A143156 - OEIS

A143156

Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

1, 3, 2, 4, 3, 1, 7, 6, 4, 3, 8, 7, 5, 4, 1, 10, 9, 7, 6, 3, 2, 11, 10, 8, 7, 4, 3, 1, 15, 14, 12, 11, 8, 7, 5, 4, 16, 15, 13, 12, 9, 8, 6, 5, 1, 18, 17, 15, 14, 11, 10, 8, 7, 3, 2, 19, 18, 16, 15, 12, 11, 9, 8, 4, 3, 1, 22, 21, 19, 18, 15, 14, 12, 11, 7, 6, 4, 3

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Row sums give A143157.

Left border gives A005187.

Right border gives A001511.

LINKS

Table of n, a(n) for n=1..78.

FORMULA

Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

From Kevin Ryde, Oct 07 2021: (Start)

T(n,k) = A005187(n) - A005187(k-1).

G.f.: (V(x) - V(x*y)) * y/((1-x)*(1-y)) where V(x) is the g.f. of A001511.

(End)

EXAMPLE

First few rows of the triangle =

k=1 k=2 k=3 k=4 k=5 k=6 k=7

n=1: 1;

n=2: 3, 2;

n=3: 4, 3, 1;

n=4: 7, 6, 4, 3;

n=5: 8, 7, 5, 4, 1;

n=6: 10, 9, 7, 6, 3, 2;

n=7: 11, 10, 8, 7, 4, 3, 1;

...

Row 6 = (10, 9, 7, 6, 3, 2) = partial sums of the first 6 terms of the ruler sequence, starting from the right: (1, 2, 1, 3, 1, 2,...).

PROG

(PARI) T(n, k) = k--; 2*(n-k) - hammingweight(n) + hammingweight(k); \\ Kevin Ryde, Oct 07 2021

CROSSREFS

Cf. A001511, A000012, A091512, A005187.

Sequence in context: A334662 A016559 A104566 * A227471 A101403 A025509

Adjacent sequences: A143153 A143154 A143155 * A143157 A143158 A143159

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, Jul 27 2008

STATUS

approved