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A255842
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a(n) = 2*n^2 + 12.
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1
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12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430
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OFFSET
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0,1
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COMMENTS
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This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
Equivalently, numbers m such that 2*m-24 is a square.
For n = 0..10, a(n)-1 is prime (see A092968).
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LINKS
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FORMULA
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G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
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MATHEMATICA
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Table[2 n^2 + 12, {n, 0, 50}]
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PROG
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(PARI) vector(50, n, n--; 2*n^2+12)
(Sage) [2*n^2+12 for n in (0..50)]
(Magma) [2*n^2+12: n in [0..50]];
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CROSSREFS
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Cf. similar sequences listed in A255843.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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