A289762 Triangular array T(m,k) = (m+1-k)^2 + k - 1 with m (row) >= 1 and k (column) >= 1, read by rows.

1, 1, 4, 2, 2, 4, 9, 5, 3, 3, 5, 9, 16, 10, 6, 4, 4, 6, 10, 16, 25, 17, 11, 7, 5, 5, 7, 11, 17, 25, 36, 26, 18, 12, 8, 6, 6, 8, 12, 18, 26, 36, 49, 37, 27, 19, 13, 9, 7, 7, 9, 13, 19, 27, 37, 49, 64, 50, 38, 28, 20, 14, 10, 8, 8, 10, 14, 20, 28, 38, 50, 64, 81, 65, 51, 39, 29, 21, 15, 11, 9

Offset: 1

Author

Miquel Cerda, Jul 12 2017

Keywords

• nonn
• tabf
• easy

Comments

The n-th row is of length = max(2n, 1) and the row sum is (2n^3 + 6n^2 - 2n) / 3.

Rows m = 2, 3, 5, 11, and 41 (Euler's lucky numbers) give the prime numbers generated by the famous polynomials, but twice each one between m^2.

Examples

			The m-th row start and end: T(m,1) = m^2, ..., T(m,2m) = m^2.
In general T(m,k) = T(m,2m+1-k).
m\k    1     2     3     4     5     6     7     8     9     10
1      1,    1,
2      4,    2,    2,    4
3      9,    5,    3,    3,    5,    9
4      16,   10,   6,    4,    4,    6,    10,   16
5      25,   17,   11,   7,    5,    5,    7,    11,   17,   25
6      36,   26,   18,   12,   8,    6,    6,    8,    12,   18, ...
7      49,   37,   27,   19,   13,   9,    7,    7,    9,    13, ...
8      64,   50,   38,   28,   20,   14,   10,   8,    8,    10, ...
9      81,   65,   51,   39,   29,   21,   15,   11,   9,    9, ...
10     100,  82,   66,   52,   40,   30    22,   16,   12,   10, ...
The T(m,k) sequence as an isosceles triangle:
                                     1  1
                                 4   2  2  4
                             9   5   3  3  5  9
                         16  10  6   4  4  6  10  16
                     25  17  11  7   5  5  7  11  17  25
                 36  26  18  12  8   6  6  8  12  18  26  36
             49  37  27  19  13  9   7  7  9  13  19  27  37  49
         64  50  38  28  20  14  10  8  8  1  14  20  28  38  50  64
     81  65  51  39  29  21  15  11  9  9  11 15  21  29  39  51  65  81
100  82  66  52  40  30  22  16  12  10 10 12 16  22  30  40  52  66  82  100
		

Links

• Miquel Cerda, Table of n, a(n) for n = 1..306
• Miquel Cerda, Triangle rows 1..41
• Miquel Cerda, Isosceles triangle Rows 1..41

Crossrefs

m(41, k+1) = A060566(n), left and right border gives A000290(n).

Programs

Mathematica

Table[(m + 1 - k)^2 + k - 1, {m, 0, 10}, {k, 2 m}] /. {} -> {0} // Flatten (* Michael De Vlieger, Jul 12 2017 *)

PARI

T(m,k) = (m+1-k)^2+k-1 \\ Charles R Greathouse IV, Jul 12 2017

Formula

The formula that gives the integers in the m-th rows can be expressed using quadratic polynomials:

for row m = 1, a(k) = k^2 - 3*k + 3

for row m = 2, a(k) = k^2 - 5*k + 8

for row m = 3, a(k) = k^2 - 7*k + 15

for row m = 4, a(k) = k^2 - 9*k + 24

for row m = 5, a(k) = k^2 - 11*k + 35

for row m = 6, a(k) = k^2 - 13*k + 48

etc.