A274603 Numbers k such that 2*k+1 and 3*k+1 are both triangular numbers.

45, 4455, 436590, 42781410, 4192141635, 410787098865, 40252943547180, 3944377680524820, 386508759747885225, 37873914077612227275, 3711257070846250387770, 363665319028854925774230, 35635490007756936475486815, 3491914355441150919671933685

Offset

1,1

Comments

Inspired by A274579.

a(n+1) / a(n) goes to 49 + 20*sqrt(6) when n goes to infinity.

Intersection of A045943 and A074377. - Colin Barker, Jun 30 2016

Links

Colin Barker, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Formula

From Colin Barker, Jun 30 2016: (Start)

a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n > 3.

G.f.: 45*x / ((1-x)*(1-98*x+x^2)). (End)

Sum_{n>=1} 1/a(n) = (2/9) * (5 - 2*sqrt(6)). - Amiram Eldar, Jan 27 2026

Example

45 is a term because 2*45 + 1 = 91 and 3*45 + 1 = 136 are both triangular numbers.

Mathematica

LinearRecurrence[{99, -99, 1}, {45, 4455, 436590}, 14] (* Amiram Eldar, Jan 27 2026 *)

Prog

(PARI) isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(3*n+1, 3);
(PARI) Vec(45*x/((1-x)*(1-98*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 30 2016

Crossrefs

Cf. A000217, A045943, A074377, A274579.

Sequence in context: A143004 A004707 A220094 * A036521 A328356 A375052

Adjacent sequences: A274600 A274601 A274602 * A274604 A274605 A274606

Keyword

nonn,easy

Author

Altug Alkan, Jun 30 2016

Extensions

More terms from Colin Barker, Jun 30 2016

Status

approved