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%I #73 Feb 12 2023 13:30:39
%S 1,33,185,553,1233,2321,3913,6105,8993,12673,17241,22793,29425,37233,
%T 46313,56761,68673,82145,97273,114153,132881,153553,176265,201113,
%U 228193,257601,289433,323785,360753,400433,442921,488313,536705,588193,642873,700841
%N Haüy rhombic dodecahedral numbers.
%C 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
%C Named after the French priest and mineralogist René Just Haüy (1743-1822). - _Amiram Eldar_, Jun 22 2021
%D H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.
%H G. C. Greubel, <a href="/A046142/b046142.txt">Table of n, a(n) for n = 1..5000</a>
%H R.-J. Haüy, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k1060890">Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées</a>, 1784.
%H Jonathan Vos Post, <a href="http://magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a> which lists Haüy rhombic dodecahedral numbers as "RhoDod(n)."
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HauyConstruction.html">Haüy Construction</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RhombicDodecahedralNumber.html">Rhombic Dodecahedral Number</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = (2*n - 1)*(8*n^2 - 14*n + 7).
%F G.f.: x*(7*x^3 +59*x^2 +29*x +1)/(1-x)^4. - _Colin Barker_, Oct 26 2012
%F a(n) = A016755(n) + A069072(n-1). - _Luciano Ancora_, Mar 23 2015
%F a(n) = A016755(n) + 6*A000447(n-1). - _Luciano Ancora_, Mar 23 2015
%F 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
%F E.g.f.: (-7 +8*x +12*x^2 +16*x^3)*exp(x) + 7. - _G. C. Greubel_, Nov 04 2017
%p A046142:=n->(2*n-1)*(8*n^2-14*n+7): seq(A046142(n), n=1..50); # _Wesley Ivan Hurt_, Mar 02 2016
%t Table[(2 n - 1) (8 n^2 - 14 n + 7), {n, 40}] (* _Vincenzo Librandi_, Mar 29 2015 *)
%t LinearRecurrence[{4, -6, 4, -1}, {1, 33, 185, 553}, 20] (* _Eric W. Weisstein_, Sep 27 2017 *)
%t CoefficientList[Series[(1 + 29 x + 59 x^2 + 7 x^3)/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 27 2017 *)
%o (PARI) Vec(x*(7*x^3+59*x^2+29*x+1)/(x-1)^4 + O(x^50)) \\ _Michel Marcus_, Mar 24 2015
%o (Magma) [(2*n-1)*(8*n^2-14*n+7): n in [1..40]]; // _Vincenzo Librandi_, Mar 29 2015
%Y Cf. A000447, A001845, A016755, A069072, A254473.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
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