A155137 a(n) = nonnegative value y such that (A155135(n), y) is a solution to the Diophantine equation x^3+28*x^2 = y^2.
0, 27, 48, 57, 48, 15, 0, 48, 147, 288, 477, 720, 1023, 1392, 1833, 2352, 2955, 3648, 4437, 5328, 6327, 7440, 8673, 10032, 11523, 13152, 14925, 16848, 18927, 21168, 23577, 26160, 28923, 31872, 35013, 38352, 41895, 45648, 49617, 53808, 58227
Offset: 1
Examples
(A155135(3), a(3)) = (-24, 48) is a solution: (-24)^3+28*(-24)^2 = -13824+16128 = 2304 = 48^2. (A155135(7), a(8)) = (0, 0) is a solution: 0^3+28*0^2 = 0 = 0^2. (A155135(8), a(8)) = (8, 48) is a solution: 8^3+28*8^2 = 512+1792 = 2304 = 48^2.
Links
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
[ Integers()!SquareRoot(a) : n in [ -30..1500] | IsSquare(a) where a is n^3+28*n^2 ];
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Mathematica
CoefficientList[Series[3x (9-20x+9x^2+16x^5+x^6-19x^7+x^8+5x^9)/(1-x)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,27,48,57,48,15,0,48,147,288,477},50] (* Harvey P. Dale, Sep 02 2021 *) -
PARI
a(n)=if(n>6, n^3 - 3*n^2 - 25*n + 27, [0, 27, 48, 57, 48, 15, 0][n+1]) \\ Charles R Greathouse IV, Oct 18 2022
Formula
G.f.: 3*x*(9-20*x+9*x^2+16*x^5+x^6-19*x^7+x^8+5*x^9)/(1-x)^4.
Comments