A367844 Triangle read by rows: T(n, k) = (n+5)*n/2 + 1 + (n^2 mod 3) - 3*k for 0 <= k <= n.

1, 5, 2, 9, 6, 3, 13, 10, 7, 4, 20, 17, 14, 11, 8, 27, 24, 21, 18, 15, 12, 34, 31, 28, 25, 22, 19, 16, 44, 41, 38, 35, 32, 29, 26, 23, 54, 51, 48, 45, 42, 39, 36, 33, 30, 64, 61, 58, 55, 52, 49, 46, 43, 40, 37, 77, 74, 71, 68, 65, 62, 59, 56, 53, 50, 47, 90, 87, 84, 81, 78, 75, 72, 69, 66, 63, 60, 57

Offset: 0

Author

Werner Schulte, Dec 02 2023

Keywords

• nonn
• easy
• tabl

Comments

This triangle read by rows yields a permutation of the natural numbers.

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n\k :    0    1   2   3   4   5   6   7   8   9  10  11  12
===========================================================
 0  :    1
 1  :    5    2
 2  :    9    6   3
 3  :   13   10   7   4
 4  :   20   17  14  11   8
 5  :   27   24  21  18  15  12
 6  :   34   31  28  25  22  19  16
 7  :   44   41  38  35  32  29  26  23
 8  :   54   51  48  45  42  39  36  33  30
 9  :   64   61  58  55  52  49  46  43  40  37
10  :   77   74  71  68  65  62  59  56  53  50  47
11  :   90   87  84  81  78  75  72  69  66  63  60  57
12  :  103  100  97  94  91  88  85  82  79  76  73  70  67
etc.
		

Crossrefs

Cf. A006003, A038722, A109857.

Programs

Maple

gf := (t^2*x-t*x-t-2)/(3*(t^2+t+1)*(t^2*x^2+t*x+1))+(5*t^2-10*t+8)/(3*(t-1)^3* (t*x-1))+(3*t-2)/((t-1)^2*(t*x-1)^2)+1/((t-1)*(t*x-1)^3):
sert := series(gf, t, 18): px := n -> simplify(coeff(sert, t, n)):
row := n -> local k; seq(coeff(px(n), x, k), k = 0..n):
for n from 0 to 12 do row(n) od;  # Peter Luschny, Dec 02 2023

Mathematica

T[n_, k_]:=(n+5)*n/2+1+Mod [n^2 ,3]-3*k; Table[T[n,k],{n,0,11},{k,0,n}] //Flatten (* Stefano Spezia, Dec 03 2023 *)

PARI

T(n,k) = (n+5)*n/2+1+(n^2%3)-3*k

Python

def A367844Row(n):
    Tn0 = (n + 5) * n // 2 + n ** 2 % 3 + 1
    return [Tn0 - k * 3 for k in range(n + 1)]
for n in range(9): print(A367844Row(n))  # Peter Luschny, Dec 03 2023

Python

from math import isqrt, comb
def A367844(n): return ((a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(a+5)>>1)+1+a**2%3-3*(n-comb(a+1,2)) # Chai Wah Wu, Nov 12 2024

Formula

T(n, 0) = (n+5)*n/2 + 1 + (n^2 mod 3) for n >= 0.

T(n, n) = (n-1)*n/2 + 1 + (n^2 mod 3) for n >= 0.

T(2*n, n) = 2*n*(n+1) + 1 + (n^2 mod 3) for n >= 0.

T(n, k) - T(n, k+1) = m = 3 for 0 <= k < n (compare with A109857 where m = 2 and with A038722, seen as a triangle, where m = 1).

G.f. of column k = 0: F(t, 0) = Sum_{n>=0} T(n, 0) * t^n = (1 + 3*t - t^3) / ((1 - t^3) * (1 - t)^2).

G.f.: F(t, x) = Sum_{n>=0, k=0..n} T(n, k) * x^k * t^n = (F(t, 0) - x * F(x*t, 0)) / (1 - x) - 3*x*t / ((1 - t) * (1 - x*t)^2).

Row sums are A006003(n+1) + (n^2 mod 3) * (n+1) for n >= 0.