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

Author

Philippe Deléham, Feb 16 2014

Keywords

• nonn
• easy

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

Examples

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

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).

Crossrefs

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

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

Programs

Magma

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

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 *)

PARI

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

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