A060647 -id:A060647 - cp's OEIS frontend

A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.

0, 1, 2, 5, 8, 17, 26, 53, 80, 161, 242, 485, 728, 1457, 2186, 4373, 6560, 13121, 19682, 39365, 59048, 118097, 177146, 354293, 531440, 1062881, 1594322, 3188645, 4782968, 9565937, 14348906, 28697813, 43046720, 86093441, 129140162

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001

WARNING: The offset of this sequence has been changed from 0 to 1 without correcting the formulas and programs, many of them correspond to the original indexing a(0)=0, a(1)=1, ... - M. F. Hasler, Oct 06 2014
Numbers n such that no entry in n-th row of Pascal's triangle is divisible by 3, i.e., such that A062296(n) = 0.
The base 3 representation of these numbers is 222...222 or 122...222.
a(n+1) is the smallest number with ternary digit sum = n: A053735(a(n+1)) = n and A053735(m) <> n for m < a(n+1). - Reinhard Zumkeller, Sep 15 2006
A138002(a(n)) = 0. - Reinhard Zumkeller, Feb 26 2008
Also, number of terms in S(n), where S(n) is defined in A114482. - N. J. A. Sloane, Nov 13 2014
a(n+1) is also the Moore lower bound on the order of a (4,g)-cage. - Jason Kimberley, Oct 30 2011

			The first rows in Pascal's triangle with no multiples of 3 are:
row 0: 1;
row 1: 1, 1;
row 2: 1, 2,  1;
row 5: 1, 5, 10, 10,  5,  1;
row 8: 1, 8, 28, 56, 70, 56, 28, 8, 1;

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023. See p. 9.
Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A062296, A024023, A048473, A114482. Pairwise sums of A052993.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), this sequence (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A037233 (actual order of a (4,g)-cage).
Smallest number whose base b sum of digits is n: A000225 (b=2), this sequence (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10).

I:=[0,1,2]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2) -3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 20 2012

A062318 :=proc(n)
    if n mod 2 = 1 then
        3^((n-1)/2)-1
    else
        2*3^(n/2-1)-1
    fi
end proc:
seq(A062318(n), n=1..37); # Emeric Deutsch, Feb 03 2005, offset updated

CoefficientList[Series[x^2*(1+x)/((1-x)*(1-3*x^2)),{x,0,40}],x] (* Vincenzo Librandi, Apr 20 2012 *)
A062318[n_]:= (1/3)*(Boole[n==0] -3 +3^(n/2)*(2*Mod[n+1,2] +Sqrt[3] *Mod[n, 2]));
Table[A062318[n], {n, 50}] (* G. C. Greubel, Apr 17 2023 *)

PARI
```
a(n)=3^(n\2)<M. F. Hasler, Oct 06 2014
```

def A062318(n): return (1/3)*(int(n==0) - 3 + 2*((n+1)%2)*3^(n/2) + (n%2)*3^((n+1)/2))
[A062318(n) for n in range(1,41)] # G. C. Greubel, Apr 17 2023

a(n) = 2*3^(n/2-1)-1 if n is even; a(n) = 3^(n/2-1/2)-1 if n is odd. - Emeric Deutsch, Feb 03 2005, offset updated.
From Paul Curtz, Feb 21 2008: (Start)
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3).
Partial sums of A108411. (End)
G.f.: x^2*(1+x)/((1-x)*(1-3*x^2)). - Colin Barker, Apr 02 2012
a(2n+1) = 3*a(2n-1) + 2;  a(2n) = ( a(2n-1) + a(2n+1) )/2. See A060647 for case where a(1)= 1. - Richard R. Forberg, Nov 30 2013
a(n) = 2^((1+(-1)^n)/2) * 3^((2*n-3-(-1)^n)/4) - 1. - Luce ETIENNE, Aug 29 2014
a(n) = A052993(n-1) + A052993(n-2). - R. J. Mathar, Sep 10 2021
E.g.f.: (1 - 3*cosh(x) + 2*cosh(sqrt(3)*x) - 3*sinh(x) + sqrt(3)*sinh(sqrt(3)*x))/3. - Stefano Spezia, Apr 06 2022
a(n) = (1/3)*([n=0] - 3 + (1+(-1)^n)*3^(n/2) + ((1-(-1)^n)/2)*3^((n+1)/2)). - G. C. Greubel, Apr 17 2023

