A063490 a(n) = (2*n - 1)*(7*n^2 - 7*n + 6)/6.

1, 10, 40, 105, 219, 396, 650, 995, 1445, 2014, 2716, 3565, 4575, 5760, 7134, 8711, 10505, 12530, 14800, 17329, 20131, 23220, 26610, 30315, 34349, 38726, 43460, 48565, 54055, 59944, 66246, 72975, 80145, 87770, 95864, 104441, 113515, 123100, 133210, 143859, 155061

Offset: 1

Author

N. J. A. Sloane, Aug 01 2001

Keywords

• nonn
• easy

Comments

From Omar E. Pol, Oct 23 2019: (Start)

a(n) is also the sum of terms that are in the n-th finite row and in the n-th finite column of the square [1,n]x[1,n] of the natural number array A000027; e.g., the [1,3]x[1,3] square is

1..3..6

2..5..9

4..8..13,

so that a(1) = 1, a(2) = 2 + 3 + 5 = 10, a(3) = 4 + 6 + 8 + 9 + 13 = 40.

Hence the partial sums give A185505. (End)

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022. (See p. 14).
• T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A064788, A185505.

Programs

Magma

[(2*n-1)*(7*n^2-7*n+6)/6: n in [1..30]]; // G. C. Greubel, Dec 01 2017

Mathematica

Table[(2*n-1)*(7*n^2-7*n+6)/6, {n,1,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,10,40,105}, 50] (* G. C. Greubel, Dec 01 2017 *)

PARI

a(n) = { (2*n - 1)*(7*n^2 - 7*n + 6)/6 } \\ Harry J. Smith, Aug 23 2009

PARI

my(x='x+O('x^30)); Vec(serlaplace((-6 + 12*x + 21*x^2 + 14*x^3 )*exp(x)/6 + 1)) \\ G. C. Greubel, Dec 01 2017

Formula

G.f.: x*(1+x)*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012

a(n) = Sum_{k = n^2-2*n+2..n^2} A064788(k). - Lior Manor, Jan 13 2013

From G. C. Greubel, Dec 01 2017: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

E.g.f.: (-6 + 12*x + 21*x^2 + 14*x^3)*exp(x)/6 + 1. (End)