|
|
A347026
|
|
Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.
|
|
0
|
|
|
1, 1, 1, 3, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 17, 15, 13, 11, 9, 7, 5, 3, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
The terms of this sequence are the numbers in an irregular triangle corresponding to the addition of rows when multiplying two large numbers via a novel method (see Links).
Sums of the rising diagonals yield sequence A007980.
When the 2n terms in row n are used as the coefficients of a (2n-1)st-order polynomial in x, dividing that polynomial by x+1 produces a (2n-2)nd-order polynomial whose coefficients are the n-th row of A004737 (if that sequence is taken as an irregular triangle with 2n-1 terms in its n-th row). E.g., for n=3, (x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1)/(x+1) = x^4 + 2x^3 + 3x^2 + 2x + 1.
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = 2k - 1 for 1 <= k <= n,
4n - 2k + 1 for n+1 <= k <= 2n.
|
|
EXAMPLE
|
Triangle begins:
1, 1;
1, 3, 3, 1;
1, 3, 5, 5, 3, 1;
1, 3, 5, 7, 7, 5, 3, 1;
1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1;
1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1;
1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1;
...
|
|
MATHEMATICA
|
Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* Michael De Vlieger, Aug 18 2021 *)
|
|
PROG
|
(C)
#include <stdio.h>
int main()
{
int n, k;
for (n=1; n<=13; n++)
{
for (k=1; k<=n; k++)
{
printf("%d ", 2*k - 1);
}
for (k=n+1; k<=2*n; k++)
{
printf("%d ", 4*n - 2*k + 1);
}
printf("\n");
}
return 0;
}
(PARI) row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ Michel Marcus, Aug 17 2021
|
|
CROSSREFS
|
Row lengths give nonzero terms of A005843.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|