A145124 -id:A145124 - cp's OEIS frontend

A145123 Numbers n such that there exists x in N : (x+1)^3-x^3=19*n^2.

1, 301, 90901, 27451801, 8290353001, 2503659154501, 756096774306301, 228338722181348401, 68957538001992910801, 20824948137879677713501, 6289065380101660676566501, 1899276919842563644645369801

Offset: 1

Richard Choulet, Oct 02 2008

			a(1)=1 because 3^3-2^3=19*1.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (302,-1).

Cf. A145124.

CoefficientList[Series[(1 - x)/(1 - 302 x + x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Oct 18 2014 *)

Vec(-x*(-1+x)/(1-302*x+x^2) + O(x^30)) \\ Colin Barker, Oct 18 2014

a(n+2) = 302*a(n+1)-a(n).

G.f.: -x*(-1+x) / ( 1-302*x+x^2 ). - R. J. Mathar, Nov 27 2011

Editing and more terms from Colin Barker, Oct 18 2014

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

Colin Barker, Jul 13 2016

nonn

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

Ray Chandler, Table of n, a(n) for n = 1..10000
Dario A. Alpern, Quadratic two integer variable equation solver

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.

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 *)

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

A145122 Numbers X such that (X+19)^3-X^3 is a square.

Original entry on oeis.org

38, 14383, 4346478, 1312624823, 396408352918, 119714009959263, 36153234599347358, 10918157134992945703, 3297247301533270257798, 995757766905912624912143, 300715548358284079453212238

Offset: 1

Richard Choulet, Oct 02 2008

			a(1)=38 because 57^3-38^3=361^2

Index entries for linear recurrences with constant coefficients, signature (303,-303,1).

LinearRecurrence[{303,-303,1},{38,14383,4346478},20] (* Harvey P. Dale, Sep 24 2016 *)

a(n+2)=302*a(n+1)-a(n)+2850

a(n)=19*A145124(n). G.f.: 19*x*(-2-151*x+3*x^2)/((x-1)*(x^2-302*x+1)). - R. J. Mathar, Oct 26 2009.

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A145123 Numbers n such that there exists x in N : (x+1)^3-x^3=19*n^2.

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

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A145122 Numbers X such that (X+19)^3-X^3 is a square.

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