A107448 Irregular triangle T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1, read by rows.
5, 7, 11, 17, 13, 17, 23, 31, 41, 53, 67, 83, 101, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033
Offset: 1
Examples
The irregular triangle begins as: 5; 7, 11, 17; 13, 17, 23, 31, 41, 53, 67, 83, 101; 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;
References
- Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 155
Links
- G. C. Greubel, Rows n = 1..10 of the irregular triangle, flattened
Programs
-
Magma
b:= func< n | n eq 1 select 2 else 2^(n-3)*(9-(-1)^n) >; A107448:= func< n,k | b(n) +k^2 +k +1 >; [A107448(n,k): k in [1..b(n)-1], n in [1..8]]; // G. C. Greubel, Mar 23 2024
-
Mathematica
(* First program *) a[1] = 3; a[2] = 5; a[3] = 11; a[n_]:= a[n]= Abs[1-4*a[n-2]] -2; euler= Table[a[n], {n,10}]; Table[k^2 + k + euler[[n]], {n,7}, {k,euler[[i]] -2}]//Flatten (* Second program *) b[n_]:= 2^(n-3)*(9-(-1)^n) - Boole[n==1]/2; T[n_, k_]:= b[n] +k^2+k+1; Table[T[n,k], {n,8}, {k,b[n]-1}]//Flatten (* G. C. Greubel, Mar 23 2024 *) -
SageMath
def b(n): return 2^(n-3)*(9-(-1)^n) - int(n==1)/2 def A107448(n,k): return b(n) + k^2+k+1; flatten([[A107448(n,k) for k in range(1,b(n))] for n in range(1,8)]) # G. C. Greubel, Mar 23 2024
Formula
T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1. - G. C. Greubel, Mar 23 2024
Extensions
Edited by G. C. Greubel, Mar 23 2024
Comments