A254473 -id:A254473 - cp's OEIS frontend

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

Offset: 1

Eric W. Weisstein

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

H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.

G. C. Greubel, Table of n, a(n) for n = 1..5000
R.-J. Haüy, Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées, 1784.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000 which lists Haüy rhombic dodecahedral numbers as "RhoDod(n)."
Eric Weisstein's World of Mathematics, Haüy Construction.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000447, A001845, A016755, A069072, A254473.

[(2*n-1)*(8*n^2-14*n+7): n in [1..40]]; // Vincenzo Librandi, Mar 29 2015

A046142:=n->(2*n-1)*(8*n^2-14*n+7): seq(A046142(n), n=1..50); # Wesley Ivan Hurt, Mar 02 2016

Table[(2 n - 1) (8 n^2 - 14 n + 7), {n, 40}] (* Vincenzo Librandi, Mar 29 2015 *)
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 *)

Vec(x*(7*x^3+59*x^2+29*x+1)/(x-1)^4 + O(x^50)) \\ Michel Marcus, Mar 24 2015

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) = A016755(n) + A069072(n-1). - Luciano Ancora, Mar 23 2015
a(n) = A016755(n) + 6*A000447(n-1). - Luciano Ancora, Mar 23 2015
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

cp's OEIS Frontend

A046142 Haüy rhombic dodecahedral numbers.

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