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Search: id:A086541
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| A086541 |
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a(1) = 1, a(2) = 4; a(n) = smallest square of the form k*a(n-1) + a(n-2). |
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+0
2
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| 1, 4, 9, 49, 2116, 1104601, 1220041748025, 73506264463383837985201, 152589000107917580345020742323132226398704361, 16697690042926481335094757278121739739304599165141249657182619309750788555320629\ 50740889
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The sequence is infinite. Proof: In a(n) = k*a(n-1)+a(n-2), One ( and the largest) value of k is a(n-1)-2{a(n-2)}^(1/2). which gives a(n) = {a(n-1)-{a(n-2)^(1/2)}^2.
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EXAMPLE
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a(3) = 4*2 +1 = 9, a(4) = 5*9 +4 = 49, a(5) = 43*49 + 9 = 2116= 46^2.
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PROGRAM
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(PARI) A = vector(11); A[1] = 1; A[2] = 2; B = vector(11, i, A[i]^2); for (n = 3, 11, z = znstar(B[n - 1]); l = length(z[2]); c = vector(l, i, z[3][i]^(z[2][i]/2)); v = vector(2^l, i, A[n - 2]*prod(j = 1, l, c[j]^(i\2^(l - j)%2))); v = vecsort(lift(v)); print(B[n - 2], vector(2^l, i, v[i]^2%B[n - 1])); A[n] = if (v[1] == A[n - 2], v[2], v[1]); B[n] = A[n]^2; print(B[n])); (Wasserman)
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CROSSREFS
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Cf. A086542.
Sequence in context: A053967 A028945 A082875 this_sequence A053965 A058444 A053925
Adjacent sequences: A086538 A086539 A086540 this_sequence A086542 A086543 A086544
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 23 2003
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EXTENSIONS
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More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 28 2003
More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Mar 21 2005
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