A274757 Numbers k such that 6*k+1 is a triangular number (A000217).

0, 9, 15, 42, 54, 99, 117, 180, 204, 285, 315, 414, 450, 567, 609, 744, 792, 945, 999, 1170, 1230, 1419, 1485, 1692, 1764, 1989, 2067, 2310, 2394, 2655, 2745, 3024, 3120, 3417, 3519, 3834, 3942, 4275, 4389, 4740, 4860, 5229, 5355, 5742, 5874, 6279, 6417

Offset

1,2

Comments

Numbers of the type floor(3*m*(m+1)/4) for which floor(3*m*(m+1)/4) = 3*floor(m*(m+1)/4). A014601 lists the values of m. - Bruno Berselli, Jan 13 2017

Numbers of the form 3*k*(4*k + 1) for k in Z. - Peter Bala, Nov 21 2024

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

G.f.: 3*x^2*(3 + 2*x + 3*x^2) / ((1 - x)^3*(1 + x)^2).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.

a(n) = 3*(2*n - 1)*(2*n + (-1)^n - 1)/4. Therefore:

a(n) = 3*n*(2*n - 1)/2 for n even,

a(n) = 3*(n-1)*(2*n - 1)/2 for n odd.

Mathematica

Table[3 (2 n - 1) (2 n + (-1)^n - 1)/4, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)

Prog

(PARI) isok(n) = ispolygonal(6*n+1, 3)
(PARI) select(n->ispolygonal(6*n+1, 3), vector(7000, n, n-1))
(PARI) concat(0, Vec(3*x^2*(3+2*x+3*x^2)/((1-x)^3*(1+x)^2) + O(x^60)))

Crossrefs

Cf. A000217, A014601.

Cf. A000096 (k+1), A074377 (2*k+1), A045943 (3*k+1), A274681 (4*k+1), A085787 (5*k+1).

Cf. similar sequences listed in A274830.

Sequence in context: A128687 A379052 A193579 * A146789 A194946 A020172

Adjacent sequences: A274754 A274755 A274756 * A274758 A274759 A274760

Keyword

nonn,easy

Author

Colin Barker, Jul 04 2016

Status

approved