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A059845 a(n) = n*(3*n + 11)/2.

%I A059845 #56 Dec 30 2024 22:43:38
%S A059845 0,7,17,30,46,65,87,112,140,171,205,242,282,325,371,420,472,527,585,
%T A059845 646,710,777,847,920,996,1075,1157,1242,1330,1421,1515,1612,1712,1815,
%U A059845 1921,2030,2142,2257,2375,2496,2620,2747,2877,3010,3146,3285,3427,3572,3720
%N A059845 a(n) = n*(3*n + 11)/2.
%C A059845 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 A059845 Harry J. Smith, <a href="/A059845/b059845.txt">Table of n, a(n) for n = 0..2000</a>
%H A059845 Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 5.
%H A059845 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 A059845 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A059845 a(n) = 3*n + a(n-1) + 4 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F A059845 G.f.: x*(7 - 4*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011
%F A059845 E.g.f.: (1/2)*(3*x^2 + 14*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%p A059845 A059845:=n->n*(3*n + 11)/2: seq(A059845(n), n=0..100); # _Wesley Ivan Hurt_, Jan 15 2017
%t A059845 Table[n (3n+11)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,17},50] (* _Harvey P. Dale_, Mar 19 2017 *)
%o A059845 (PARI) a(n) = n*(3*n + 11)/2 \\ _Harry J. Smith_, Jun 29 2009
%Y A059845 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 A059845 easy,nonn
%O A059845 0,2
%A A059845 _Jason Earls_, Mar 10 2001