A350519 a(n) = A(n,n) where A(1,n) = A(n,1) = prime(n+1) and A(m,n) = A(m-1,n) + A(m,n-1) + A(m-1,n-1) for m > 1 and n > 1.

3, 13, 63, 325, 1719, 9237, 50199, 275149, 1518263, 8422961, 46935819, 262512929, 1472854451, 8285893713, 46723439019, 264009961733, 1494486641911, 8473508472009, 48112827862527, 273541139290857, 1557023508876891, 8872219429659729, 50605041681538595, 288897992799897481

Offset

1,1

Comments

Replacing prime(n+1) by other functions f(n) we get:

A001850 (with f(n) = 1),

A002002 (with f(n) = n+1),

A050151 (with f(n) = n/2), and

A344576 (with f(n) = Fibonacci(n)).

Links

Table of n, a(n) for n=1..24.

Example

The two-dimensional recurrence A(m,n) can be depicted in matrix form as
   3   5   7   11   13    17    19 ...
   5  13  25   43   67    97   133 ...
   7  25  63  131  241   405   635 ...
  11  43 131  325  697  1343  2383 ...
  13  67 241  697 1719  3759  7485 ...
  17  97 405 1343 3759  9237 20481 ...
  19 133 635 2383 7485 20481 50199 ...
  ...
and then a(n) is the main diagonal of this matrix, A(n,n).

Mathematica

f[1, 1]=3; f[m_, 1]:=Prime[m+1]; f[1, n_]:=Prime[n+1]; f[m_, n_]:=f[m, n]=f[m-1, n]+f[m, n-1]+f[m-1, n-1]; Table[f[n, n], {n, 25}] (* Giorgos Kalogeropoulos, Jan 03 2022 *)

Prog

(MATLAB)
clear all
close all
sz = 14
f = zeros(sz, sz);
pp = primes(50);
f(1, :) = pp(2:end);
f(:, 1) = pp(2:end);
for m=2:sz
    for  n=2:sz
        f(m, n) = f(m-1, n-1)+f(m, n-1)+f(m-1, n);
    end
end
an = []
for n=1:sz
    an = [an f(n, n)];
end
S = sprintf('%i, ', an);
S = S(1:end-1)

Crossrefs

Cf. A000040, A001850, A002002, A050151, A344576 (see comments).

Sequence in context: A001850 A130525 A362744 * A243280 A000259 A007855

Adjacent sequences: A350516 A350517 A350518 * A350520 A350521 A350522

Keyword

nonn

Author

Yigit Oktar, Jan 02 2022

Status

approved