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%I #49 Sep 04 2023 19:58:52
%S 3,10,31,94,283,850,2551,7654,22963,68890,206671,620014,1860043,
%T 5580130,16740391,50221174,150663523,451990570,1355971711,4067915134,
%U 12203745403,36611236210,109833708631,329501125894,988503377683,2965510133050,8896530399151
%N a(n) = 3^(n+1) + (3^n-1)/2.
%C a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
%C The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - _R. J. Mathar_, Jan 08 2020
%C First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - _Klaus Purath_, Apr 13 2020
%C This is a subsequence of A017209 for n > 1. See formula. - _Klaus Purath_, Jul 03 2020
%H Colin Barker, <a href="/A237930/b237930.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphRadius.html">Graph Radius</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiCarpetGraph.html">Sierpinski Carpet Graph</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F G.f.: (3-2*x)/((1-x)*(1-3*x)).
%F a(n) = A000244(n+1) + A003462(n).
%F a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - _N. J. A. Sloane_, Dec 06 2019)
%F a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
%F a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
%F a(n) = A199109(n) - 1.
%F a(n) = (7*3^n - 1)/2. - _Eric W. Weisstein_, Mar 13 2018
%F From _Klaus Purath_, Apr 13 2020: (Start)
%F a(n) = A057198(n+1) + A024023(n).
%F a(n) = A029858(n+2) - A024023(n).
%F a(n) = A052919(n+1) + A029858(n+1).
%F a(n) = (A000244(n+1) + A171498(n))/2.
%F a(n) = 7*A003462(n) + 3.
%F a(n) = A116952(n) + 2. (End)
%F a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - _Klaus Purath_, Jul 03 2020
%F E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - _Stefano Spezia_, Aug 28 2023
%e Ternary....................Decimal
%e 10...............................3
%e 101.............................10
%e 1011............................31
%e 10111...........................94
%e 101111.........................283
%e 1011111........................850
%e 10111111......................2551
%e 101111111.....................7654, etc.
%t (* Start from _Eric W. Weisstein_, Mar 13 2018 *)
%t Table[(7 3^n - 1)/2, {n, 0, 20}]
%t (7 3^Range[0, 20] - 1)/2
%t LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
%t CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
%t (* End *)
%o (PARI) Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ _Colin Barker_, Nov 27 2019
%o (Magma) [3^(n+1) + (3^n-1)/2: n in [0..40]]; // _Vincenzo Librandi_, Jan 09 2020
%Y Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.
%Y Cf. A024023, A029858, A057198, A116952, A171498.
%K nonn,easy
%O 0,1
%A _Philippe Deléham_, Feb 16 2014
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