A350236 a(n) is the sum of the entries in an n X n X n 3D matrix whose elements start at 1 in the corner cells and increase by 1 with each step towards the center.
1, 8, 54, 160, 425, 864, 1666, 2816, 4617, 7000, 10406, 14688, 20449, 27440, 36450, 47104, 60401, 75816, 94582, 116000, 141561, 170368, 204194, 241920, 285625, 333944, 389286, 450016, 518897, 594000, 678466, 770048, 872289, 982600, 1104950, 1236384, 1381321
Offset: 1
Examples
For n=3: we have the following 3D matrix: (sliced for each Z surface)
(z=1): 1 2 1
2 3 2
1 2 1
(z=2): 2 3 2
3 4 3
2 3 2
(z=3): 1 2 1
2 3 2
1 2 1
The sum of all elements is: (3/4)*n^2 * (n^2 - 2/3*n + (n mod 2)) = 54.
Links
- Saeed Barari, In depth (pdf)
- Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
Crossrefs
Cf. A317614.
Programs
-
Maple
a:=n->(3/4)*n^2 * (n^2 - (2/3)*n + modp(n, 2)): seq(a(n), n=1..50);
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Mathematica
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{1,8,54,160,425,864,1666,2816},35] (* Stefano Spezia, May 19 2022 *) -
Python
for n in range(1, nmax): sum = round(3/4*n**2 * (n**2 - 2/3*n + n % 2)) print(sum, end=', ')
Formula
a(n) = (3/4)*n^2 * (n^2 - 2/3*n + (n mod 2)).
From Stefano Spezia, May 19 2022: (Start)
O.g.f.: x*(1 + 6*x + 36*x^2 + 42*x^3 + 45*x^4 + 12*x^5 + 2*x^6)/((1 - x)^5*(1 + x)^3).
E.g.f.: x*((4 + 15*x + 16*x^2 + 3*x^3)*cosh(x) + (1 + 18*x + 16*x^2 + 3*x^3)*sinh(x))/4.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5)- 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 8. (End)
Extensions
Python program and a(23), a(34) corrected by Georg Fischer, Sep 30 2022
Comments