A026532 -id:A026532 - cp's OEIS frontend

A026565 a(n) = 6*a(n-2), starting with 1, 3, 9.

1, 3, 9, 18, 54, 108, 324, 648, 1944, 3888, 11664, 23328, 69984, 139968, 419904, 839808, 2519424, 5038848, 15116544, 30233088, 90699264, 181398528, 544195584, 1088391168, 3265173504, 6530347008, 19591041024, 39182082048

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A026532, A026534, A026551, A026552, A026567.

[1] cat [n le 2 select 3^n else 6*Self(n-2): n in [1..35]]; // G. C. Greubel, Dec 17 2021

Table[(1/4)*6^(n/2)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)) - (1/2)*Boole[n==0], {n, 0, 35}] (* G. C. Greubel, Dec 17 2021 *)

def A026565(n): return ( (3/2)*6^(n/2) if (n%2==0) else 3*6^((n-1)/2) ) - bool(n==0)/2
[A026565(n) for n in (0..30)] # G. C. Greubel, Dec 17 2021

a(n) = Sum_{j=0..2*n} A026552(n, j).
G.f.: (1+3*x+3*x^2)/(1-6*x^2). - Ralf Stephan, Feb 03 2004
a(0)=1, a(1)=3; a(n) = 3*a(n-1) if n is even, a(n) = 2*a(n-1) if n is odd. - Vincenzo Librandi, Nov 19 2010
a(n) = (1/4)*6^(n/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) - (1/2)*[n=0]. - G. C. Greubel, Dec 17 2021

Better name from Ralf Stephan, Jul 17 2013

A208131 Partial products of A052901.

Original entry on oeis.org

1, 3, 6, 12, 36, 72, 144, 432, 864, 1728, 5184, 10368, 20736, 62208, 124416, 248832, 746496, 1492992, 2985984, 8957952, 17915904, 35831808, 107495424, 214990848, 429981696, 1289945088, 2579890176, 5159780352, 15479341056, 30958682112, 61917364224, 185752092672

Offset: 0

Reinhard Zumkeller, Apr 04 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,12).

Cf. A001222, A052901, A026549, A026532.

a208131 n = a208131_list !! n
a208131_list = scanl (*) 1 $ a052901_list
-- Reinhard Zumkeller, Mar 29 2012

FoldList[Times,1,PadRight[{},30,{3,2,2}]] (* Harvey P. Dale, Mar 19 2013 *)

a(n+1) = a(n) * A052901(n).
A001222(a(n)) = n.
a(n) = 12^floor(n/3)*(r+1)*(r+2)/2 with r = n mod 3. G.f.: -(6*x^2+3*x+1) / (12*x^3-1). - Alois P. Heinz, Apr 05 2012
Sum_{n>=0} 1/a(n) = 18/11. - Amiram Eldar, Feb 13 2023

A121123 Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).

Original entry on oeis.org

1, 3, 12, 63, 342, 1998, 11772, 70308, 420552, 2521368, 15120432, 90710928, 544218912, 3265243488, 19591180992, 117546666048, 705278316672, 4231667380608, 25389994205952, 152339950119168, 914039640248832, 5484237750793728, 32905426141965312, 197432556307596288

Offset: 2

N. J. A. Sloane, Aug 13 2006

J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 12, q=9, alpha=0.
Index entries for linear recurrences with constant coefficients, signature (6,6,-36).

# Exhibit 1
Hra := proc(r::integer,a::integer,q::integer)
    binomial(r-1,a-1)*(q-3)+binomial(r-1,a) ;
    %*(q-3)^(r-a-1) ;
end proc:
Jra := proc(r::integer,a::integer,q::integer)
    binomial(r-2,a-2)*(q-3)^2 +2*binomial(r-2,a-1)*(q-3) +binomial(r-2,a) ;
    %*(q-3)^(r-a-2) ;
end proc:
# Exhibit 2, I_m
A121123 := proc(r::integer)
    local q,a,f ;
    q := 9 ;
    a := 0 ;
    f := 1 +(-1)^(r+a) +(1+(-1)^a) *(1-(-1)^r) *floor((q-3)/2) /2 ;
    Jra(r,a,q)+binomial(2,r-a)+f*Hra(floor(r/2),floor(a/2),q) ;
    %/4 ;
end proc:
seq(A121123(n),n=2..30) ; # R. J. Mathar, Aug 01 2019

