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

%I A367844 #19 Nov 13 2024 17:16:23
%S A367844 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,
%T A367844 19,16,44,41,38,35,32,29,26,23,54,51,48,45,42,39,36,33,30,64,61,58,55,
%U A367844 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
%N A367844 Triangle read by rows: T(n, k) = (n+5)*n/2 + 1 + (n^2 mod 3) - 3*k for 0 <= k <= n.
%C A367844 This triangle read by rows yields a permutation of the natural numbers.
%F A367844 T(n, 0) = (n+5)*n/2 + 1 + (n^2 mod 3) for n >= 0.
%F A367844 T(n, n) = (n-1)*n/2 + 1 + (n^2 mod 3) for n >= 0.
%F A367844 T(2*n, n) = 2*n*(n+1) + 1 + (n^2 mod 3) for n >= 0.
%F A367844 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).
%F A367844 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).
%F A367844 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).
%F A367844 Row sums are A006003(n+1) + (n^2 mod 3) * (n+1) for n >= 0.
%e A367844 Triangle T(n, k) for 0 <= k <= n starts:
%e A367844 n\k :    0    1   2   3   4   5   6   7   8   9  10  11  12
%e A367844 ===========================================================
%e A367844  0  :    1
%e A367844  1  :    5    2
%e A367844  2  :    9    6   3
%e A367844  3  :   13   10   7   4
%e A367844  4  :   20   17  14  11   8
%e A367844  5  :   27   24  21  18  15  12
%e A367844  6  :   34   31  28  25  22  19  16
%e A367844  7  :   44   41  38  35  32  29  26  23
%e A367844  8  :   54   51  48  45  42  39  36  33  30
%e A367844  9  :   64   61  58  55  52  49  46  43  40  37
%e A367844 10  :   77   74  71  68  65  62  59  56  53  50  47
%e A367844 11  :   90   87  84  81  78  75  72  69  66  63  60  57
%e A367844 12  :  103  100  97  94  91  88  85  82  79  76  73  70  67
%e A367844 etc.
%p A367844 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):
%p A367844 sert := series(gf, t, 18): px := n -> simplify(coeff(sert, t, n)):
%p A367844 row := n -> local k; seq(coeff(px(n), x, k), k = 0..n):
%p A367844 for n from 0 to 12 do row(n) od;  # _Peter Luschny_, Dec 02 2023
%t A367844 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 *)
%o A367844 (PARI) T(n,k) = (n+5)*n/2+1+(n^2%3)-3*k
%o A367844 (Python)
%o A367844 def A367844Row(n):
%o A367844     Tn0 = (n + 5) * n // 2 + n ** 2 % 3 + 1
%o A367844     return [Tn0 - k * 3 for k in range(n + 1)]
%o A367844 for n in range(9): print(A367844Row(n))  # _Peter Luschny_, Dec 03 2023
%o A367844 (Python)
%o A367844 from math import isqrt, comb
%o A367844 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
%Y A367844 Cf. A006003, A038722, A109857.
%K A367844 nonn,easy,tabl
%O A367844 0,2
%A A367844 _Werner Schulte_, Dec 02 2023