A214659 a(n) = n*(7*n^2 - 3*n - 1)/3.

0, 1, 14, 53, 132, 265, 466, 749, 1128, 1617, 2230, 2981, 3884, 4953, 6202, 7645, 9296, 11169, 13278, 15637, 18260, 21161, 24354, 27853, 31672, 35825, 40326, 45189, 50428, 56057, 62090, 68541, 75424, 82753, 90542, 98805, 107556, 116809, 126578, 136877

Offset: 0

Author

Reinhard Zumkeller, Jul 25 2012

Keywords

• nonn
• easy

Comments

a(n) = the sum of the n X n matrices of A204008. For example, for n = 3, the sum of the 9 elements of the 3 X 3 submatrix of A204008 is 1 + 4 + 7 + 4 + 5 + 8 + 7 + 8 + 9 = 53. - J. M. Bergot, Jul 15 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002378, A051673, A204008, A214604, A214675.

Programs

Haskell

a214659 n = ((7 * n - 3) * n - 1) * n `div` 3

Magma

[(7*n^3-3*n^2-n)/3 : n in [0..50]]; // Wesley Ivan Hurt, Apr 11 2015

Maple

A214659:=n->(7*n^3-3*n^2-n)/3: seq(A214659(n), n=0..50); # Wesley Ivan Hurt, Apr 11 2015

Mathematica

Table[(7 n^3 -3 n^2 -n)/3, {n,0,50}] (* Wesley Ivan Hurt, Apr 11 2015 *)
LinearRecurrence[{4,-6,4,-1}, {0,1,14,53}, 51] (* G. C. Greubel, Mar 09 2024 *)

SageMath

[(7*n^3-3*n^2-n)/3 for n in range(51)] # G. C. Greubel, Mar 09 2024

Formula

a(n) = Sum_{k=0..n} A214604(n, k) for n > 0 (row sums).

a(n) = A002378(n) + A051673(n).

From Wesley Ivan Hurt, Apr 11 2015: (Start)

a(n) = (7*n^3 - 3*n^2 - n)/3.

G.f.: x*(1+10*x+3*x^2)/(1-x)^4.

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

E.g.f.: (x/3)*(3 + 18*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 09 2024