A180039 Combinatorial tetrahedral numbers

1, 112, 111223, 1111222334, 111112222333445, 111111222223333444556, 1111111222222333334444555667, 111111112222222333333444445555666778, 111111111222222223333333444444555556666777889

Offset

1,2

Comments

These are partial sums in adding base-1 triangular numbers (1 + 111 + 111111 + 1111111111,...).

Third row of:

1,2,3,4,5,...

1,12,123,1234,12345,...

1,112,111223,1111222334,111112222333445,...

1,1112,1111112223,11111111112222223334,11111111111111122222222223333334445,..

Links

Table of n, a(n) for n=1..9.

Formula

a(n) = Sum_{i=1..n} A002275(A000217(i)). - R. J. Mathar, Nov 02 2016

Example

For n=4, a(4)= 1 + 111 + 111111 + 1111111111 = 1111222334.
Combinatorially:
1,12,123,1234,12345,123456,...
1,21,321,4321,54321,654321,...
----------------------------- X
1,112,111223,1111222334.......
For n =(4),a(4)= 4 x 1 + 3 x 2 + 2 x 3 + 1 x 4 = 1111 + 222 + 33 + 4 = 1111222334.

Mathematica

Accumulate[Table[FromDigits[PadRight[{}, n, 1]], {n, Accumulate[Range[10]]}]] (* Harvey P. Dale, Jun 12 2021 *)

Crossrefs

Cf. A000292, A014824, A180027.

Sequence in context: A184898 A270113 A275052 * A261949 A159432 A151645

Adjacent sequences: A180036 A180037 A180038 * A180040 A180041 A180042

Keyword

nonn,easy,base

Author

Mark Dols, Aug 07 2010

Status

approved