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%I #22 Jun 29 2023 18:01:55
%S 1,3,10,21,190,1711,15400,138601,1247410,11226691,101040220,909361981,
%T 8184257830,73658320471,662924884240,5966323958161,53696915623450,
%U 483272240611051,4349450165499460,39145051489495141,352305463405456270
%N a(1) = 1, a(n) = smallest triangular number of the form k*a(n-1) + 1 for some positive integer k.
%H Vincenzo Librandi, <a href="/A068082/b068082.txt">Table of n, a(n) for n = 0..200</a>
%H Robert Israel, <a href="/A068082/a068082.pdf">Proof of conjectured formula</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -9).
%F Conjectures from _Colin Barker_, May 30 2016: (Start)
%F a(n) = (169*9^(n-3)-1)/8 for n>2.
%F a(n) = 10*a(n-1)-9*a(n-2) for n>4.
%F G.f.: (1-7*x-11*x^2-52*x^3+70*x^4) / ((1-x)*(1-9*x)).
%F (End)
%F Conjectures confirmed by _Robert Israel_, Jun 05 2018 (see link).
%e a(5) = 190 = 9*21 + 1 = 9*a(4) + 1 and none of the values of the form k*21 + 1 for 0 < k < 9 are triangular.
%p readlib(issqr): istri := proc(m) if issqr(8*m+1) then RETURN(1) else RETURN(0) fi: end: L := [seq(0,i=1..40)]: L[1] := 1: for n from 2 to 40 do for k from 2 to 100 do if istri(k*L[n-1]+1)=1 then L[n] := k*L[n-1]+1; break; fi:od: od:
%t ntn[n_]:=Module[{k=1},While[!IntegerQ[(Sqrt[8(k*n+1)+1]-1)/2],k++]; k*n+ 1]; NestList[ntn,1,20] (* _Harvey P. Dale_, Jul 23 2014 *)
%K nonn
%O 0,2
%A _Amarnath Murthy_, Feb 18 2002
%E Edited and extended by _James A. Sellers_, Feb 20 2002
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