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

%I A124101 #33 Sep 08 2022 08:45:28
%S A124101 1,56,2193,74200,2322401,69294456,2002105393,56527314200,
%T A124101 1568580924801,42944117148856,1163113467888593,31226091614554200,
%U A124101 832210422221287201,22042655816999563256,580763882378429351793,15231836751090861794200,397901671409627547409601,10358079848649863260537656
%N A124101 Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (7, 24, 25).
%D A124101 G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 197.
%H A124101 Vincenzo Librandi, <a href="/A124101/b124101.txt">Table of n, a(n) for n = 0..720</a>
%H A124101 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (56,-943,4200).
%F A124101 a(m) = (x^(m+2)*(z-y) + y^(m+2)*(x-z) + z^(m+2)*(y-x))/((x-y)*(y-z)*(z-x)).
%F A124101 From _Chai Wah Wu_, Sep 24 2016: (Start)
%F A124101 a(n) = 56*a(n-1) - 943*a(n-2) + 4200*a(n-3) for n > 2.
%F A124101 G.f.: 1/((1 - 7*x)*(1 - 24*x)*(1 - 25*x)). (End)
%F A124101 a(n) = 5^(2*n+4)/18 + 7^(n+2)/306 - 2^(3*n+6)*3^(n+2)/17. - _Vaclav Kotesovec_, Sep 25 2016
%e A124101 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.
%p A124101 seq(sum(7^(m-n)*sum(24^p*25^(n-p),p=0..n),n=0..m),m=0..N);
%t A124101 CoefficientList[Series[1/((1 - 7 x) (1 - 24 x) (1 - 25 x)), {x, 0, 15}], x] (* _Michael De Vlieger_, Sep 25 2016 *)
%t A124101 LinearRecurrence[{56, -943, 4200}, {1, 56, 2193}, 50] (* _Vincenzo Librandi_, Aug 18 2018 *)
%o A124101 (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
%o A124101 (PARI) x='x+O('x^99); Vec(1/((1-7*x)*(1-24*x)*(1-25*x))) \\ _Altug Alkan_, Sep 26 2016
%o A124101 (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
%Y A124101 Cf. A019682, A020000, A020341, A020346, A021664, A021684, A021844, A025942.
%K A124101 nonn,easy
%O A124101 0,2
%A A124101 _Giorgio Balzarotti_ and _Paolo P. Lava_, Nov 26 2006