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
Examples
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.
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).
Programs
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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
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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
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Maple
seq(sum(7^(m-n)*sum(24^p*25^(n-p),p=0..n),n=0..m),m=0..N);
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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 *) -
PARI
x='x+O('x^99); Vec(1/((1-7*x)*(1-24*x)*(1-25*x))) \\ Altug Alkan, Sep 26 2016
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