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A130284
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Integers n>0 such (2n+1)^2(m^2-1)+1 is a square for some integer m>1.
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5
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7, 17, 31, 49, 71, 97, 104, 127, 161, 199, 241, 287, 337, 391, 449, 511, 577, 594, 647, 721, 799, 881, 967, 1057, 1151, 1249, 1351, 1455, 1457, 1567, 1681, 1799, 1921, 1952, 2047, 2177, 2311, 2449, 2591, 2737, 2887, 3041, 3199, 3361, 3527, 3697, 3871, 4049
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| All terms > 4 in A130283 are odd squares, but not all odd squares are in that sequence: This sequence here gives the exceptions as (2a(n)+1)^2. The sequence consists mainly of the subsequences: (1) A056220(k) = 2k^2-1 with k>1: {7,17,31,49,...}, for which m=k gives (1+2*A056220(k))^2(k^2-1)+1 = k^2(4k^2-3)^2; (2) 2*A079414(k) = 2k^2(4k^2-3) with k>1: {104,594,1952,4850,...}, for which m=k gives (1+4*A079414(k))^2(k^2-1)+1 = k^2(16k^4-20k^2+5)^2. A third subsequence starts {1455,20195,...}; up to 20195, all terms are in one of these subsequences.
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FORMULA
| A130284 = { P[k](m) ; k=1,2,3,..., m=2,3,4,... } where P[k] = (sqrt((X^2 Q[k]^2 -1)/(X^2 -1))-1)/2 and Q[0] = Q[ -1] = 1, Q[k+1] = (4X^2 -2)*Q[k]-Q[k-1]. Furthermore, (2P[k](m)+1)^2 (m^2 -1)+1 = m^2 Q[k](m)^2, thus A130280(P[k](m)) <= m. So far, no case is known where we have strict inequality.
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EXAMPLE
| Up to k=17, a(k)=P[1](k+1) with P[1]=2x^2 -1, A130280(a(k))=k+1.
a(18) = P[2](2) < P[1](19) with P[2]=2x^2 (4x^2 -3), A130280(a(18))=2.
a(106) = P[1](100) < a(107) = P[3](3) < a(108) = P[4](2) < a(109) = P[1](101).
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PROG
| (PARI) A130284( LIM=9999, START=1 )={ local(N); for( n=START, LIM, N=(2*n+1)^2; for( m=2, sqrtint(n>>1+1), if(!issquare( N*(m^2-1)+1 ), next); print1(n", "); next(2))) }
(PARI) {Q(k, x=x)=if(m>0, (4*x^2-2)*Q(k-1, x)-Q(k-2, x), 1)} {P(k, x=x)=if(type(x=(x^2*Q(k, x)^2-1)/(x^2-1))!="t_POL", sqrtint(x)\2, ((-1)^k*Pol(sqrt(x))-1)/2)}
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CROSSREFS
| Cf. A084702, A130280, A130284, A130288.
Cf. A130280, A130283, A130281.
Sequence in context: A094080 A046118 A120092 * A056220 A024840 A024835
Adjacent sequences: A130281 A130282 A130283 * A130285 A130286 A130287
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KEYWORD
| nonn
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AUTHOR
| M. F. Hasler (Maximilian.Hasler(AT)gmail.com), May 24 2007, May 29 2007
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