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 A191597 #22 Dec 16 2023 15:51:13 %S A191597 0,1,6,21,85,342,1365,5461,21846,87381,349525,1398102,5592405, %T A191597 22369621,89478486,357913941,1431655765,5726623062,22906492245, %U A191597 91625968981,366503875926,1466015503701,5864062014805,23456248059222,93824992236885,375299968947541 %N A191597 Expansion of x*(1+3*x)/ ( (1-4*x)*(1+x+x^2)). %C A191597 a(n) and successive differences define a square array T(0,k) = a(k), T(n,k) = T(n-1,k+1) - T(n-1,k): %C A191597 0, 1, 6, 21, 85, 342,... %C A191597 1, 5, 15, 64, 257, 1023,... %C A191597 4, 10, 49, 193, 766, 3073,... %C A191597 As with any sequence which obeys a homogeneous linear recurrence (we say it once, only once and we shall not repeat it), the recurrence is also valid for the rows of such arrays of higher order differences. %H A191597 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,4). %F A191597 a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3), n >= 3. %F A191597 a(n) = A024495(2*n). %F A191597 a(n) = A113405(2*n) + A113405(2*n+1). %F A191597 a(n+1) - 4*a(n) = A132677(n). %F A191597 a(n+3) - a(n) = 21*4^n. %F A191597 a(n) = A178872(n) + 3*A178872(n-1) = (4^n-A061347(n+1))/3. - _R. J. Mathar_, Jun 08 2011 %p A191597 A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc: %p A191597 A191597 := proc(n) (4^n-A061347(n+1))/3 ; end proc: %p A191597 seq(A191597(n),n=0..30) ; # _R. J. Mathar_, Jun 08 2011 %o A191597 (PARI) a(n)=([0,1,0; 0,0,1; 4,3,3]^n*[0;1;6])[1,1] \\ _Charles R Greathouse IV_, Jul 06 2017 %K A191597 nonn,easy %O A191597 0,3 %A A191597 _Paul Curtz_, Jun 08 2011