Join[{1}, LinearRecurrence[{6, 6, -36}, {3, 12, 63}, 23]] (* Jean-François Alcover, Mar 31 2020 *)

From Colin Barker, Aug 30 2013: (Start)
a(n) = 6*a(n-1)+6*a(n-2)-36*a(n-3) for n>5.
G.f.: x^2 -3*x^3*(-1+2*x+9*x^2) / ( (6*x-1)*(6*x^2-1) ). (End)
a(n) = A026532(n+1)/12 +6^(n-2)/4, n>2. - R. J. Mathar, Aug 01 2019

A203161 (n-1)-st elementary symmetric function of the first n terms of (3,1,2,3,1,2,3,1,2,...).

Original entry on oeis.org

1, 4, 11, 39, 57, 132, 432, 540, 1188, 3780, 4428, 9504, 29808, 33696, 71280, 221616, 244944, 513216, 1586304, 1726272, 3592512, 11057472, 11897280, 24634368, 75582720, 80621568, 166281984, 508923648, 539156736, 1108546560, 3386105856

Offset: 1

Clark Kimberling, Dec 29 2011

From R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of the first n terms of 3,1,2,3,1,2.., form a triangle T(n,k), 0<=k<=n, n>=0:
1
3
4 3
6 11 6
9 29 39 18
10 38 68 57 18
12 58 144 193 132 36
15 94 318 625 711 432 108
16 109 412 943 1336 1143 540 108
18 141 630 1767 3222 3815 2826 1188 216
21 195 1053 3657 8523 13481 14271 9666 3780 648
This here is the first subdiagonal. The diagonal is a stuttered version of A026532. The 2nd column is A047231 (or A144429). (End)

			Let esf abbreviate "elementary symmetric function".  Then
0th esf of {3}:  1,
1st esf of {3,1}:  3+1=4,
2nd esf of {3,1,2} is 3*1+3*1+1*2=11.

Index entries for linear recurrences with constant coefficients, signature (0,0,12,0,0,-36).

Cf. A010882, A203160, A203162.

f[k_] := 1 + Mod[k + 1, 3]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 33}] (* A203161 *)

Vec(x*(3*x+1)*(3*x^3+8*x^2+x+1)/(6*x^3-1)^2 + O(x^100)) \\ Colin Barker, Aug 15 2014

G.f.: x*(3*x+1)*(3*x^3+8*x^2+x+1) / (6*x^3-1)^2. - Colin Barker, Aug 15 2014

A329115 a(n) = floor(A026549(n)/5).

Original entry on oeis.org

0, 0, 1, 2, 7, 14, 43, 86, 259, 518, 1555, 3110, 9331, 18662, 55987, 111974, 335923, 671846, 2015539, 4031078, 12093235, 24186470, 72559411, 145118822, 435356467, 870712934, 2612138803, 5224277606, 15672832819, 31345665638, 94036996915, 188073993830

Offset: 1

Clark Kimberling, Nov 10 2019

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-6).

Cf. A026532, A329115.

s[1] = 1; s[n_] := If[IntegerQ[n/2], 2*s[n - 1], 3*s[n - 1]]
Table[s[n], {n, 1, 20}] (* A026532 *)
Table[Floor[s[n]/5], {n, 1, 50}] (* A329115 *)

a(n+1) = 2*a(n) if n is odd, a(n+1) = 3*a(n)+1 if n is even.
a(n) = f(2^f(n/2) * 3^f((n-1)/2) / 5), where f = floor.
G.f.: (x^2 (1 + 2 x))/((-1 + x) (1 + x) (-1 + 6 x^2)).
a(n) = 7*a(n-2) - 6*a(n-4).

A068179 Product_{i=1..3} (i+x) / Product_{i=1..3} (i-x) = Sum_{n>=0} (a(n)/b(n))*x^n.

Original entry on oeis.org

1, 11, 121, 971, 6721, 43331, 269641, 1648091, 9981841, 60176051, 361921561, 2174145611, 13052763361, 78340331171, 470113403881, 2820895001531, 16926014399281, 101558020876691, 609353931324601, 3656141011383851

Offset: 0

Benoit Cloitre, Mar 12 2002

If n mod 10 == 1, 2, or 4 then a(n)==0 (mod 11). - Bruno Berselli, Aug 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A026532.

