A124101 Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (7, 24, 25).

1, 56, 2193, 74200, 2322401, 69294456, 2002105393, 56527314200, 1568580924801, 42944117148856, 1163113467888593, 31226091614554200, 832210422221287201, 22042655816999563256, 580763882378429351793, 15231836751090861794200, 397901671409627547409601, 10358079848649863260537656

Offset

0,2

References

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 197.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..720

Index entries for linear recurrences with constant coefficients, signature (56,-943,4200).

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) = 56*a(n-1) - 943*a(n-2) + 4200*a(n-3) for n > 2.

G.f.: 1/((1 - 7*x)*(1 - 24*x)*(1 - 25*x)). (End)

a(n) = 5^(2*n+4)/18 + 7^(n+2)/306 - 2^(3*n+6)*3^(n+2)/17. - Vaclav Kotesovec, Sep 25 2016

Example

a(2) = 2193 because x^2 + y^2 + z^2 + x*y + x*z + y*z = 7^2 + 24^2 + 25^2 + 7*24 + 7*25 + 24*25 = 2193 and x^2 + y^2 = z^2.

Maple

seq(sum(7^(m-n)*sum(24^p*25^(n-p), p=0..n), n=0..m), m=0..N);

Mathematica

CoefficientList[Series[1/((1 - 7 x) (1 - 24 x) (1 - 25 x)), {x, 0, 15}], x] (* Michael De Vlieger, Sep 25 2016 *)
LinearRecurrence[{56, -943, 4200}, {1, 56, 2193}, 50] (* Vincenzo Librandi, Aug 18 2018 *)

Prog

(Magma) [5^(2*n+4)/18 + 7^(n+2)/306 - 2^(3*n+6)*3^(n+2)/17 : n in [0..20]]; // Wesley Ivan Hurt, Sep 26 2016
(PARI) x='x+O('x^99); Vec(1/((1-7*x)*(1-24*x)*(1-25*x))) \\ Altug Alkan, Sep 26 2016
(Magma) I:=[1, 56, 2193]; [n le 3 select I[n] else 56*Self(n-1)-943*Self(n-2)+4200*Self(n-3): n in [1..90]]; // Vincenzo Librandi, Aug 18 2018

Crossrefs

Cf. A019682, A020000, A020341, A020346, A021664, A021684, A021844, A025942.

Sequence in context: A111781 A221398 A280803 * A198948 A003696 A199709

Adjacent sequences: A124098 A124099 A124100 * A124102 A124103 A124104

Keyword

nonn,easy

Author

Giorgio Balzarotti and Paolo P. Lava, Nov 26 2006

Status

approved