More terms from Emeric Deutsch, Feb 03 2005

Entry revised by N. J. A. Sloane, Jul 29 2011

A068911 Number of n-step walks (each step +-1 starting from 0) which are never more than 2 or less than -2.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 36, 54, 108, 162, 324, 486, 972, 1458, 2916, 4374, 8748, 13122, 26244, 39366, 78732, 118098, 236196, 354294, 708588, 1062882, 2125764, 3188646, 6377292, 9565938, 19131876, 28697814, 57395628, 86093442, 172186884, 258280326, 516560652

Offset: 0

Henry Bottomley, Mar 06 2002

From Johannes W. Meijer, May 29 2010: (Start)
a(n) is the number of ways White can force checkmate in exactly (n+1) moves, n >= 0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6, g5 and h6; Black Ke8, Nh8, pawns b3, c7, d3, f7, g6 and h7. (After Noam D. Elkies, see link; diagram 5).
Counts all paths of length n, n >= 0, starting at the third node on the path graph P_5, see the Maple program. (End)
From Alec Jones, Feb 25 2016: (Start)
The a(n) are the n-th terms in a "Fibonacci snake" drawn on a rectilinear grid. The n-th term is computed as the sum of the previous terms in cells adjacent to the n-th cell (diagonals included). (This sequence excludes the first term of the snake.)
For example:
  1 ...  1      1  ...   1 4      1 4 6 ...  1 4 6       1 4 6   ...  and so on.
         1 ...  1 2      1 2 ...  1 2        1 2 12 ...  1 2 12 18 (End)
From Gus Wiseman, Oct 06 2023: (Start)
Also the number of subsets of {1..n} containing no two distinct elements summing to n. The a(0) = 1 through a(4) = 12 subsets are:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,3}  {4}
                  {2,3}  {1,2}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}
For n+1 instead of n we have A038754, complement A167762.
Including twins gives A117855, complement A366131.
The complement is counted by A365544.
For all subsets (not just pairs) we have A365377, complement A365376. (End)

			The a(3) = 6 walks: (0,-1,-2,-1), (0,-1,0,-1), (0,-1,0,1), (0,1,0,-1), (0,1,0,1), (0,1,2,1). - _Gus Wiseman_, Oct 10 2023

Alois P. Heinz, Table of n, a(n) for n = 0..4191
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
Robert Dorward et al., A Generalization of Zeckendorf's Theorem via Circumscribed m-gons, arXiv:1508.07531 [math.NT], 2015. See Example 1.3 p. 4.
Noam D. Elkies, New Directions in Enumerative Chess Problems, arXiv:math/0508645 [math.CO], 2005; The Electronic Journal of Combinatorics, 11 (2), 2004-2005.
Sean A. Irvine, Walks on Graphs.
D. Panario, M. Sahin, Q. Wang, and W. Webb, General conditional recurrences, Applied Mathematics and Computation, Volume 243, Sep 15 2014, Pages 220-231.
Noriaki Sannomiya, H. Katsura, and Y. Nakayama, Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion, arXiv preprint arXiv:1612.02285 [cond-mat.str-el], 2016-2017. See Table I, line 3.
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000007, A016116 (without initial term), A068912, A068913 for similar.
Equals A060647(n-1)+1.
Cf. also A028495, A038754, A048328, A078038, A124302, A306293.
First differences are A117855.
Cf. A004526, A004737, A008967, A046663, A088809, A365376, A365377, A365381, A365541, A365544, A366130.

