A085377 -id:A085377 - cp's OEIS frontend

A085409 Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.

0, 12, 84, 270, 624, 1200, 2052, 3234, 4800, 6804, 9300, 12342, 15984, 20280, 25284, 31050, 37632, 45084, 53460, 62814, 73200, 84672, 97284, 111090, 126144, 142500, 160212, 179334, 199920, 222024, 245700, 271002, 297984, 326700, 357204, 389550, 423792, 459984

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 13 2003

Parametric representation of the solution is (x, y, z) = (6n^3, 3n^3, 3n^2), thus getting a(n) = 9n^3 + 3n^2.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A085377.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 6*x*(2 + 6*x + x^2) /(1 - x)^4)); // Marius A. Burtea, Oct 25 2019

Mathematica
```
Table[9n^3 + 3n^2, {n, 0, 34}]
```

concat(0, Vec(6*x*(2 + 6*x + x^2) /(1 - x)^4 + O(x^40))) \\ Colin Barker, Oct 25 2019

a(n) = 9*n^3 + 3*n^2.
From Colin Barker, Oct 25 2019: (Start)
G.f.: 6*x*(2 + 6*x + x^2) /(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(3)*Pi/6 + 3*log(3)/2 - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 - Pi/sqrt(3) - 2*log(2) + 3. (End)

More terms from Robert G. Wilson v, Aug 16 2003

A085479 Product of three solutions of the Diophantine equation x^3 - y^3 = z^2.

Original entry on oeis.org

728, 93184, 1592136, 11927552, 56875000, 203793408, 599539304, 1526726656, 3482001432, 7280000000, 14186660488, 26085556224, 45680920376, 76741030912, 124385625000, 195421011968, 298726553944, 445696183296, 650738625992

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 15 2003

Parametric representation of the solution is (x,y,z) = (8n^2, 7n^2, 13n^3), thus getting a(n) = 728*n^7.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001015 (n^7), A085377.

728*Range[20]^7 (* Harvey P. Dale, May 27 2012 *)

Vec(728*x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6) / (1 - x)^8 + O(x^25)) \\ Colin Barker, Oct 25 2019

a(n) = 728*n^7.
From Colin Barker, Oct 25 2019: (Start)
G.f.: 728*x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

More terms from Matthew Conroy, Jan 16 2006

cp's OEIS Frontend

A085409 Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.

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