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 A237930 a(n) = 3^(n+1) + (3^n-1)/2. 8
 3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020 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 This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Graph Radius. Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph. Index entries for linear recurrences with constant coefficients, signature (4,-3). FORMULA G.f.: (3-2*x)/((1-x)*(1-3*x)). a(n) = A000244(n+1) + A003462(n). 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) a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10. a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1. a(n) = A199109(n) - 1. a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018 From Klaus Purath, Apr 13 2020: (Start) a(n) = A057198(n+1) + A024023(n). a(n) = A029858(n+2) - A024023(n). a(n) = A052919(n+1) + A029858(n+1). a(n) = (A000244(n+1) + A171498(n))/2. a(n) = 7*A003462(n) + 3. a(n) = A116952(n) + 2. (End) a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020 E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023 EXAMPLE Ternary....................Decimal 10...............................3 101.............................10 1011............................31 10111...........................94 101111.........................283 1011111........................850 10111111......................2551 101111111.....................7654, etc. MATHEMATICA (* Start from Eric W. Weisstein, Mar 13 2018 *) Table[(7 3^n - 1)/2, {n, 0, 20}] (7 3^Range[0, 20] - 1)/2 LinearRecurrence[{4, -3}, {10, 31}, {0, 20}] CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x] (* End *) PROG (PARI) Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019 (Magma) [3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020 CROSSREFS Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774. Cf. A024023, A029858, A057198, A116952, A171498. Sequence in context: A339032 A033121 A180432 * A192337 A106517 A363780 Adjacent sequences: A237927 A237928 A237929 * A237931 A237932 A237933 KEYWORD nonn,easy AUTHOR Philippe Deléham, Feb 16 2014 STATUS approved

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Last modified December 5 20:45 EST 2023. Contains 367594 sequences. (Running on oeis4.)