A022265 a(n) = n*(7*n + 1)/2.

0, 4, 15, 33, 58, 90, 129, 175, 228, 288, 355, 429, 510, 598, 693, 795, 904, 1020, 1143, 1273, 1410, 1554, 1705, 1863, 2028, 2200, 2379, 2565, 2758, 2958, 3165, 3379, 3600, 3828, 4063, 4305, 4554, 4810

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n >= 4, a(n) is the sum of the numbers appearing in the 4th row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

Examples

			From _Bruno Berselli_, Oct 27 2017: (Start)
After 0:
4  =       -(1)       +             (2 + 3).
15 =     -(1 + 2)     +         (3 + 4 + 5 + 6).
33 =   -(1 + 2 + 3)   +     (4 + 5 + 6 + 7 + 8 + 9).
58 = -(1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + 9 + 10 + 11 + 12). (End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Leo Tavares, Illustration: Square-sided Triangles.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001106, A022264, A024966, A110449, A174738, A179986, A186029, A218471.

Cf. similar sequences listed in A022289.

Cf. A000217, A000290.

Programs

Maple

seq(binomial(7*n+1,2)/7, n=0..37); # Zerinvary Lajos, Jan 21 2007
seq(binomial(6*n+1,2)/3-binomial(5*n+1,2)/5, n=0..42); # Zerinvary Lajos, Jan 21 2007

Mathematica

Table[n (7 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)
LinearRecurrence[{3,-3,1},{0,4,15},40] (* Harvey P. Dale, Oct 09 2018 *)

PARI

a(n)=n*(7*n+1)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = A110449(n, 3) for n>2.

a(n) = A049453(n) - A005475(n). - Zerinvary Lajos, Jan 21 2007

a(n) = 7*n + a(n-1) - 3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=0, a(1)=4, a(2)=15. - Philippe Deléham, Mar 26 2013

a(n) = A174738(7n+3). - Philippe Deléham, Mar 26 2013

a(n) = A000217(4*n) - A000217(3*n). - Bruno Berselli, Oct 13 2016

G.f.: x*(4 + 3*x)/(1 - x)^3. - Ilya Gutkovskiy, Oct 13 2016

E.g.f.: (x/2)*(7*x + 8)*exp(x). - G. C. Greubel, Aug 23 2017

a(n) = A000217(n) + 3*A000290(n). - Leo Tavares, Mar 15 2025