A329774 -id:A329774 - cp's OEIS frontend

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

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

Offset: 0

Philippe Deléham, Feb 16 2014

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

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

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).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* 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 *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

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

A337270 Number of regions formed at generation n when the Conant "warp and woof" construction is applied to the base and left side of an equilateral triangle.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 20, 36, 57, 108, 185, 355, 637, 1246, 2344, 4595, 8895, 17532, 34592, 68287, 136053, 269046, 539516, 1068111, 2147477, 4254870, 8567392, 16982215, 34213477, 67850054, 136710948, 271162515, 546323617, 1083843471

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Aug 27 2020

nonn

This sequence completes a set of four. (1) The original Conant warp and woof construction used dissection lines that alternated between the base and left side of a square (see A328078).
(2) Robert Fathauer observed that if the construction starts with an equilateral triangle, and the dissection lines start from each of the three sides in rotation, the resulting structure in generation 3n converges to the Sierpinski Gasket fractal (see A329774).
(3) If the construction is applied to a square, and the dissection lines start from each of the four sides in rotation, we obtain the structures shown in A335703. To our surprise, these is no apparent fractal structure.
(4) The remaining case, an equilateral triangle with the dissection lines alternating between the base and the left side, is the subject of the present sequence.

Rémy Sigrist, Illustration for a(0)
Rémy Sigrist, Illustration for a(1)
Rémy Sigrist, Illustration for a(2)
Rémy Sigrist, Illustration for a(3)
Rémy Sigrist, Illustration for a(4)
Rémy Sigrist, Illustration for a(5)
Rémy Sigrist, Illustration for a(6)
Rémy Sigrist, Illustration for a(7)
Rémy Sigrist, Illustration for a(8)
Rémy Sigrist, Illustration for a(9)
Rémy Sigrist, Illustration for a(10)
Rémy Sigrist, Illustration for a(11)
Rémy Sigrist, Illustration for a(12)
Rémy Sigrist, Illustration for a(13)
Rémy Sigrist, Illustration for a(14)
Rémy Sigrist, Illustration for a(15)
Rémy Sigrist, Illustration for a(16)
Rémy Sigrist, Illustration of generations 0 through 16 (animated gif)
Rémy Sigrist, C++ program for A337270
N. J. A. Sloane, Illustration of generations 0 through 7. (The colored tick marks on the edges indicate the generation.)

Cf. A328078, A329774, A335703.

cp's OEIS Frontend

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

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