A356351 Partial sums of the ziggurat sequence A347186.

1, 5, 11, 27, 39, 76, 96, 160, 196, 286, 328, 489, 545, 701, 808, 1064, 1154, 1488, 1598, 2006, 2208, 2550, 2706, 3403, 3610, 4072, 4384, 5169, 5409, 6385, 6657, 7681, 8127, 8883, 9324, 10910, 11290, 12220, 12824, 14560, 15022, 16863, 17369, 19175, 20276, 21608, 22208, 25129, 25849, 27669

Offset: 1

Author

Omar E. Pol, Oct 15 2022

Keywords

• nonn

Comments

a(n) is the volume (or the number of cubes) in a polycube whose base is the symmetric representation of A024916(n) which is formed with the first n 3D-Ziggurats described in A347186.

a(n) is also the total number of cubes in a three-dimensional spiral formed with the first n 3D-Ziggurats described in A347186 (see example). The base of the 3D-spiral is the spiral formed with the symmetric representation of sigma of the first n positive integers as shown in the example section of A239660.

Examples

			For n = 16 the figure shows the top view of a three-dimensional spiral formed with the first 16 3D-Ziggurats described in A347186. There are four 3D-Ziggurats in every quadrant:
.
                  _ _ _ _ _ _ _ _
                 |_|_|_|_|_|_|_|_|_ _ _ _ _ _ _
                 |_|             |_|_|_|_|_|_|_|
                _|_|                           |
               |_|_|  _ _ _ _ _ _              |_ _
            _ _|     |_|_|_|_|_|_|_ _ _ _ _        |_
      _ _ _|_|      _|_|         |_|_|_|_|_|         |
     |_|_|_|_|    _|_|_|                   |_ _      |_ _ _
     |_|      _ _|_|      _ _ _ _          |_|_|         |_|
     |_|     |_|_|_|    _|_|_|_|_|_ _ _      |_|_ _      |_|
     |_|     |_|      _|_|_|     |_|_|_|         |_|     |_|
     |_|     |_|     |_|_|_|           |_ _      |_|     |_|
     |_|     |_|     |_|      _ _        |_|     |_|     |_|
     |_|     |_|     |_|     |_|_|_      |_|     |_|     |_|
    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
   |_|     |_|     |_|     |_|         |_|     |_|     |_|     |_|
   |_|     |_|     |_|     |_|_ _     _|_|     |_|     |_|     |_|
   |_|     |_|     |_|       |_|_|_ _|_|_|     |_|     |_|     |_|
   |_|     |_|     |_|_          |_|_|_|    _ _|_|     |_|     |_|
   |_|     |_|         |_                 _|_|_|_|     |_|     |_|
   |_|     |_|_ _        |_ _ _ _        |_|_|    _ _ _|_|     |_|
   |_|           |_      |_|_|_|_|_ _ _ _|_|    _|_|_|_|_|     |_|
   |_|_ _ _        |_            |_|_|_|_|_|  _|_|_|_|    _ _ _|_|
         |_|_ _      |                       |_|_|_|_|   |_|_|_|_|
         |_|_|_|     |_ _ _ _ _ _            |_|_|_|    _|_|
           |_|_|_    |_|_|_|_|_|_|_ _ _ _ _ _|_|      _|_|_|
             |_|_|               |_|_|_|_|_|_|_|  _ _|_|_|
                 |                               |_|_|_|
                 |_ _ _ _ _ _ _ _                |_|
                 |_|_|_|_|_|_|_|_|_ _ _ _ _ _ _ _|_|
                                 |_|_|_|_|_|_|_|_|_|
.
The number of square cells in the top view of the n-th 3D-Ziggurat equals A000203(n).
The total number of square cells in the top view of the 3D-Spiral with the first n 3D-Ziggurats equals A024916(n).
In the above figure the total number of square cells equals A024916(16) = 220.
a(16) = 1064 is the total number of cubes in the 3D-Spiral with the first 16 3D-Ziggurats.
		

Crossrefs

Cf. A000203, A024916, A196020, A235791, A236104, A237270, A237591, A237593, A239660, A239931, A239932, A239933, A239934, A347186, A296508, A299778, A347186, A347263, A347367, A347529, A351819.