A195026 a(n) = 7*n*(2*n + 1).

0, 21, 70, 147, 252, 385, 546, 735, 952, 1197, 1470, 1771, 2100, 2457, 2842, 3255, 3696, 4165, 4662, 5187, 5740, 6321, 6930, 7567, 8232, 8925, 9646, 10395, 11172, 11977, 12810, 13671, 14560, 15477, 16422, 17395, 18396, 19425, 20482, 21567, 22680, 23821, 24990

Offset: 0

Author

Omar E. Pol, Oct 13 2011

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 21, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Semi-diagonal opposite to A195320 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Sum of the numbers from 6*n to 8*n. - Wesley Ivan Hurt, Dec 23 2015

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

Cf. A014105, A144555, A152760, A195019, A195020, A195021, A195023, A195024, A195025, A195320.

Cf. A185019, A193053, A198017.

Programs

Magma

[14*n^2 +7*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Maple

A195026:=n->7*n*(2*n+1): seq(A195026(n), n=0..50); # Wesley Ivan Hurt, Dec 23 2015

Mathematica

Table[7*n*(2*n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Dec 23 2015 *)
LinearRecurrence[{3,-3,1},{0,21,70},50] (* Harvey P. Dale, Apr 26 2017 *)

PARI

a(n)=7*n*(2*n+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 14*n^2 + 7*n.

a(n) = 7*A014105(n). - Bruno Berselli, Oct 13 2011

From Colin Barker, Apr 09 2012: (Start)

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

G.f.: 7*x*(3+x)/(1-x)^3. (End)

a(n) = Sum_{i=6*n..8*n} i. - Wesley Ivan Hurt, Dec 23 2015

E.g.f.: 7*exp(x)*x*(3 + 2*x). - Elmo R. Oliveira, Dec 29 2024