login
A068082
a(1) = 1, a(n) = smallest triangular number of the form k*a(n-1) + 1 for some positive integer k.
1
1, 3, 10, 21, 190, 1711, 15400, 138601, 1247410, 11226691, 101040220, 909361981, 8184257830, 73658320471, 662924884240, 5966323958161, 53696915623450, 483272240611051, 4349450165499460, 39145051489495141, 352305463405456270
OFFSET
0,2
FORMULA
Conjectures from Colin Barker, May 30 2016: (Start)
a(n) = (169*9^(n-3)-1)/8 for n>2.
a(n) = 10*a(n-1)-9*a(n-2) for n>4.
G.f.: (1-7*x-11*x^2-52*x^3+70*x^4) / ((1-x)*(1-9*x)).
(End)
Conjectures confirmed by Robert Israel, Jun 05 2018 (see link).
EXAMPLE
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.
MAPLE
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:
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A286067 A337623 A158030 * A058077 A161672 A190092
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Feb 18 2002
EXTENSIONS
Edited and extended by James A. Sellers, Feb 20 2002
STATUS
approved