A118005 - OEIS

%I #23 Feb 16 2025 08:33:00

%S 1,4,61,424,4441,36844,347221,3046864,27812401,248358484,2244991981,

%T 20156099704,181649037961,1633620638524,14708689262341,

%U 132347685782944,1191281759937121,10720772899980964,96490770797094301,868397863687520584,7815676140619325881,70340608428415729804,633067860041532583861,5697598819444838176624

%N a(n) = ((-1)^n*5^(n+1) + 9^(n+1))/14.

%C Number of black cells after n iterations of Haferman's carpet.

%H Colin Barker, <a href="/A118005/b118005.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HafermanCarpet.html">Haferman Carpet</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,45).

%F From _Colin Barker_, Jun 08 2013: (Start)

%F a(n) = 4*a(n-1) + 45*a(n-2).

%F G.f.: 1 / ((1+5*x)*(1-9*x)). (End)

%F E.g.f.: (1/14)*(5*exp(-5*x) + 9*exp(9*x)). - _G. C. Greubel_, Nov 12 2024

%t LinearRecurrence[{4,45}, {1,4}, 41] (* _G. C. Greubel_, Nov 12 2024 *)

%o (PARI) Vec(1 / ((1 + 5*x)*(1 - 9*x)) + O(x^40)) \\ _Colin Barker_, Feb 26 2020

%o (Magma) [n le 2 select 4^(n-1) else 4*Self(n-1) +45*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Nov 12 2024

%o (Python)

%o def A118005(n): return (9^(n+1) +(-1)^n*5^(n+1))//14

%o print([A118005(n) for n in range(41)]) # _G. C. Greubel_, Nov 12 2024

%K nonn,easy,changed

%O 0,2

%A _Eric W. Weisstein_, Apr 09 2006