A046142 Haüy rhombic dodecahedral numbers.

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, 762193, 827025

Offset

1,2

Comments

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

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 125.

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

Links

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).

Formula

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

a(n) = A001845(2*n-2) + 8*A002412(n-1). - Derek Delk, Jan 17 2026

Maple

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

Mathematica

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 *)

Prog

(PARI) Vec(x*(7*x^3+59*x^2+29*x+1)/(x-1)^4 + O(x^50)) \\ Michel Marcus, Mar 24 2015
(Magma) [(2*n-1)*(8*n^2-14*n+7): n in [1..40]]; // Vincenzo Librandi, Mar 29 2015

Crossrefs

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

Sequence in context: A279867 A230650 A231360 * A135827 A189180 A209531

Adjacent sequences: A046139 A046140 A046141 * A046143 A046144 A046145

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved