A124100 Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

1, 40, 1089, 25160, 531521, 10625640, 204744769, 3844391560, 70827391041, 1286290883240, 23101397290049, 411249127989960, 7269184506192961, 127745926316548840, 2234231991096868929, 38920247688751940360

Offset

0,2

References

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 196.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..811

Index entries for linear recurrences with constant coefficients, signature (40, -511, 2040).

Formula

a(m) = (x^(m+2)*(z-y) + y^(m+2)*(x-z) + z^(m+2)*(y-x))/((x-y)*(y-z)*(z-x)).

From Chai Wah Wu, Sep 24 2016: (Start)

a(n) = 40*a(n-1) - 511*a(n-2) + 2040*a(n-3) for n > 2.

G.f.: 1/((1 - 8*x)*(1 - 15*x)*(1 - 17*x)). (End)

a(n) = 2^(3*n+6)/63 - 15^(n+2)/14 + 17^(n+2)/18. - Vaclav Kotesovec, Sep 25 2016

Example

a(2) = 1089 because x^2 + y^2 + z^2 + x*y + x*z + y*z = 8^2 + 15^2 + 17^2 + 8*15 + 8*17 + 15*17 = 1089 and x^2 + y^2 = z^2.

Maple

seq(sum(8^(m-n)*sum(15^p*17^(n-p), p=0..n), n=0..m), m=0..N);

Mathematica

LinearRecurrence[{40, -511, 2040}, {1, 40, 1089}, 30] (* Harvey P. Dale, May 25 2025 *)

Prog

(PARI) a(n)=2^(3*n+6)/63-15^(n+2)/14+17^(n+2)/18 \\ Charles R Greathouse IV, Jun 03 2026

Crossrefs

Cf. A019682, A020000, A020340-A020342, A020344-A020346, A021664, A021684, A021844, A025942, A077515.

Sequence in context: A075907 A062143 A284838 * A071952 A331906 A144914

Adjacent sequences: A124097 A124098 A124099 * A124101 A124102 A124103

Keyword

nonn,easy

Author

Giorgio Balzarotti and Paolo P. Lava, Nov 26 2006

Status

approved