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

1,1

Comments

Former title: Triangular form sequence made from a version of A082605 Euler extension.

References

Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 155

Links

G. C. Greubel, Rows n = 1..10 of the irregular triangle, flattened

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A056486, A082605.

Sequence in context: A072055 A314300 A189320 * A147853 A111226 A168224

Adjacent sequences: A107445 A107446 A107447 * A107449 A107450 A107451

Keyword

nonn

Author

Roger L. Bagula, May 26 2005

Extensions

Edited by G. C. Greubel, Mar 23 2024

Status

approved