A244082 a(n) = 32*n^2.

0, 32, 128, 288, 512, 800, 1152, 1568, 2048, 2592, 3200, 3872, 4608, 5408, 6272, 7200, 8192, 9248, 10368, 11552, 12800, 14112, 15488, 16928, 18432, 20000, 21632, 23328, 25088, 26912, 28800, 30752, 32768, 34848, 36992, 39200, 41472, 43808, 46208, 48672, 51200

Offset: 0

Author

Wesley Ivan Hurt, Jun 19 2014

Keywords

• nonn
• easy

Comments

Geometric connections of a(n) to the area and perimeter of a square.

Area:

. half the area of a square with side 8n (cf. A008590);

. area of a square with diagonal 8n (cf. A008590);

. twice the area of a square with side 4n (cf. A008586);

. four times the area of a square with diagonal 4n (cf. A008586);

. eight times the area of a square with side 2n (cf. A005843);

. sixteen times the area of a square with diagonal 2n (cf. A005843);

. thirty two times the area of a square with side n (cf. A001477);

. sixty four times the area of a square with diagonal n (cf. A001477).

Perimeter:

. perimeter of a square with side 8n^2 (cf. A139098);

. twice the perimeter of a square with side 4n^2 (cf. A016742);

. four times the perimeter of a square with side 2n^2 (cf. A001105);

. eight times the perimeter of a square with side n^2 (cf. A000290).

Sequence found by reading the line from 0, in the direction 0, 32, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, May 10 2018

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Pentasection of A077221, A181900.

Cf. A000290, A001105, A010021, A016742, A016802, A139098.

Cf. A001477, A005843, A008586, A008590, A139098, A174312.

Programs

Magma

[32*n^2 : n in [0..50]];

Maple

A244082:=n->32*n^2; seq(A244082(n), n=0..50);

Mathematica

32 Range[0, 50]^2 (* or *)
Table[32 n^2, {n, 0, 50}] (* or *)
CoefficientList[Series[32 x (1 + x)/(1 - x)^3, {x, 0, 30}], x]

PARI

a(n)=32*n^2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: 32*x*(1+x)/(1-x)^3.

a(n) = 2 * A016802(n).

a(n) = 4 * A139098(n).

a(n) = 8 * A016742(n).

a(n) = 16 * A001105(n).

a(n) = 32 * A000290(n).

a(n) = A010021(n) - 2 for n > 0. - Bruno Berselli, Jun 24 2014

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Nov 19 2021

From Elmo R. Oliveira, Dec 02 2024: (Start)

E.g.f.: 32*x*(1 + x)*exp(x).

a(n) = n*A174312(n) = A139098(2*n). (End)