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 A118057 #55 Sep 08 2022 08:45:24
%S A118057 1,21,57,109,177,261,361,477,609,757,921,1101,1297,1509,1737,1981,
%T A118057 2241,2517,2809,3117,3441,3781,4137,4509,4897,5301,5721,6157,6609,
%U A118057 7077,7561,8061,8577,9109,9657,10221,10801,11397,12009,12637,13281,13941,14617
%N A118057 a(n) = 8*n^2 - 4*n - 3.
%C A118057 In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(1)=A001652(1)=3 and z(1)=A001653(1)=5; cf. A000290, A118058-A118061.
%C A118057 Sequence found by reading the segment (1, 21) together with the line from 21, in the direction 21, 57, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011
%H A118057 Harvey P. Dale, <a href="/A118057/b118057.txt">Table of n, a(n) for n = 1..1000</a>
%H A118057 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A118057 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+18*x-3*x^2)/(1-x)^3. - _Colin Barker_, Jul 01 2012
%F A118057 a(n)+(a(n)+1)+...+(a(n)+8n+5)=(a(n)+8n+6)+...+a(n+1)-1; a(n+1)-1=a(n)+16n+3.
%F A118057 a(n)+(a(n)+1)+...+(a(n)+8n+5)=(4n-1)(4n+1)(4n+3); e.g., 21+22+...+56=693=7*9*11.
%F A118057 a(n) = 16*n+a(n-1)-12 (with a(1)=1). - _Vincenzo Librandi_, Nov 13 2010
%F A118057 a(n) = A139098(n) - A004767(n). - _Omar E. Pol_, Sep 18 2012
%e A118057 a(3)=8*3^2-4*3-3=57, a(4)=8*4^2-4*4-3=109 and 57+58+...+86=87+...+108.
%t A118057 Table[8n^2-4n-3,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,57},50] (* _Harvey P. Dale_, Sep 18 2012 *)
%o A118057 (PARI) a(n)=8*n^2-4*n-3 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A118057 (Magma) [8*n^2-4*n-3 : n in [1..60]]; // _Wesley Ivan Hurt_, Jan 28 2021
%Y A118057 Cf. A004767, A139098.
%Y A118057 Cf. A000290, A001652, A001653, A118058-A118061.
%K A118057 nonn,easy
%O A118057 1,2
%A A118057 _Charlie Marion_, Apr 26 2006