[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
3
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
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
Sequence in context: A072055 A314300 A189320 * A147853 A111226 A168224
KEYWORD
nonn
AUTHOR
Roger L. Bagula, May 26 2005
EXTENSIONS
Edited by G. C. Greubel, Mar 23 2024
STATUS
approved