A347026 - OEIS

%I #77 Aug 31 2024 17:26:14

%S 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,

%T 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,

%U 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

%N 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.

%C 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).

%C Sums of the rising diagonals yield sequence A007980.

%C 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.

%H Harvey P. Dale, <a href="/A347026/b347026.txt">Table of n, a(n) for n = 1..1000</a>

%H Eddie Gutierrez, <a href="http://oddwheel.com/Mathx.html">A Novel Method for Multiplying Large Numbers</a>.

%H Eddie Gutierrez, <a href="http://oddwheel.com/CAT.html">The Column Addition Triangle (CAT)-A Pascal Analog (Part I)</a>.

%H Eddie Gutierrez, <a href="http://oddwheel.com/CAT2.html">The Column Addition Triangle (CAT) and Polynomials</a>.

%F T(n,k) = 2k - 1      for   1 <= k <= n,

%F          4n - 2k + 1 for n+1 <= k <= 2n.

%e Triangle begins:

%e   1,  1;

%e   1,  3,  3,  1;

%e   1,  3,  5,  5,  3,  1;

%e   1,  3,  5,  7,  7,  5,  3,  1;

%e   1,  3,  5,  7,  9,  9,  7,  5,  3,  1;

%e   1,  3,  5,  7,  9, 11, 11,  9,  7,  5,  3,  1;

%e   1,  3,  5,  7,  9, 11, 13, 13, 11,  9,  7,  5,  3,  1;

%e   1,  3,  5,  7,  9, 11, 13, 15, 15, 13, 11,  9,  7,  5,  3,  1;

%e ...

%t Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* _Michael De Vlieger_, Aug 18 2021 *)

%t Flatten[Table[Join[Range[1,2n+1,2],Range[2n+1,1,-2]],{n,0,10}]] (* _Harvey P. Dale_, Aug 31 2024 *)

%o (C)

%o #include <stdio.h>

%o int main()

%o {

%o    int n, k;

%o    for (n=1; n<=13; n++)

%o    {

%o       for (k=1; k<=n; k++)

%o       {

%o          printf("%d ", 2*k - 1);

%o       }

%o       for (k=n+1; k<=2*n; k++)

%o       {

%o          printf("%d ", 4*n - 2*k + 1);

%o       }

%o       printf("\n");

%o    }

%o    return 0;

%o }

%o (PARI) row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ _Michel Marcus_, Aug 17 2021

%Y Even-indexed rows of A157454.

%Y Antidiagonal sums give A007980.

%Y Row lengths give nonzero terms of A005843.

%Y Cf. A004737.

%K nonn,tabf

%O 1,4

%A _Eddie Gutierrez_, Aug 11 2021

%E Better definition from _Omar E. Pol_, Aug 14 2021