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

0, 7, 17, 30, 46, 65, 87, 112, 140, 171, 205, 242, 282, 325, 371, 420, 472, 527, 585, 646, 710, 777, 847, 920, 996, 1075, 1157, 1242, 1330, 1421, 1515, 1612, 1712, 1815, 1921, 2030, 2142, 2257, 2375, 2496, 2620, 2747, 2877, 3010, 3146, 3285, 3427, 3572, 3720

Offset: 0

Author

Jason Earls, Mar 10 2001

Keywords

• easy
• nonn

Comments

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

Links

• Harry J. Smith, Table of n, a(n) for n = 0..2000
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
• John Nash, The Imbedding Problem For Riemannian Manifolds, Annals of Mathematics, Vol. 63, No. 1, 1956, pp. 20-63.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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.

Programs

Maple

A059845:=n->n*(3*n + 11)/2: seq(A059845(n), n=0..100); # Wesley Ivan Hurt, Jan 15 2017

Mathematica

Table[n (3n+11)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,17},50] (* Harvey P. Dale, Mar 19 2017 *)

PARI

a(n) = n*(3*n + 11)/2 \\ Harry J. Smith, Jun 29 2009

Formula

a(n) = 3*n + a(n-1) + 4 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010

G.f.: x*(7 - 4*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011

E.g.f.: (1/2)*(3*x^2 + 14*x)*exp(x). - G. C. Greubel, Jul 17 2017