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%I #49 Nov 10 2019 12:27:43
%S 0,7,17,30,46,65,87,112,140,171,205,242,282,325,371,420,472,527,585,
%T 646,710,777,847,920,996,1075,1157,1242,1330,1421,1515,1612,1712,1815,
%U 1921,2030,2142,2257,2375,2496,2620,2747,2877,3010,3146,3285,3427,3572,3720
%N a(n) = n*(3*n + 11)/2.
%C Maximum dimension of Euclidean spaces which suffice for every smooth compact Riemannian n-manifold to be realizable as a sub-manifold. - comment edited by _Gene Ward Smith_, Jan 15 2017
%H Harry J. Smith, <a href="/A059845/b059845.txt">Table of n, a(n) for n = 0..2000</a>
%H John Nash, <a href="http://www.jstor.org/stable/1969989">The Imbedding Problem For Riemannian Manifolds</a>, Annals of Mathematics, Vol. 63, No. 1, 1956, pp. 20-63.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*n + a(n-1) + 4 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F G.f.: x*(7 - 4*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011
%F E.g.f.: (1/2)*(3*x^2 + 14*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%p A059845:=n->n*(3*n + 11)/2: seq(A059845(n), n=0..100); # _Wesley Ivan Hurt_, Jan 15 2017
%t Table[n (3n+11)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,17},50] (* _Harvey P. Dale_, Mar 19 2017 *)
%o (PARI) for (n=0, 2000, write("b059845.txt", n, " ", n*(3*n + 11)/2); ) \\ _Harry J. Smith_, Jun 29 2009
%Y The generalized pentagonal numbers b*n + 3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A059845, A115067, A140090, A140091, A140672, A140673, A140674, A140675, A151542.
%K easy,nonn
%O 0,2
%A _Jason Earls_, Mar 10 2001
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