A076921 Smallest number such that the highest common factor of pair of successive terms follows the pattern 1, 1, 2, 2, 3, 3, 4, 4, ..., i.e., b(2n-1) = b(2n) = n given by A004526.

1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784

Offset

1,3

Comments

(1) a(2n) = n^2, a(2n-1) = n(n+1) = twice the n-th triangular number.

(2) Geometric mean of the successive squares interleaved between them.

Essentially the same as A002620.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

a(n+1) + a(n) = binomial(n+1,2), a(1) = a(2) = 1. - G. C. Greubel, Oct 29 2017

From Stefano Spezia, Nov 16 2024: (Start)

a(n) = (2*n^2 + (-1)^n - 1)/8 for n > 1.

G.f.: x*(1 - x + 2*x^2 - x^4)/((1 - x)^3*(1 + x)).

E.g.f.: (4*x + x*(1 + x)*cosh(x) - (1 - x - x^2)*sinh(x))/4. (End)

From Amiram Eldar, Dec 11 2025: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/6 + 2 = A013661 + 2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - Pi^2/6 (A152416). (End)

Mathematica

Join[{1}, LinearRecurrence[{2, 0, -2, 1}, {1, 2, 4, 6}, 50]] (* G. C. Greubel, Oct 29 2017 *)

Crossrefs

Cf. A002620, A004526, A013661, A152416.

Sequence in context: A088900 A371972 A083392 * A002620 A087811 A025699

Adjacent sequences: A076918 A076919 A076920 * A076922 A076923 A076924

Keyword

nonn,easy

Author

Amarnath Murthy, Oct 17 2002

Extensions

More terms from Philippe Deléham, Jun 20 2005

Status

approved