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A001533
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a(n) = (8*n+1)*(8*n+7).
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3
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7, 135, 391, 775, 1287, 1927, 2695, 3591, 4615, 5767, 7047, 8455, 9991, 11655, 13447, 15367, 17415, 19591, 21895, 24327, 26887, 29575, 32391, 35335, 38407, 41607, 44935, 48391, 51975, 55687, 59527, 63495, 67591, 71815, 76167, 80647, 85255, 89991, 94855
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OFFSET
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0,1
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COMMENTS
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This is A028560(8*n+1), and thus a(n) + 9 is a square. (See formulas.)
7 is the only prime number of this sequence in which all odd prime factors occur.
Each prime factor p appears exactly twice in any interval of p consecutive terms. If a(m) and a(n) are within such an interval containing p, then m + n == -1 (mod p). (End)
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LINKS
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FORMULA
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a(n) + 9 = ((a(n+1) - a(n-1))/32)^2 = A017113(n)^2.
a(2*n) = (a(n+1) - a(n-1))*n + 7. (End)
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(cot(Pi/16)) + sin(Pi/8) * log(cot(3*Pi/16)))/12.
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/8)*cos(sqrt(5/2)*Pi/4).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/8)*cos(sqrt(2)*Pi/4). (End)
G.f.: -(7+114*x+7*x^2)/(x-1)^3 . - R. J. Mathar, Apr 23 2024
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MATHEMATICA
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a[n_] := (8 n + 1)*(8 n + 7); Array[a, 40, 0] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{3, -3, 1}, {7, 135, 391}, 40] (* Harvey P. Dale, Jan 07 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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