A195320 7 times hexagonal numbers: a(n) = 7*n*(2*n-1).

0, 7, 42, 105, 196, 315, 462, 637, 840, 1071, 1330, 1617, 1932, 2275, 2646, 3045, 3472, 3927, 4410, 4921, 5460, 6027, 6622, 7245, 7896, 8575, 9282, 10017, 10780, 11571, 12390, 13237, 14112, 15015, 15946, 16905, 17892, 18907, 19950, 21021, 22120, 23247, 24402, 25585

Offset: 0

Author

Omar E. Pol, Sep 18 2011

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277.

Also sequence found by reading the same line (mentioned above) in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5]. - Omar E. Pol, Oct 13 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A024966.

Cf. A000384, A118277, A152746, A152750, A185019, A193053, A195019, A195020, A198017, A316466.

Programs

Magma

[7*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Sep 28 2011

Mathematica

Table[7*n*(2*n - 1), {n, 0, 50}] (* Paolo Xausa, Jan 29 2025 *)
7*PolygonalNumber[6, Range[0, 50]] (* Paolo Xausa, Jan 29 2025 *)

PARI

a(n)=7*n*(2*n-1) \\ Charles R Greathouse IV, Sep 28 2015

Formula

a(n) = 14*n^2 - 7*n = 7*A000384(n).

G.f.: -7*x*(1+3*x)/(x-1)^3. - R. J. Mathar, Sep 27 2011

From Elmo R. Oliveira, Dec 27 2024: (Start)

E.g.f.: 7*exp(x)*x*(2*x + 1).

a(n) = A316466(n) - n = A024966(2*n+1).

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