[5*2^(n+1)+6^(n+1)-5*3^(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 29 2011

Table[5*2^(n+1)+6^(n+1)-5*3^(n+1), {n,0,20}] (* G. C. Greubel, Nov 10 2018 *)
LinearRecurrence[{11,-36,36},{1,11,121},20] (* Harvey P. Dale, Aug 16 2021 *)

vector(20, n, n--; 5*2^(n+1)+6^(n+1)-5*3^(n+1)) \\ G. C. Greubel, Nov 10 2018

for n in range(0,20): print(5*2**(n+1)+6**(n+1)-5*3**(n+1), end=', ') # Stefano Spezia, Nov 12 2018

b(n) = A026532(2*n-1) for n >= 1.
Lim_{n -> infinity} a(n)/b(n) = 12.
From Yalcin Aktar, Aug 10 2011: (Start)
a(n) = 5*2^(n+1) + 6^(n+1) - 5*3^(n+1).
a(n)/b(n) = 12 - 30/2^n + 20/3^n.
General case: lim_{m-->+oo} a_n(m)/b_n(m) = A002378(n) where
Product_{i=1..d} (x+i)/Product_{i=1..d} (i-x) = Sum_{n>=0} (a_d(n)/b_d(n))*x^n) = ((-1)^d) * (1 + Sum_{j>=1} (Sum_{k=1..d} ((-1)^k/k^j) * binomial(2*k,k) * binomial(d+k,2*k)) * x^j). (End)
G.f.: (1+36*x^2)/((1-2*x)*(1-3*x)*(1-6*x)). - Bruno Berselli, Aug 26 2011
E.g.f.: 10*exp(2*x) - 15*exp(3*x) + 6*exp(6*x). - G. C. Greubel, Nov 10 2018

A103279 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 9, 22, 1, 3, 10, 27, 60, 1, 3, 11, 34, 81, 164, 1, 3, 12, 43, 116, 243, 448, 1, 3, 13, 54, 171, 396, 729, 1224, 1, 3, 14, 67, 252, 683, 1352, 2187, 3344, 1, 3, 15, 82, 365, 1188, 2731, 4616, 6561, 9136, 1, 3, 16, 99, 516, 2019, 5616, 10923, 15760

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005

Consider the matrix M = [1,1,1;1,N,1;1,1,1]; Characteristic polynomial of M is x^3 + (-N - 2)*x^2 + (2*N - 2)*x.
Now (M^n)[1,1] is equivalent to the recursion a(1) = 1, a(2) = 3, a(n) = (N+2)a(n-1)+(2N-2)a(n-2). (This also holds for negative N and fractional N.)
a(n+1)/a(n) converges to the upper root of the characteristic polynomial ((N + 2) + sqrt((N - 2)^2 + 8))/2 for n to infinity.
Columns of array follow the polynomials:
1,
3,
N + 8,
N^2 + 4*N + 22,
N^3 + 4*N^2 + 16*N + 60,
N^4 + 4*N^3 + 18*N^2 + 56*N + 164,
N^5 + 4*N^4 + 20*N^3 + 68*N^2 + 188*N + 448,
N^6 + 4*N^5 + 22*N^4 + 80*N^3 + 248*N^2 + 608*N + 1224,
N^7 + 4*N^6 + 24*N^5 + 92*N^4 + 312*N^3 + 864*N^2 + 1920*N + 3344,
N^8 + 4*N^7 + 26*N^6 + 104*N^5 + 380*N^4 + 1152*N^3 + 2928*N^2 + 5952*N + 9136,
etc.

			Array begins:
1,3,8,22,60,164,448,1224,3344,9136,...
1,3,9,27,81,243,729,2187,6561,19683,...
1,3,10,34,116,396,1352,4616,15760,53808,...
1,3,11,43,171,683,2731,10923,43691,174763,...
1,3,12,54,252,1188,5616,26568,125712,594864,...
...

Cf. A103280 (for (M^n)[1, 2]), A028859 (for N=0), A000244 (for N=1), A007052 (for N=2), A007583 (for N=3), A083881 (for N=4), A026581 (for N=-1), A026532 (for N=-2), A026568.

