A124100
Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).
Original entry on oeis.org
1, 40, 1089, 25160, 531521, 10625640, 204744769, 3844391560, 70827391041, 1286290883240, 23101397290049, 411249127989960, 7269184506192961, 127745926316548840, 2234231991096868929, 38920247688751940360
Offset: 0
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.
- G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 196.
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seq(sum(8^(m-n)*sum(15^p*17^(n-p),p=0..n),n=0..m),m=0..N);
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LinearRecurrence[{40,-511,2040},{1,40,1089},30] (* Harvey P. Dale, May 25 2025 *)
A124101
Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (7, 24, 25).
Original entry on oeis.org
1, 56, 2193, 74200, 2322401, 69294456, 2002105393, 56527314200, 1568580924801, 42944117148856, 1163113467888593, 31226091614554200, 832210422221287201, 22042655816999563256, 580763882378429351793, 15231836751090861794200, 397901671409627547409601, 10358079848649863260537656
Offset: 0
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.
- G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 197.
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[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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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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seq(sum(7^(m-n)*sum(24^p*25^(n-p),p=0..n),n=0..m),m=0..N);
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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 *)
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x='x+O('x^99); Vec(1/((1-7*x)*(1-24*x)*(1-25*x))) \\ Altug Alkan, Sep 26 2016
A124099
Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (5, 12, 13).
Original entry on oeis.org
1, 30, 619, 10920, 177061, 2726130, 40547359, 588485820, 8387148121, 117876868230, 1638536364499, 22574666496720, 308755233696781, 4197234089634330, 56765041887676039, 764357559726523620
Offset: 0
a(2)=619 because Sum_(x^i*y^j*z^k) = x^2 + y^2 + z^2 + x*y + x*z + y*z = 5^2 + 12^2 + 13^2 + 5*12 + 5*13 + 12*13 = 619 and x^2 + y^2 = z^2.
- G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 196.
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