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A217482
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Quarter-square tetrahedrals: a(n) = k*(k - 1)*(k - 2)/6, k = A002620(n).
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0
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0, 0, 0, 0, 4, 20, 84, 220, 560, 1140, 2300, 4060, 7140, 11480, 18424, 27720, 41664, 59640, 85320, 117480, 161700, 215820, 287980, 374660, 487344, 620620, 790244, 988260, 1235780, 1521520, 1873200, 2275280, 2763520, 3317040, 3981264, 4728720, 5616324, 6608580
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OFFSET
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0,5
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COMMENTS
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Observation: (3/2)*a(n) + 2 is a power of 2 up to n = 6 (giving {2, 2, 2, 2, 8, 32, 128}).
Conjecture: There are no other tetrahedral numbers (Tetra_n = A000292) > 84 such that (3/2)*Tetra_n + 2 is a power of 2. This is true to at least 1.41*10^1505 per computer check by Charles R Greathouse IV on Physics Forums (Nov 2010).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
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FORMULA
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a(n) = (1/6)*floor(n^2/4)*(floor(n^2/4)-1)*(floor(n^2/4)-2).
G.f.: -4*x^4*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^7*(x+1)^5). - Colin Barker, Oct 11 2012
Sum_{n>=4} 1/a(n) = Pi^2/2 - 5/12 - 3*Pi*cot(sqrt(2)*Pi)/(2*sqrt(2)) - 6*Pi*tan(sqrt(5)*Pi/2)/sqrt(5). - Amiram Eldar, Feb 17 2024
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MAPLE
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a:= n-> binomial(floor(n^2/4), 3):
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MATHEMATICA
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(#*(#-1)*(#-2)/6)& /@ Table[Floor[n^2/4], {n, 0, 20}] (* Amiram Eldar, Feb 17 2024 *)
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PROG
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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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EXTENSIONS
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STATUS
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approved
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