A364602 Triangle T(n,k) with rows of length 2*n-1, generated by T(1,1)=0, T(n,1)=T(n-1,1)+2, T(n,2)=4*(n-1)-1, and for k>=3, T(n,k)=4*T(n-1,k-2)+1.
0, 2, 3, 1, 4, 7, 9, 13, 5, 6, 11, 17, 29, 37, 53, 21, 8, 15, 25, 45, 69, 117, 149, 213, 85, 10, 19, 33, 61, 101, 181, 277, 469, 597, 853, 341, 12, 23, 41, 77, 133, 245, 405, 725, 1109, 1877, 2389, 3413, 1365, 14, 27, 49, 93, 165, 309, 533, 981, 1621, 2901
Offset: 1
Examples
Triangle T(n,k) begins: n/k 1| 2| 3| 4| 5| 6| 7| 8| 9| 10| 11| 1| 0 2| 2 3 1 3| 4 7 9 13 5 4| 6 11 17 29 37 53 21 5| 8 15 25 45 69 117 149 213 85 6| 10 19 33 61 101 181 277 469 597 853 341 7| 12 ...
Programs
-
PARI
my(N=8, v=Vec([0, 2, 3, 1], N^2), p=4); for(n=3, N, my(K=2*n-1); for(k=1, K, v[p+k]=if(k<=2, v[p-K+k+2]+2^k, 4*v[p-K+k]+1)); p+=K); v
-
PARI
T(n, k) = 2^k*(n-(6*k+3-(-1)^k)/12)-1/3;
-
PARI
n_of_x(x) = my(n=0); while(1==x%4, x>>=2; n++); n + if(x%2,(x+1)/4, x/2) + 1;
-
PARI
k_of_x(x) = valuation(3*x+1,2) + 1;
Formula
For n>1, T(n,k) = T(n-1,k) + 2^k, so T(n,1) = 2*(n-1).
T(n,2) = 4*(n-1)-1 = 2*T(n,1)-1, so T(2,2) = 3.
For n>1 and k>2, T(n,k) = 4*T(n-1,k-2)+1, so T(2,3) = 1.
For i>=0, a(i^2+1) = T(i+1,1).
T(n, k) = 2^k * (n - (6*k + 3 - (-1)^k)/12) - 1/3.
T(n,1) == 0 (mod 2); T(n,2) == 3 (mod 4); T(n,k>=3) == 1 (mod 4).
k = v2(3*T(n,k)+1) + 1, where v2(x) = A007814(x) is the 2-adic valuation of x.
Comments