A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

0, 2, 5, 11, 17, 27, 36, 50, 62, 80, 95, 117, 135, 161, 182, 212, 236, 270, 297, 335, 365, 407, 440, 486, 522, 572, 611, 665, 707, 765, 810, 872, 920, 986, 1037, 1107, 1161, 1235, 1292, 1370, 1430, 1512, 1575, 1661, 1727, 1817, 1886, 1980, 2052, 2150, 2225

Offset

1,2

Comments

From Peter Bala, Nov 21 2024: (Start)

Numbers of the form n*(7*n + 3)/2 for n in Z. Cf. A057570.

The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-2)) * (1 + x^(7*n-5)) = 1 + x^2 + x^5 + x^11 + x^17 + x^27 + x^36 + .... Cf. A363800. (End)

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.: x^2*(2 + 3*x + 2*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) = (14*(n - 1)*n + (2*n - 1)*(-1)^n + 1)/16. Therefore:

a(n) = n*(7*n - 6)/8 for n even,

a(n) = (n - 1)*(7*n - 1)/8 for n odd.

E.g.f.: (x*(7*x -1)*cosh(x) + (7*x^2 + x + 1)*sinh(x))/8. - Stefano Spezia, Nov 26 2024

Mathematica

Table[(14 (n - 1) n + (2 n - 1) (-1)^n + 1)/16, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)

Prog

(PARI) select(n->ispolygonal(7*n+1, 3), vector(3000, n, n-1))
(PARI) concat(0, Vec(x^2*(2+3*x+2*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

Crossrefs

Cf. A000217, A057570, A363800.

Cf. similar sequences where k*n+1 is a triangular number: A000096 (k=1), A074377 (k=2), A045943 (k=3), A274681 (k=4), A085787 (k=5), A274757 (k=6).

Sequence in context: A115057 A228344 A157421 * A038390 A048210 A153222

Adjacent sequences: A274827 A274828 A274829 * A274831 A274832 A274833

Keyword

nonn,easy

Author

Colin Barker, Jul 08 2016

Extensions

Edited by Bruno Berselli, Jul 08 2016

Status

approved