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A322830
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a(n) = 32*n^3 + 48*n^2 + 18*n + 1.
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2
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1, 99, 485, 1351, 2889, 5291, 8749, 13455, 19601, 27379, 36981, 48599, 62425, 78651, 97469, 119071, 143649, 171395, 202501, 237159, 275561, 317899, 364365, 415151, 470449, 530451, 595349, 665335, 740601, 821339, 907741, 999999, 1098305, 1202851, 1313829, 1431431, 1555849
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) + sqrt(a(n)^2 - 1) = (sqrt(n+1) + sqrt(n))^6.
a(n) - sqrt(a(n)^2 - 1) = (sqrt(n+1) - sqrt(n))^6.
a(n) = A144129(2*n+1) = T_3(2*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind.
G.f.: (1 + x)*(1 + 94*x + x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
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EXAMPLE
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(sqrt(2) + sqrt(1))^6 = 99 + 70*sqrt(2) = 99 + sqrt(99^2 - 1). So a(1) = 99.
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MAPLE
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MATHEMATICA
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CoefficientList[Series[(1 + x) (1 + 94 x + x^2)/(1 - x)^4, {x, 0, 36}], x] (* or *)
Table[32n^3+48n^2+18n+1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 99, 485, 1351}, 40] (* Harvey P. Dale, Mar 11 2019 *)
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PROG
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(PARI) {a(n) = 32*n^3+48*n^2+18*n+1}
(PARI) {a(n) = polchebyshev(3, 1, 2*n+1)}
(PARI) Vec((1 + x)*(1 + 94*x + x^2) / (1 - x)^4 + O(x^40)) \\ Colin Barker, Dec 27 2018
(GAP) a:=List([0..40], n->32*n^3+48*n^2+18*n+1);; Print(a); # Muniru A Asiru, Jan 02 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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