A116970 a(n) = (3^n - 7)/2.

1, 10, 37, 118, 361, 1090, 3277, 9838, 29521, 88570, 265717, 797158, 2391481, 7174450, 21523357, 64570078, 193710241, 581130730, 1743392197, 5230176598, 15690529801, 47071589410, 141214768237, 423644304718, 1270932914161

Offset: 2

Author

N. J. A. Sloane, Apr 01 2006

Keywords

• nonn
• easy

Comments

Number of moves to solve Type 1 Zig-Zag puzzle.

(3^(p+1) - 7)/2 = a(p+1) == 1 (mod p) since (3^(p-1) - 1)/2 = A003462(p-1) == 0 (mod p), for primes p > 7 (see comment by Alexander Adamchuck in A003462); in addition, a(4) == 1 (mod 3) and a(6) == 1 (mod 5). - Hartmut F. W. Hoft, Aug 22 2018

References

• Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.
• Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A003462, A063679.

Programs

Magma

[(3^n-7)/2: n in [2..30]]; // Vincenzo Librandi, Mar 30 2015

Maple

a[1]:=1:for n from 2 to 50 do a[n]:=3^n+a[n-1] od: seq(a[n], n=1..25); # Zerinvary Lajos, Mar 09 2008

Mathematica

Table[(3^n - 7)/2, {n, 2, 30}] (* Stefan Steinerberger, Apr 02 2006 *)
LinearRecurrence[{4,-3},{1,10},30] (* Harvey P. Dale, Jan 17 2013 *)
CoefficientList[Series[(1 + 6 x) / ((1 - 3 x) (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 30 2015 *)

PARI

a(n)=(3^n-7)/2 \\ Charles R Greathouse IV, Sep 04 2014

Formula

a(n) = 3*a(n-1) + 7 with n > 2, a(2)=1. - Vincenzo Librandi, Aug 02 2010

a(2)=1, a(3)=10; for n > 3, a(n) = 4*a(n-1) - 3*a(n-2). - Harvey P. Dale, Jan 17 2013

G.f.: x^2*(1+6*x)/((1-3*x)*(1-x)). - Vincenzo Librandi, Mar 30 2015

From Hartmut F. W. Hoft, Aug 22 2018: (Start)

a(2) = 1; a(n) = a(n-1) + 3^(n-1) for n > 2. -

a(n) = A003462(n) - 3, n >= 2. (End)