[Floor((5-(-1)^n)*3^Floor(n/2)/3): n in [0..40]]; // Bruno Berselli, Feb 26 2016, after Charles R Greathouse IV

with(GraphTheory): G:= PathGraph(5): A:=AdjacencyMatrix(G): nmax:=34; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[3,k], k=1..5) od: seq(a(n), n=0..nmax); # Johannes W. Meijer, May 29 2010
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, n+1, (4-irem(n, 2))/2*a(n-1)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 03 2019

Join[{1},Transpose[NestList[{Last[#],3First[#]}&,{2,4},40]][[1]]] (* Harvey P. Dale, Oct 24 2011 *)
Table[Length[Select[Subsets[Range[n]],FreeQ[Total/@Subsets[#,{2}],n]&]],{n,0,15}] (* Gus Wiseman, Oct 06 2023 *)

a(n)=[4,6][n%2+1]*3^(n\2)\3 \\ Charles R Greathouse IV, Feb 26 2016

def A068911(n): return 3**(n>>1)<<1 if n&1 else (3**(n-1>>1)<<2 if n else 1) # Chai Wah Wu, Aug 30 2024

a(n) = A068913(2, n) = 2*A038754(n-1) = 3*a(n-2) = a(n-1)*a(n-2)/a(n-3) starting with a(0)=1, a(1)=2, a(2)=4 and a(3)=6.
For n>0: a(2n) = 4*3^(n-1) = 2*a(2n-1); a(2n+1) = 2*3^n = 3*a(2n)/2 = 2*a(2n)-A000079(n-2).
From Paul Barry, Feb 17 2004: (Start)
G.f.: (1+x)^2/(1-3x^2).
a(n) = 2*3^((n+1)/2)*((1-(-1)^n)/6 + sqrt(3)*(1+(-1)^n)/9) - (1/3)*0^n.
The sequence 0, 1, 2, 4, ... has a(n) = 2*3^(n/2)*((1+(-1)^n)/6 + sqrt(3)*(1-(-1)^n)/9) - (2/3)*0^n + (1/3)*Sum_{k=0..n} binomial(n, k)*k*(-1)^k. (End)
a(n) = 2^((3 + (-1)^n)/2)*3^((2*n - 3 - (-1)^n)/4) - ((1 - (-1)^(2^n)))/6. - Luce ETIENNE, Aug 30 2014
For n > 2, indexing from 0, a(n) = a(n-1) + a(n-2) if n is odd, a(n-1) + a(n-2) + a(n-3) if n is even. - Alec Jones, Feb 25 2016
a(n) = 2*a(n-1) if n is even, a(n-1) + a(n-2) if n is odd. - Vincenzo Librandi, Feb 26 2016
E.g.f.: (4*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) - 1)/3. - Stefano Spezia, Feb 17 2022

A210799 Triangle of coefficients of polynomials u(n,x) jointly generated with A210800; see the Formula section.

Original entry on oeis.org

1, 3, 1, 5, 4, 2, 11, 13, 9, 3, 17, 32, 32, 17, 5, 35, 77, 96, 72, 32, 8, 53, 164, 254, 243, 153, 59, 13, 107, 353, 641, 739, 579, 313, 107, 21, 161, 704, 1496, 2042, 1938, 1305, 623, 192, 34, 323, 1433, 3440, 5348, 5898, 4774, 2831, 1213, 341, 55, 485

Offset: 1

Clark Kimberling, Mar 27 2012

Row n starts with A060647(n) and ends with F(n), where F=A000045 (Fibonacci numbers).

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3....1
5....4....2
11...13...9....3
17...32...32...17...5
First three polynomials u(n,x): 1, 3 + x, 5 + 4x + 2x^2.

Cf. A210800, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 1; h = 2; p = -1; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210799 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210800 *)

u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2) + a(k) with a(0) = 2, a(1) = -1, a(k) = 0 if k>1, T(1,0) = T(2,1) = 1, T(2,0) = 3 and T(n,k) = 0 if k<0 or if k>=n.

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A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.

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A068911 Number of n-step walks (each step +-1 starting from 0) which are never more than 2 or less than -2.

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A210799 Triangle of coefficients of polynomials u(n,x) jointly generated with A210800; see the Formula section.

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