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A166929
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Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some coprime integers a, b, c.
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4
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1, 13, 1525, 2165017, 42422452969, 7658246457672229, 15512114571284835412957, 452005526897888844293504165425, 126314830357375266295717376544111167953, 368440923990671763222767414151367493861848396861
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OFFSET
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1,2
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COMMENTS
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Values of m in coprime solutions to 2m^4 = c^4 + d^2 (so that a, b = (c^2 +- d)/2).
Corresponding values of c are given in A167437.
Terms with positive a,b,c are given in A166930.
This is a generalized Somos-4 sequence. - Michael Somos, Jan 29 2023
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LINKS
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FORMULA
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a/m^2 = (-8*u^2 + 32*v + u^4 + 4*v^2 - 4*u^3 + 16*u*v)/(8 + u^2 + 4*u)^2 and b/m^2 = (4*u^3 - 8*u^2 - 4*v*u^2 - 16*u*v + 64)/(8 + u^2 + 4*u)^2 where (u,v) is a rational point on the elliptic curve v^2 = u^3 + 8*u.
a(n) = a(1-n) = (1764*a(n-1)*a(n-3) - 3107*a(n-2)^2)/a(n-4) = A360187(2*n-1) for all n in Z. - Michael Somos, Jan 29 2023
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PROG
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(PARI) {a(n) = my(A); if(n<1, n=1-n); A = vector(max(4, n+2)); A[1] = 13; A[2] = 1; A[3] = 1; A[4] = 13; for(k=5, n+2, A[k] = (1764*A[k-1]*A[k-3] - 3107*A[k-2]^2)/A[k-4]); A[n+2]}; /* Michael Somos, Jan 29 2023 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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