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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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

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

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.

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

Sequence in context: A322510 A021707 A126560 * A064213 A245518 A217462

Adjacent sequences:  A289759 A289760 A289761 * A289763 A289764 A289765

KEYWORD

nonn,tabf,easy

AUTHOR

Miquel Cerda, Jul 12 2017

STATUS

approved

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Last modified October 22 17:39 EDT 2019. Contains 328319 sequences. (Running on oeis4.)