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A254473
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24-hedral numbers: a(n) = (2*n + 1)*(8*n^2 + 14*n + 7).
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2
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7, 87, 335, 847, 1719, 3047, 4927, 7455, 10727, 14839, 19887, 25967, 33175, 41607, 51359, 62527, 75207, 89495, 105487, 123279, 142967, 164647, 188415, 214367, 242599, 273207, 306287, 341935, 380247, 421319, 465247, 512127, 562055, 615127, 671439, 731087
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OFFSET
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0,1
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COMMENTS
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This sequence is very close to the A046142 sequence: a(n) is asymptotic to A046142(n) as n tends to infinity.
The formula for A046142, the Haüy rhombic dodecahedral number, is remarkably similar, (2*n-1)*(8*n^2-14*n+7), where the first factor of the dodecahedral formula has "+1" versus "-1" in the 24-hedral formula, and the second factor of the former has "-14n" versus the latter of "+14n". Note that the rhombic dodecahedron has 24 faces, further explaining the relationship. The difference of these sequences is diff(n)=72*n^2 + 14. - Peter M. Chema, Jan 09 2016
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LINKS
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FORMULA
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G.f.: (7 + 59*x + 29*x^2 + x^3)/(1 - x)^4.
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) - a(n-4).
a(n) = 6*Sum_{k=0..n} (2*k+1)^2 + (2*n+1)^3. - Robert FERREOL, Nov 13 2023
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MAPLE
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seq((2*n + 1)*(8*n^2 + 14*n + 7), n=0..100); # Robert Israel, Jan 11 2016
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MATHEMATICA
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Table[(2 n + 1) (8 n^2 + 14 n + 7), {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {7, 87, 335, 847}, 40]
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PROG
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(PARI) vector(40, n, n--; (2*n+1)*(8*n^2+14*n+7)) \\ Bruno Berselli, Mar 27 2015
(Sage) [(2*n+1)*(8*n^2+14*n+7) for n in (0..40)] # Bruno Berselli, Mar 27 2015
(Magma) [(2*n+1)*(8*n^2+14*n+7): n in [0..40]]; // Bruno Berselli, Mar 27 2015
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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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