A080859 -id:A080859 - cp's OEIS frontend

A214731 a(n) = n^3 - 2*n^2 - 1.

-2, -1, 8, 31, 74, 143, 244, 383, 566, 799, 1088, 1439, 1858, 2351, 2924, 3583, 4334, 5183, 6136, 7199, 8378, 9679, 11108, 12671, 14374, 16223, 18224, 20383, 22706, 25199, 27868, 30719, 33758, 36991, 40424, 44063, 47914, 51983, 56276, 60799, 65558, 70559

Offset: 1

Marco Piazzalunga, Jul 27 2012

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

Cf. A080859, A085490, A144390 (first differences), A152619.

Similar sequences: A152015 (of the type m^3+2m^2-1), A081437 (m^3-2m^2+1).

[n^3-2*n^2-1: n in [1..50]]; // Vincenzo Librandi, Jul 29 2012

A214731:=n->n^3-2*n^2-1: seq(A214731(n), n=1..60); # Wesley Ivan Hurt, Apr 18 2017

Table[n^3 - 2 n^2 - 1, {n, 50}] (* Vincenzo Librandi, Jul 29 2012 *)

a(n)=n^3-2*n^2-1 \\ Charles R Greathouse IV, Jul 27 2012

[n^2*(n-2)-1 for n in range(1,51)] # G. C. Greubel, Dec 31 2023

From Bruno Berselli, Jul 27 2012: (Start)
G.f.: -x*(2-7*x-x^3)/(1-x)^4.
a(n) = A085490(n-1) + 2.
a(n) = A152619(n-2) - 1 for n>1.
a(n) - a(n-2) = A080859(n-2) - 1 for n>2. (End)
E.g.f.: 1 - (1-x)*(1+x)^2*exp(x). - G. C. Greubel, Dec 31 2023

a(3) corrected by Charles R Greathouse IV, Jul 27 2012

A372220 Four-column table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that a > (a + b + c - d)/2 = 2n(n + 1) and b = c.

Original entry on oeis.org

17, 20, 20, 33, 31, 42, 42, 67, 49, 72, 72, 113, 71, 110, 110, 171, 97, 156, 156, 241, 127, 210, 210, 323, 161, 272, 272, 417, 199, 342, 342, 523, 241, 420, 420, 641, 287, 506, 506, 771, 337, 600, 600, 913, 391, 702, 702, 1067, 449, 812, 812, 1233, 511, 930, 930, 1411, 577, 1056, 1056, 1601

Offset: 2

Miguel-Ángel Pérez García-Ortega, Apr 22 2024

A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.

			Table begins:
  n=2:   17,   20,    20,    33;
  n=3:   31,   42,    42,    67;
  n=4:   49,   72,    72,   113;
  n=5:   71,  110,   110,   171;
  n=6:   97,  156,   156,   241;

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Miguel-Ángel Pérez García-Ortega, Teorema 10.13

Cf. A372219, A056220 (first column), A002943 (second column), A080859 (fourth column).

cuaternas={};Do[cuaternas=Join[cuaternas,{2n^2+4n+1,4n^2+2n,4n^2+2n,6n^2+4n+1}],{n,2,35}];cuaternas

Row n = (a, b, c, d) = (2n^2 + 4n + 1, 4n^2 + 2n, 4n^2 + 2n, 6n^2 + 4n + 1).

cp's OEIS Frontend

A214731 a(n) = n^3 - 2*n^2 - 1.

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