A068082 a(1) = 1, a(n) = smallest triangular number of the form k*a(n-1) + 1 for some positive integer k.

1, 3, 10, 21, 190, 1711, 15400, 138601, 1247410, 11226691, 101040220, 909361981, 8184257830, 73658320471, 662924884240, 5966323958161, 53696915623450, 483272240611051, 4349450165499460, 39145051489495141, 352305463405456270

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Robert Israel, Proof of conjectured formula

Index entries for linear recurrences with constant coefficients, signature (10, -9).

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

Adjacent sequences: A068079 A068080 A068081 * A068083 A068084 A068085

Keyword

nonn

Author

Amarnath Murthy, Feb 18 2002

Extensions

Edited and extended by James A. Sellers, Feb 20 2002

Status

approved