T11(N, n) = if(n==1,1,if(n==2,3,(N+2)*r1(N,n-1)-(2*N-2)*r1(N,n-2))) for(k=0,10,print1(k,": ");for(i=1,10,print1(T11(k,i),","));print())

T(N, 1)=1, T(N, 2)=3, T(N, n)=(N+2)*T(N, n-1)-(2*N-2)*T(N, n-2).

A176812 Expansion of 3(1+x)/(1-2x-5*x^2).

Original entry on oeis.org

3, 9, 33, 111, 387, 1329, 4593, 15831, 54627, 188409, 649953, 2241951, 7733667, 26677089, 92022513, 317430471, 1094973507, 3777099369, 13029066273, 44943629391, 155032590147, 534783327249, 1844729605233, 6363375846711

Offset: 0

Roger L. Bagula, Apr 26 2010

Binomial transform of A026532 after dropping A026532(0). [From R. J. Mathar, Apr 27 2010]

Index entries for linear recurrences with constant coefficients, signature (2,5).

a[n_] = 2^n*(((3 + Sqrt[ 6])/2)*((1 + Sqrt[6])/2)^n + ((3 - Sqrt[6])/2)*((1 - Sqrt[6])/2)^n); Table[FullSimplify[a[n]], {n, 0, 30}]
CoefficientList[Series[(-3(1+x))/(5x^2+2x-1),{x,0,40}],x]  (* Harvey P. Dale, Feb 24 2011 *)

Binet form: a(n)=2^n*(((3 + Sqrt[6])/2)*((1 + Sqrt[6])/2)^n + ((3 - Sqrt[6])/2)*((1 - Sqrt[6])/2)^n) = 3*A180168(n).

A068154 Numerators of coefficients in power series for (Product_{i=1..5} (x+i)) / (Product_{i=1..5} (i-x)) = Sum_{n>=0} a(n)/b(n)*x^n.

Original entry on oeis.org

1, 137, 18769, 1799603, 140815861, 9800649707, 638003187109, 40003144104683, 2456948367146821, 149230625474121227, 9010618306714845349, 542390253445959003563, 32597040868332220933381, 1957452401279697344559947, 117496474687502028535109989

Offset: 0

Benoit Cloitre, Mar 12 2002

If n > 0, denominators b(n) = 10^n*A026532(2n-1).

Lim_{n->infinity} a(n)/b(n) = 30.

Offset corrected and more terms from Sean A. Irvine, Jan 30 2024

A137457 Consider a row of standard dice as a counter. This sequence enumerates the number of changes (one face rotated over an edge to an adjacent face) from n-1 to n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1

Offset: 0

Robert G. Wilson v, Apr 18 2008

nonn

Most counters 'zero' out at '0' but the dice 'zero' out at '1' which is the initial state. So to increment 1 -> 2 requires 1 move, 2 -> 3 requires 1 move, 3 -> 4 requires 2 moves, 4 -> 5 requires 1 move, 5 -> 6 requires 1 move and 6 -> 0 requires 2 moves.

First occurrence of k (A026532): 1, 3, 6, 18, 36, 108, 216, 648, 1296, 3888, ....

Eric Weisstein's World of Mathematics, Dice.
Wikipedia, Dice.

Cf. A057436, A060852, A026532, A133794.

f[n_] := Block[{a = IntegerDigits[n - 1, 6] + 1, b = IntegerDigits[n, 6] + 1, c}, If[Length@b > Length@a, a = Prepend[a, 1]]; c = Transpose[{a, b}] /. {{d_, d_} -> 0, {1, 2} -> 1, {2, 3} -> 1, {3, 4} -> 2, {4, 5} -> 1, {5, 6} -> 1, {6, 1} -> 2}; Plus @@ c]; Array[f, 105]

cp's OEIS Frontend

A026565 a(n) = 6*a(n-2), starting with 1, 3, 9.

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Magma

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A208131 Partial products of A052901.

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Haskell

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A121123 Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).

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Maple

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A203161 (n-1)-st elementary symmetric function of the first n terms of (3,1,2,3,1,2,3,1,2,...).

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A329115 a(n) = floor(A026549(n)/5).

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Mathematica

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A068179 Product_{i=1..3} (i+x) / Product_{i=1..3} (i-x) = Sum_{n>=0} (a(n)/b(n))*x^n.

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A103279 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

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A176812 Expansion of 3*(1+x)/(1-2*x-5*x^2).

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A176812 Expansion of 3(1+x)/(1-2x-5*x^2).