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 A077995 #34 Jan 11 2025 18:10:08
%S A077995 1,1,4,11,31,88,249,705,1996,5651,15999,45296,128241,363073,1027924,
%T A077995 2910235,8239391,23327176,66043369,186980481,529374876,1498754083,
%U A077995 4243238399,12013359840,34011950561,96293859201,272624979364,771849627691,2185243073311,6186810381368
%N A077995 Expansion of (1 - x)/(1 - 2*x - 2*x^2 - x^3).
%C A077995 Equals INVERT transform of (1, 3, 4, 4, 4, ...). - _Gary W. Adamson_, Jan 03 2009
%H A077995 Harvey P. Dale, <a href="/A077995/b077995.txt">Table of n, a(n) for n = 0..1000</a>
%H A077995 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,1).
%F A077995 a(n) = Sum_{m=1..n} Sum_{i=0..n-m} C(m+i-1,m-1)*Sum_{j=0..m} C(j,n-3*m +2*j-i) * C(m,j)*2^(n-3*m+2*j-i), n>0, a(0)=1. - _Vladimir Kruchinin_, May 12 2011
%F A077995 G.f.: 1 + x/(G(0)-x) where G(k) = 1 - x*(2*k+2)/(1 - 1/(1 + (2*k+2)/G(k+1)));(continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 17 2012
%F A077995 a(n) = 2*a(n-1) + 2*a(n-2) + a(n-3); a(0)=1, a(1)=1, a(2)=4. - _Harvey P. Dale_, Sep 11 2013
%t A077995 CoefficientList[Series[(1-x)/(1-2x-2x^2-x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,1},{1,1,4},30] (* _Harvey P. Dale_, Sep 11 2013 *)
%o A077995 (Maxima) a(n):=sum(sum(binomial(m+i-1,m-1)*sum(binomial(j,n-3*m+2*j-i) *binomial(m,j) *2^(n-3*m+2*j-i),j,0,m) ,i,0,n-m) ,m,1,n); /* _Vladimir Kruchinin_, May 12 2011 */
%o A077995 (PARI) Vec((1-x)/(1-2*x-2*x^2-x^3)+O(x^30)) \\ _Charles R Greathouse IV_, Sep 24 2012
%o A077995 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/( 1-2*x-2*x^2-x^3) )); // _G. C. Greubel_, Jun 27 2019
%o A077995 (Sage) ((1-x)/(1-2*x-2*x^2-x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 27 2019
%o A077995 (GAP) a:=[1,1,4];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Jun 27 2019
%K A077995 nonn,easy
%O A077995 0,3
%A A077995 _N. J. A. Sloane_, Nov 17 2002