This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A022265 #58 Mar 15 2025 17:34:45
%S A022265 0,4,15,33,58,90,129,175,228,288,355,429,510,598,693,795,904,1020,
%T A022265 1143,1273,1410,1554,1705,1863,2028,2200,2379,2565,2758,2958,3165,
%U A022265 3379,3600,3828,4063,4305,4554,4810
%N A022265 a(n) = n*(7*n + 1)/2.
%C A022265 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
%H A022265 G. C. Greubel, <a href="/A022265/b022265.txt">Table of n, a(n) for n = 0..5000</a>
%H A022265 Leo Tavares, <a href="/A022265/a022265.jpg">Illustration: Square-sided Triangles</a>.
%H A022265 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A022265 a(n) = A110449(n, 3) for n>2.
%F A022265 a(n) = A049453(n) - A005475(n). - _Zerinvary Lajos_, Jan 21 2007
%F A022265 a(n) = 7*n + a(n-1) - 3 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 04 2010
%F A022265 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
%F A022265 a(n) = A174738(7n+3). - _Philippe Deléham_, Mar 26 2013
%F A022265 a(n) = A000217(4*n) - A000217(3*n). - _Bruno Berselli_, Oct 13 2016
%F A022265 G.f.: x*(4 + 3*x)/(1 - x)^3. - _Ilya Gutkovskiy_, Oct 13 2016
%F A022265 E.g.f.: (x/2)*(7*x + 8)*exp(x). - _G. C. Greubel_, Aug 23 2017
%F A022265 a(n) = A000217(n) + 3*A000290(n). - _Leo Tavares_, Mar 15 2025
%e A022265 From _Bruno Berselli_, Oct 27 2017: (Start)
%e A022265 After 0:
%e A022265 4 = -(1) + (2 + 3).
%e A022265 15 = -(1 + 2) + (3 + 4 + 5 + 6).
%e A022265 33 = -(1 + 2 + 3) + (4 + 5 + 6 + 7 + 8 + 9).
%e A022265 58 = -(1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + 9 + 10 + 11 + 12). (End)
%p A022265 seq(binomial(7*n+1,2)/7, n=0..37); # _Zerinvary Lajos_, Jan 21 2007
%p A022265 seq(binomial(6*n+1,2)/3-binomial(5*n+1,2)/5, n=0..42); # _Zerinvary Lajos_, Jan 21 2007
%t A022265 Table[n (7 n + 1)/2, {n, 0, 40}] (* _Bruno Berselli_, Oct 13 2016 *)
%t A022265 LinearRecurrence[{3,-3,1},{0,4,15},40] (* _Harvey P. Dale_, Oct 09 2018 *)
%o A022265 (PARI) a(n)=n*(7*n+1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A022265 Cf. A001106, A022264, A024966, A110449, A174738, A179986, A186029, A218471.
%Y A022265 Cf. similar sequences listed in A022289.
%Y A022265 Cf. A000217, A000290.
%K A022265 nonn,easy
%O A022265 0,2
%A A022265 _N. J. A. Sloane_