A355802 A variant of Pascal's triangle (A007318) where new rows are added cyclically below, to the right and to the left.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 5, 6, 5, 3, 2, 1, 1, 2, 3, 5, 7, 7, 5, 3, 2, 1, 1, 2, 3, 5, 7, 8, 7, 5, 3, 2, 1, 1, 2, 3, 5, 8, 11, 11, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 12, 14, 12, 8, 5, 3, 2, 1

Offset

0,5

Comments

The procedure to build the present triangle is as follows:

- row 0 contains a single 1:

. 1

- row 1 is added below, each new term is the sum of the adjacent prior terms:

. 1

. ---

. 1 1

- row 2 is added to the right, each new term is the sum of the adjacent prior terms:

. 1

. \

. 1 \ 2

. \

. 1 1 \ 1

- row 3 is added to the left, each new term is the sum of the adjacent prior terms:

. 1

. /

. 2 / 1

. /

. 2 / 1 2

. /

. 1 / 1 1 1

- row 4 is added below, each new term is the sum of the adjacent prior terms:

. 1

.

. 2 1

.

. 2 1 2

.

. 1 1 1 1

. ---------------

. 1 2 2 2 1

- and so on.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Rémy Sigrist, Illustration of the odd terms among the first 2^9 rows

Rémy Sigrist, PARI program

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(n, 0) = T(n, n) = 1.

T(n, k) = T(n, n-k).

Example

Triangle begins:
    1;
    1, 1;
    1, 2, 1;
    1, 2, 2, 1;
    1, 2, 2, 2, 1;
    1, 2, 3, 3, 2, 1;
    1, 2, 3, 4, 3, 2, 1;
    1, 2, 3, 4, 4, 3, 2, 1;
    1, 2, 3, 5, 6, 5, 3, 2, 1;
    1, 2, 3, 5, 7, 7, 5, 3, 2, 1;
    1, 2, 3, 5, 7, 8, 7, 5, 3, 2, 1;
    1, 2, 3, 5, 8, 11, 11, 8, 5, 3, 2, 1;
    1, 2, 3, 5, 8, 12, 14, 12, 8, 5, 3, 2, 1;
    ...

Prog

(PARI) See Links section.

Crossrefs

Cf. A007318, A355806.

Sequence in context: A348042 A143209 A163994 * A156593 A206498 A184848

Adjacent sequences: A355799 A355800 A355801 * A355803 A355804 A355805

Keyword

nonn,tabl

Author

Rémy Sigrist, Jul 17 2022

Status

approved