A274972 -id:A274972 - cp's OEIS frontend

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

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

A275060 Numbers n such that there exists x in N : (x+1)^3 - x^3 = 61*n^2.

Original entry on oeis.org

1, 973, 947701, 923059801, 899059298473, 875682833652901, 852914180918627101, 830737536531909143473, 809137507667898587115601, 788099101730996691941451901, 767607715948483110052387035973, 747649127234720818194333031585801

Offset: 1

Colin Barker, Jul 15 2016

			973 is in the sequence because 973^2 = 946729 = ((4387+1)^3-4387^3)/61.

Colin Barker, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (974,-1).

Cf. A274972.

LinearRecurrence[{974,-1},{1,973}, 50] (* G. C. Greubel, Jul 15 2016 *)

PARI
```
Vec(x*(1-x)/(1-974*x+x^2) + O(x^20))
```

G.f.: x*(1-x) / (1-974*x+x^2).
a(n) = 974*a(n-1) - a(n-2) for n>2.
a(n) = ((487 + 36*sqrt(183))^(-n)*(2 - 9*sqrt(3/61) + (2+9*sqrt(3/61))* (487 + 36*sqrt(183))^(2*n)))/4.

cp's OEIS Frontend

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

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