A058895 -id:A058895 - cp's OEIS frontend

A204767 Quadruples (a,b,c,d) of the form ( n(n^3-1), n^3-1, 2n^3+1, n*(n^3+2) ).

0, 0, 3, 3, 14, 7, 17, 20, 78, 26, 55, 87, 252, 63, 129, 264, 620, 124, 251, 635, 1290, 215, 433, 1308, 2394, 342, 687, 2415, 4088, 511, 1025, 4112, 6552, 728, 1459, 6579, 9990, 999, 2001, 10020, 14630, 1330, 2663, 14663, 20724, 1727, 3457, 20760, 28548, 2196, 4395, 28587

Offset: 1

Vincenzo Librandi, Mar 04 2012

Four consecutive (a,b,c,d) in the sequence are solutions to a^3+b^3+c^3 = d^3, that is a(4k+1)^3+a(4k+2)^3+a(4k+3)^3 = a(4k+4)^3.
Also, A058895(n)^3 + A068601(n)^3 + A033562(n)^3 = A185065(n)^3.
The sequence corresponds to the case m=1 in the identity (n*(n^3-m^3))^3+(m*(n^3-m^3))^3+(m*(2*n^3+m^3))^3 = (n*(n^3+2*m^3))^3.
G. H. Hardy and E. M. Wright gave this identity in their "An Introduction to the Theory of Numbers" together with (n*(n^3-2*m^3))^3+(m*(n^3+m^3))^3+(m*(2*n^3-m^3))^3 = (n*(n^3+m^3))^3 (see References). - Bruno Berselli, Mar 13 2012

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 2008 (Sixth edition), Par. 13.7.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A058895, A068601, A033562, A185065.

&cat[[n*(n^3-1), n^3-1, 2*n^3+1, n*(n^3+2)]: n in [1..40]];

Flatten[Table[{n^4 - n, n^3 - 1, 2 n^3 + 1, n^4 + 2 n}, {n, 1, 40}]] (* Vincenzo Librandi, Jan 02 2014 *)

A365372 Array read by ascending antidiagonals: A(n, k) = n(kn^2 - 1) with k > 0.

Original entry on oeis.org

0, 6, 1, 24, 14, 2, 60, 51, 22, 3, 120, 124, 78, 30, 4, 210, 245, 188, 105, 38, 5, 336, 426, 370, 252, 132, 46, 6, 504, 679, 642, 495, 316, 159, 54, 7, 720, 1016, 1022, 858, 620, 380, 186, 62, 8, 990, 1449, 1528, 1365, 1074, 745, 444, 213, 70, 9, 1320, 1990, 2178, 2040, 1708, 1290, 870, 508, 240, 78, 10

Offset: 1

Stefano Spezia, Sep 02 2023

			The array begins:
    0,   1,   2,   3,    4,    5, ...
    6,  14,  22,  30,   38,   46, ...
   24,  51,  78, 105,  132,  159, ...
   60, 124, 188, 252,  316,  380, ...
  120, 245, 370, 495,  620,  745, ...
  210, 426, 642, 858, 1074, 1290, ...
  ...

Cf. A007531, A017137, A035328 (k=4), A058895 (main diagonal), A365373 (antidiagonal sums).

A[n_,k_]:=n(k n^2-1); Table[A[n-k+1,k],{n,11},{k,n}]//Flatten

G.f.: x*y*(x^2*y + y - 2*x*(y - 3))/((1 - x)^4*(1 - y)^2).
1st column: A(n, 1) = A007531(n+1).
2nd row: A(2, n) = A017137(n-1).

cp's OEIS Frontend

A204767 Quadruples (a,b,c,d) of the form ( n(n^3-1), n^3-1, 2n^3+1, n*(n^3+2) ).

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A365372 Array read by ascending antidiagonals: A(n, k) = n(kn^2 - 1) with k > 0.

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cp's OEIS Frontend

A204767 Quadruples (a,b,c,d) of the form ( n*(n^3-1), n^3-1, 2*n^3+1, n*(n^3+2) ).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

A365372 Array read by ascending antidiagonals: A(n, k) = n*(k*n^2 - 1) with k > 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A204767 Quadruples (a,b,c,d) of the form ( n(n^3-1), n^3-1, 2n^3+1, n*(n^3+2) ).

A365372 Array read by ascending antidiagonals: A(n, k) = n(kn^2 - 1) with k > 0.