A376468 Triangle T read by rows: T(n, k) = (n^2 - 2*n + 3 - (-1)^n + n^2 mod 8) / 2 + 4*k.

1, 2, 6, 3, 7, 11, 4, 8, 12, 16, 5, 9, 13, 17, 21, 10, 14, 18, 22, 26, 30, 15, 19, 23, 27, 31, 35, 39, 20, 24, 28, 32, 36, 40, 44, 48, 25, 29, 33, 37, 41, 45, 49, 53, 57, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96

Offset: 0

Author

Werner Schulte, Sep 23 2024

Keywords

• nonn
• easy
• tabl

Comments

This triangle seen as a sequence yields a permutation of the natural numbers. For similar triangles see A000027 (seen as a triangle), A074147, and A367844 (row reversed).

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n \k :   0   1   2   3   4   5   6   7   8   9  10  11
======================================================
   0 :   1
   1 :   2   6
   2 :   3   7  11
   3 :   4   8  12  16
   4 :   5   9  13  17  21
   5 :  10  14  18  22  26  30
   6 :  15  19  23  27  31  35  39
   7 :  20  24  28  32  36  40  44  48
   8 :  25  29  33  37  41  45  49  53  57
   9 :  34  38  42  46  50  54  58  62  66  70
  10 :  43  47  51  55  59  63  67  71  75  79  83
  11 :  52  56  60  64  68  72  76  80  84  88  92  96
  etc.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Cf. A001844

Programs

Mathematica

Table[Range[#, #+n*4, 4] & [(Mod[n^2, 8] + n*(n-2) - (-1)^n + 3)/2], {n, 0, 15}] (* Paolo Xausa, Nov 13 2024 *)

PARI

T(n,k)=(n^2-2*n+3-(-1)^n+n^2%8)/2+4*k

Python

from math import comb, isqrt
def A376468(n): return ((a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(a-2)+3+(1 if a&1 else -1)+(a**2&7)>>1)+(n-comb(a+1,2)<<2) # Chai Wah Wu, Nov 12 2024

Formula

T(n, k) = T(n, k-1) + 4.

T(n+4, 0) = T(n, n) + 4 for n > 3.

T(2*n, n) = 2 * (n^2 + n + 1) - (-1)^n = A001844(n) + 1 - (-1)^n.