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 A257448 #62 Apr 02 2025 05:03:11
%S A257448 1,9,37,111,283,657,1441,3051,6319,12909,26149,52695,105859,212265,
%T A257448 425161,851043,1702903,3406725,6814477,13630095,27261451,54524289,
%U A257448 109050097,218101851,436205503,872412957,1744828021,3489658311,6979319059,13958640729
%N A257448 a(n) = 13*(2^n - 1) - 3*n^2 - 9*n.
%C A257448 These numbers belong to a family of sequences obtained as follows:
%C A257448 . A000225: 1*(2^n-1);
%C A257448 . A050488: 3*(2^n-1) - 2*n;
%C A257448 . a(n): 13*(2^n-1) - 3*n^2 - 9*n;
%C A257448 . A257449: 75*(2^n-1) - 4*n^3 - 18*n^2 - 52*n;
%C A257448 . A257450: 541*(2^n-1) - 5*n^4 - 30*n^3 - 130*n^2 - 375*n,
%C A257448 where the sequence 1, 3, 13, 75, 541, ... is A000670 (after the first term), and A208744 gives the triangle of coefficients:
%C A257448 2;
%C A257448 3, 9;
%C A257448 4, 18, 52;
%C A257448 5, 30, 130, 375;
%C A257448 6, 45, 260, 1125, 3246;
%C A257448 7, 63, 455, 2625, 11361, 32781, etc.
%C A257448 Also, the antidiagonal sums in the array are given by the formula (6*n^2 + 6*k*n + (k-1)*k)*(k+n)!/((k+3)!*(n-1)!) for k = 0, 1, 2, 3, 4, ... (see Example field).
%H A257448 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5, -9, 7, -2).
%F A257448 G.f.: x*(1+4*x+x^2)/((1-x)^3*(1-2*x)).
%F A257448 a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - _Ray Chandler_, Jul 25 2015
%e A257448 By the second comment, the array begins (antidiagonals in A046902):
%e A257448 k=0: 1, 8, 27, 64, 125, 216, ... A000578
%e A257448 k=1: 1, 9, 36, 100, 225, 441, ... A000537
%e A257448 k=2: 1, 10, 46, 146, 371, 812, ... A024166
%e A257448 k=3: 1, 11, 57, 203, 574, 1386, ... A101094
%e A257448 k=4: 1, 12, 69, 272, 846, 2232, ... A101097
%e A257448 k=5: 1, 13, 82, 354, 1200, 3432, ... A101102
%e A257448 k=6: 1, 14, 96, 450, 1650, 5082, ... A254469
%e A257448 ...
%e A257448 See also A254469 (Example field).
%t A257448 Table[13 (2^n - 1) - 3 n^2 - 9n, {n, 30}]
%t A257448 CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^3*(1 - 2 x)), {x, 0, 30}], x] (* _Michael De Vlieger_, Nov 14 2016 *)
%o A257448 (Magma) [13*(2^n-1)-3*n^2-9*n: n in [1..30]]; // _Vincenzo Librandi_, Apr 24 2015
%Y A257448 Cf. A000225, A000670, A050488, A208744, A257449, A257450.
%K A257448 nonn,easy
%O A257448 1,2
%A A257448 _Luciano Ancora_, Apr 23 2015
%E A257448 Edited by _Bruno Berselli_, Apr 28 2015