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A214961 a(0)=a(1)=1, a(n) = least k > a(n-1) such that k*a(n-2) is a triangular number.

%I A214961 #31 Jun 05 2025 18:13:43
%S A214961 1,1,3,6,7,11,13,21,25,30,49,59,97,117,193,233,385,465,492,596,983,
%T A214961 1191,1965,2381,2516,4761,5031,5761,6290,8466,9795,15470,15867,17403,
%U A214961 20559,24170,26945,27192,27755,30130,35235,43537,45100,56805,58717,58739,91000,117477
%N A214961 a(0)=a(1)=1, a(n) = least k > a(n-1) such that k*a(n-2) is a triangular number.
%H A214961 Robert Israel, <a href="/A214961/b214961.txt">Table of n, a(n) for n = 0..2100</a>
%p A214961 f:= proc(a,b)
%p A214961   local s;
%p A214961   s:= map(t -> rhs(op(t)), [msolve(x^2=1, 8*a)]);
%p A214961   min(select(`>`, map(t -> (t^2-1)/(8*a), s), b))
%p A214961 end proc:
%p A214961 A[0]:= 1: A[1]:= 1:
%p A214961 for nn from 2 to 100 do
%p A214961   A[nn]:= f(A[nn-2],A[nn-1])
%p A214961 od:
%p A214961 seq(A[i],i=0..100); # _Robert Israel_, Jun 17 2020
%t A214961 a[0]=a[1]=1;a[n_]:=a[n]=(k=a[n-1]+1;While[!IntegerQ@Sqrt[1+8*a[n-2]k],k++];k);Array[a,50,0] (* _Giorgos Kalogeropoulos_, May 21 2021 *)
%t A214961 lktn[{a_,b_}]:=Module[{k=b+1},While[!OddQ[Sqrt[8a k+1]],k++];{b,k}]; NestList[lktn,{1,1},50][[;;,1]] (* _Harvey P. Dale_, Sep 09 2023 *)
%o A214961 (Python)
%o A214961 prpr = prev = 1
%o A214961 for n in range(1, 55):
%o A214961     print(prpr, end=', ')
%o A214961     b = k = 0
%o A214961     while k<=prev:
%o A214961         d = b*(b+1)//2
%o A214961         k = 0
%o A214961         if d%prpr==0:
%o A214961             k = d // prpr
%o A214961         b += 1
%o A214961     prpr = prev
%o A214961     prev = k
%Y A214961 Cf. A214914, A214915, A214916, A214963, A213005.
%K A214961 nonn
%O A214961 0,3
%A A214961 _Alex Ratushnyak_, Aug 03 2012