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
Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eddie Gutierrez, A Novel Method for Multiplying Large Numbers.
Eddie Gutierrez, The Column Addition Triangle (CAT)-A Pascal Analog (Part I).
Eddie Gutierrez, The Column Addition Triangle (CAT) and Polynomials.
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 *)
Flatten[Table[Join[Range[1, 2n+1, 2], Range[2n+1, 1, -2]], {n, 0, 10}]] (* Harvey P. Dale, Aug 31 2024 *)
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
KEYWORD
nonn,tabf
AUTHOR
Eddie Gutierrez, Aug 11 2021
EXTENSIONS
Better definition from Omar E. Pol, Aug 14 2021
STATUS
approved