%I #8 Mar 23 2024 20:23:12

%S 5,7,11,17,13,17,23,31,41,53,67,83,101,19,23,29,37,47,59,73,89,107,

%T 127,149,173,199,227,257,43,47,53,61,71,83,97,113,131,151,173,197,223,

%U 251,281,313,347,383,421,461,503,547,593,641,691,743,797,853,911,971,1033

%N 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.

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

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

%H G. C. Greubel, <a href="/A107448/b107448.txt">Rows n = 1..10 of the irregular triangle, flattened</a>

%F 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

%e The irregular triangle begins as:

%e 5;

%e 7, 11, 17;

%e 13, 17, 23, 31, 41, 53, 67, 83, 101;

%e 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;

%t (* First program *)

%t a[1] = 3; a[2] = 5; a[3] = 11; a[n_]:= a[n]= Abs[1-4*a[n-2]] -2;

%t euler= Table[a[n], {n,10}];

%t Table[k^2 + k + euler[[n]], {n,7}, {k,euler[[i]] -2}]//Flatten

%t (* Second program *)

%t b[n_]:= 2^(n-3)*(9-(-1)^n) - Boole[n==1]/2;

%t T[n_, k_]:= b[n] +k^2+k+1;

%t Table[T[n,k], {n,8}, {k,b[n]-1}]//Flatten (* _G. C. Greubel_, Mar 23 2024 *)

%o (Magma)

%o b:= func< n | n eq 1 select 2 else 2^(n-3)*(9-(-1)^n) >;

%o A107448:= func< n,k | b(n) +k^2 +k +1 >;

%o [A107448(n,k): k in [1..b(n)-1], n in [1..8]]; // _G. C. Greubel_, Mar 23 2024

%o (SageMath)

%o def b(n): return 2^(n-3)*(9-(-1)^n) - int(n==1)/2

%o def A107448(n,k): return b(n) + k^2+k+1;

%o flatten([[A107448(n,k) for k in range(1,b(n))] for n in range(1,8)]) # _G. C. Greubel_, Mar 23 2024

%Y Cf. A056486, A082605.

%K nonn

%O 1,1

%A _Roger L. Bagula_, May 26 2005

%E Edited by _G. C. Greubel_, Mar 23 2024