A096037 -id:A096037 - cp's OEIS frontend

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.

1, 6, 4, 14, 12, 7, 25, 23, 18, 10, 39, 37, 32, 24, 13, 56, 54, 49, 41, 30, 16, 76, 74, 69, 61, 50, 36, 19, 99, 97, 92, 84, 73, 59, 42, 22, 125, 123, 118, 110, 99, 85, 68, 48, 25, 154, 152, 147, 139, 128, 114, 97, 77, 54, 28, 186, 184, 179, 171, 160, 146, 129, 109, 86, 60, 31

Offset: 1

Gary W. Adamson, Jun 17 2004

The triangle is obtained by subtracting the triangle A094930 from
its square root (also described in A094930) and then dividing each element of column m through 3*m-1.
For the first three rows n=1 to 3 this yields for example:
4;.................2;............2......................1;
14,25;......minus..2,5;.......=..12,20;......->.divide..6,4;
30,65,64;..........2,5,8;........28,60,56;..............14;12,7;

Cf. A095794, A011379, A002411, A096037, A096036, A024212.

def A096038(n,m):
    return (3*n**2-3*m**2+5*m-4+n)//2
print( [A096038(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

T(n,1) = A095794(n).
T(n,n) = 3*n-2.
T(n,m) = A094930(n,m)/(3*m-1)-1.

Edited, T(3,2) corrected, and extended by R. J. Mathar, Oct 11 2009

A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

Original entry on oeis.org

4, 14, 25, 30, 65, 64, 52, 120, 152, 121, 80, 190, 264, 275, 196, 114, 275, 400, 462, 434, 289, 154, 375, 560, 682, 714, 629, 400, 200, 490, 744, 935, 1036, 1020, 860, 529, 252, 620, 952, 1221, 1400, 1462, 1380, 1127, 676, 310, 765, 1184, 1540, 1806, 1955, 1960

Offset: 1

Gary W. Adamson, Jun 17 2004

Matrix square of the matrix B(n,m) = 2+3*(m-1), B containing the first terms of A016789

in its row n, n>0, 1<=m<=n.

			The matrix B starts as
  2 ;
  2,5 ;
  2,5,8 ;
  2,5,8,11 ;
  2,5,8,11,14 ;
and interpreting this as a lower triangular matrix, its square T = B^2 starts
  4;
  14,25;
  30,65,64;
  52,120,152,121;

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A024212, A096037, A011379, A002411, A096038, A095794.

A094930 := proc(n,m) (3*m-1)*(3*m+3*n-2)*(n+1-m)/2 ; end: seq(seq(A094930(n,m),m=1..n),n=1..20) ; # R. J. Mathar, Oct 09 2009

T(n,m) = sum_{k=m..n} B(n,k)*B(k,m) = (3*m-1)*(3*m+3*n-2)*(n+1-m)/2.
Row sums: sum_{m=1..n} T(n,m) = A024212(n).
G.f. as triangle: x*y*(4+2*x+13*x*y-16*x^2*y+x^2*y^2-4*x^3*y^2)/((1-x)*(1-x*y))^3. - Robert Israel, May 06 2019

Edited and extended by R. J. Mathar, Oct 09 2009

cp's OEIS Frontend

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.

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A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

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cp's OEIS Frontend

A096038 Triangle T(n,m) = (3*n^2-3*m^2+5*m-4+n)/2 read by rows.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

Extensions

A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.