This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360826 #41 Mar 25 2025 10:17:45
%S A360826 1,6,91,597871,213122969971321411,
%T A360826 9680343693975641657052402556458789711774336036960631
%N A360826 a(1) = 1, a(n) = (k+1)*(2*k+1), where k = Product_{i=1..n-1} a(i).
%C A360826 A sequence of pairwise relatively prime triangular (and also hexagonal) numbers.
%C A360826 As a clarification to the problem definition by Sierpinski, here we show that only one triangular (hexagonal) seed is needed to produce such a sequence.
%C A360826 This sequence can be used for proving the infinitude of primes.
%C A360826 In general: Let m = 2*q, for any q > 0. There are infinitely many sequences of pairwise coprime m-gonal numbers, whose first term is any positive m-gonal number and whose general term is of the form a(n) = (k + 1)*((q - 1)*k + 1), where k = Product_{i=1..n-1} a(i).
%D A360826 W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #42.
%H A360826 Michel Marcus, <a href="/A360826/b360826.txt">Table of n, a(n) for n = 1..8</a>
%F A360826 a(1) = 1, a(n) = (k+1)*(2*k+1), where k = Product_{i=1..n-1} a(i).
%F A360826 a(n) ~ c^(3^n), where c = 1.1784502032269064445225839284451956694752084180050932315805089054871825498... - _Vaclav Kotesovec_, May 05 2023
%t A360826 a[1]=1; a[n_]:=Module[{k=Product[a[i],{i,1,n-1}]},(k+1)*(2*k+1)];
%t A360826 a/@Range[6]
%t A360826 Join[{1}, RecurrenceTable[{a[2] == 6, a[n+1] == (1 + a[n]*(Sqrt[1 + 8*a[n]] - 3)/4) * (1 + 2*a[n]*(Sqrt[1 + 8*a[n]] - 3)/4)}, a, {n, 2, 8}]] (* _Vaclav Kotesovec_, May 05 2023 *)
%o A360826 (PARI) a(n) = if (n==1, 1, my(k = prod(i=1,n-1, a(i))); (k+1)*(2*k+1)); \\ _Michel Marcus_, Mar 25 2025
%Y A360826 Cf. A000217, A000384, A004019, A034792, A115590.
%K A360826 nonn
%O A360826 1,2
%A A360826 _Ivan N. Ianakiev_, Feb 22 2023