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 A016153 #36 Jul 27 2025 11:15:58
%S A016153 0,1,13,133,1261,11605,105469,953317,8596237,77431669,697147165,
%T A016153 6275373061,56482551853,508359743893,4575304803901,41178011670565,
%U A016153 370603178776909,3335432903959477,30018913315504477,270170288559017029
%N A016153 a(n) = (9^n-4^n)/5.
%C A016153 a(n) is also the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y)=xF(n-1)(x,y)+yF(n-2)(x,y),F(0)(x,y)=0,F(1)(x,y)=1, when we write 13x for x and -36x^2 for y. - Mario Catalani (mario.catalani(AT)unito.it), Dec 09 2002
%H A016153 John Elias, <a href="/A016153/a016153.png">Illustration: Union of Cantor Square and Koch Snowflake fractals</a>
%H A016153 R. Flórez, R. A. Higuita and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%H A016153 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-36).
%F A016153 G.f.: x/((1-4*x)*(1-9*x)). a(n)=13*a(n-1)-36*a(n-2).
%F A016153 a(n) = A015441(2*n).
%F A016153 From _Peter Bala_, Jul 23 2025: (Start)
%F A016153 The following products telescope. Cf. A002450:
%F A016153 Product_{k >= 1} 1 + 6^k/a(k+1) = 3; Product_{k >= 1} 1 - 6^k/a(k+1) = 3/13.
%F A016153 Product_{k >= 1} 1 + (-6)^k/a(k+1) = 3/5; Product_{k >= 1} 1 - (-6)^k/a(k+1) = 15/13.
%F A016153 (End)
%t A016153 Join[{a=0,b=1},Table[c=13*b-36*a;a=b;b=c,{n,60}]](*and/or*)f[n_]:=(9^n-4^n)/5;f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)
%o A016153 (PARI) a(n)=(9^n-4^n)/5
%Y A016153 Cf. A015441, A002450.
%K A016153 nonn,easy
%O A016153 0,3
%A A016153 _N. J. A. Sloane_