A024966 7 times triangular numbers: 7*n*(n+1)/2.

0, 7, 21, 42, 70, 105, 147, 196, 252, 315, 385, 462, 546, 637, 735, 840, 952, 1071, 1197, 1330, 1470, 1617, 1771, 1932, 2100, 2275, 2457, 2646, 2842, 3045, 3255, 3472, 3696, 3927, 4165, 4410, 4662, 4921, 5187, 5460, 5740, 6027, 6321, 6622

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 7, ... and the same line from 0, in the direction 1, 21, ..., in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the main diagonal in the spiral. - Omar E. Pol, Sep 09 2011

Also sequence found by reading the same line mentioned above in the square spiral whose vertices are the generalized enneagonal numbers A118277. Axis perpendicular to A195145 in the same spiral. - Omar E. Pol, Sep 18 2011

Sequence provides all integers m such that 56*m + 49 is a square. - Bruno Berselli, Oct 07 2015

Sum of the numbers from 3*n to 4*n. - Wesley Ivan Hurt, Dec 22 2015

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A001106, A002378, A022264, A022265, A028895, A028896, A033996.

Cf. A045943, A046092, A069099, A118277, A174738, A179986, A186029, A193053.

Cf. A195019, A195020, A195145, A218471.

Programs

Magma

[ (7*n^2 + 7*n)/2 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

Maple

[seq(7*binomial(n,2), n=1..44)]; # Zerinvary Lajos, Nov 24 2006

Mathematica

7 Table[n (n + 1)/2, {n, 0, 43}] (* or *)
Table[Sum[i, {i, 3 n, 4 n}], {n, 0, 43}] (* or *)
Table[SeriesCoefficient[7 x/(1 - x)^3, {x, 0, n}], {n, 0, 43}] (* Michael De Vlieger, Dec 22 2015 *)
7*Accumulate[Range[0,50]] (* or *) LinearRecurrence[{3,-3,1},{0,7,21},50] (* Harvey P. Dale, Jul 20 2025 *)

PARI

x='x+O('x^100); concat(0, Vec(7*x/(1-x)^3)) \\ Altug Alkan, Dec 23 2015

Formula

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

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

a(n) = (7*n^2 + 7*n)/2 = 7*A000217(n). - Omar E. Pol, Dec 12 2008

a(n) = a(n-1) + 7*n with n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010

a(n) = A069099(n+1) - 1. - Omar E. Pol, Oct 03 2011

a(n) = a(-n-1), a(n+2) = A193053(n+2) + 2*A193053(n+1) + A193053(n). - Bruno Berselli, Oct 21 2011

From Philippe Deléham, Mar 26 2013: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 21.

a(n) = A174738(7*n+6).

a(n) = A179986(n) + n = A186029(n) + 2*n = A022265(n) + 3*n = A022264(n) + 4*n = A218471(n) + 5*n = A001106(n) + 6*n. (End)

a(n) = Sum_{i=3*n..4*n} i. - Wesley Ivan Hurt, Dec 22 2015

E.g.f.: (7/2)*x*(x+2)*exp(x). - G. C. Greubel, Aug 19 2017

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 2/7.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2/7)*(2*log(2) - 1). (End)

From Amiram Eldar, Feb 21 2023: (Start)

Product_{n>=1} (1 - 1/a(n)) = -(7/(2*Pi))*cos(sqrt(15/7)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (7/(2*Pi))*cosh(Pi/(2*sqrt(7))). (End)