A143971 -id:A143971 - cp's OEIS frontend

A143972 Eigentriangle by rows, A143971 * (A108300 * 0^(n-k)); 1<=k<=1.

1, 4, 1, 7, 4, 5, 10, 7, 20, 16, 13, 10, 35, 64, 53, 16, 13, 50, 112, 212, 175, 19, 16, 65, 160, 371, 700, 578, 22, 19, 80, 208, 530, 1225, 2312, 1909, 25, 28, 95, 256, 689, 1750, 4046, 7636, 6305, 28, 25, 110, 304, 848, 2275, 5780, 23363, 25220, 20824

Offset: 1

Gary W. Adamson, Sep 06 2008

Right border = A108300: (1, 1, 5, 16, 53, 175, 578,...). Row sums = (1, 5, 16, 53, 175, 578,...) = INVERT transform of (1, 4, 7, 10,...).
Sum of n-th row terms = rightmost term of next row.
Comment in A108300 states that (5, 16, 53, 175,...) is related to the numbers of hydrogen bonds in hydrocarbons.

			First few rows of the triangle =
1;
4, 1;
7, 4, 5;
10, 7, 10, 16;
13, 10, 35, 64, 53;
16, 13, 50, 112, 212, 175;
19, 16, 65, 160, 371, 700, 578;
22, 19, 80, 208, 530, 1225, 2312, 1909;
25, 22, 95, 256, 689, 1750, 4046, 7636, 6305;
... Example: row 4 = (10, 7, 20, 16) = termwise products of (10, 7, 4, 1) and (1, 1, 5, 16) = (10*1, 7*1, 4*5, 1*16), where (10, 7, 4, 1) = row 4 of triangle A143971.

A143971, Cf. A016777, A108300

Eigentriangle by rows, A143971 * (A108300 * 0^(n-k)); 1<=k<=1

Triangle A143971 = (1; 4,1; 7,4,1; 10,7,4,1;...). A108300 * 0^(n-k) = an infinite lower triangular matrix with A108300 (1, 1, 5, 16, 53, 175, 578, 1909,...) in the main diagonal and the rest zeros. By rows, = termwise products of n-th row terms of A143971 and n terms of A108300.

A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

Original entry on oeis.org

1, 4, 7, 1, 10, 4, 13, 7, 1, 16, 10, 4, 19, 13, 7, 1, 22, 16, 10, 4, 25, 19, 13, 7, 1, 28, 22, 16, 10, 4, 31, 25, 19, 13, 7, 1, 34, 28, 22, 16, 10, 4, 37, 31, 25, 19, 13, 7, 1, 40, 34, 28, 22, 16, 10, 4, 43, 37, 31, 25, 19, 13, 7, 1, 46, 40, 34, 28, 22, 16, 10

Offset: 1

Ctibor O. Zizka, Mar 11 2012

OEIS contains a lot of similar sequences, for example A152204, A122196, A173284.
Row sums for this sequence gives A006578.
In general, by given triangle with (A-B,2*A-B,...,A*n-B,...) in every column, shifted down K-times, we have the row sum s(n)= A*(n*n+K*n+nmodK)/(2*K) - B*(n+nmodK)/K. In this sequence K=2,A=3,B=2, in A152204 K=2,A=2,B=1.
No triangle with primes in every column, shifted down by K>=2 in OEIS, no row sums of it in OEIS.
From Johannes W. Meijer, Sep 28 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A143971.
The alternating row sums equal A004524(n+2) + 2*A004524(n+1).
The antidiagonal sums equal A171452(n+1). (End)

			Triangle:
1
4
7,  1
10, 4
13, 7,  1
16, 10, 4
19, 13, 7,  1
22, 16, 10, 4
25, 19, 13, 7,  1
28, 22, 16, 10, 4
...

Cf. A008315, A011973, A102541, A122196, A122197, A128099, A152198, A152204, A173284, A207538.

Cf. (Related to triangle sums) A006578, A000217, A002620, A004524, A171452.

T := (n, k) -> 3*n - 6*k + 4: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..15); # Johannes W. Meijer, Sep 28 2013

From Johannes W. Meijer, Sep 28 2013: (Start)
T(n, k) = 3*n - 6*k + 4, n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A143971(n-k+1, k), n >= 1 and 1 <= k <= floor((n+1)/2). (End)

cp's OEIS Frontend

A143972 Eigentriangle by rows, A143971 * (A108300 * 0^(n-k)); 1<=k<=1.

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