire des Mathématiciens, 11 (1904), 44-45.

Crossrefs

Cf. A144929, A350979, A350980, A350981.
Showing 1-6 of 6 results.

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

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