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A046142
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Haüy rhombic dodecahedral numbers.
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5
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1, 33, 185, 553, 1233, 2321, 3913, 6105, 8993, 12673, 17241, 22793, 29425, 37233, 46313, 56761, 68673, 82145, 97273, 114153, 132881, 153553, 176265, 201113, 228193, 257601, 289433, 323785, 360753, 400433, 442921, 488313, 536705, 588193, 642873, 700841
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OFFSET
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1,2
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COMMENTS
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The Haüy rhombic dodecahedral formula is remarkably similar to that of A254473, the 24-hedral numbers: a(n) = (2*n+1)*(8*n^2+14*n+7). Compare with (2*n-1)*(8*n^2-14*n+7); the differences are simple: (1) the first factor of the dodecahedral formula has "+1" and the 24-hedral formula has "-1"; (2) the second factor of the former has "-14n" and the latter has "+14n". Note that the rhombic dodecahedron has 24 edges. The difference between these sequences is diff(n) = 72*n^2 + 14. - Peter M. Chema, Jan 09 2016
Named after the French priest and mineralogist René Just Haüy (1743-1822). - Amiram Eldar, Jun 22 2021
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REFERENCES
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H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.
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LINKS
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FORMULA
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a(n) = (2*n - 1)*(8*n^2 - 14*n + 7).
G.f.: x*(7*x^3 +59*x^2 +29*x +1)/(1-x)^4. - Colin Barker, Oct 26 2012
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4. - Wesley Ivan Hurt, Mar 02 2016
E.g.f.: (-7 +8*x +12*x^2 +16*x^3)*exp(x) + 7. - G. C. Greubel, Nov 04 2017
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MAPLE
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {1, 33, 185, 553}, 20] (* Eric W. Weisstein, Sep 27 2017 *)
CoefficientList[Series[(1 + 29 x + 59 x^2 + 7 x^3)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)
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PROG
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(PARI) Vec(x*(7*x^3+59*x^2+29*x+1)/(x-1)^4 + O(x^50)) \\ Michel Marcus, Mar 24 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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