A025192 -id:A025192 - cp's OEIS frontend

A189315 Expansion of g.f. 5(1-3x+x^2)/(1-5x+5x^2).

5, 10, 30, 100, 350, 1250, 4500, 16250, 58750, 212500, 768750, 2781250, 10062500, 36406250, 131718750, 476562500, 1724218750, 6238281250, 22570312500, 81660156250, 295449218750, 1068945312500, 3867480468750, 13992675781250, 50625976562500, 183166503906250, 662702636718750

Offset: 0

L. Edson Jeffery, Apr 20 2011

Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,1)=
(0 1 0 0 0)
(1 0 1 0 0)
(0 1 0 1 0)
(0 0 1 0 1)
(0 0 0 2 0).
Then a(n) = Trace(A^(2*n)).
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0From Tom Copeland, Dec 08 2015: (Start)
These are also the non-vanishing traces for the adjacency matrices of the simple Lie algebras B_5 and C_5. See links for B_4, A265185, and B_3, A025192.
a(n+1) = 10 * A081567(n), and, ignoring a(0), a G.F. is 10 *(1-2*x)/(1-5*x+5*x^2) whose denominator is y^5 * A127672(5,1/y) with y = sqrt(x).
-log(1 - 5x^2 + 5x^4) = 10 x^2/2 + 30 x^4/4 + ... provides a logarithmic series for the traces of both the odd and even powers of the matrix beginning  with the first power. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A147748, A189316, A189317, A189318.

Cf. A025192, A081567, A127672, A265185.

I:=[5,10,30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015

CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2),{x,0,40}],x] (* or *)
Join[{5},LinearRecurrence[{5,-5},{10,30},40]]  (* Harvey P. Dale, Apr 25 2011 *)

Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n) = 5*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=10, a(2)=30.
a(n) = Sum_{k=1..5} (w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.
a(n) = 5*A147748(n).
E.g.f.: 1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Jul 09 2024

A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 8, 20, 18, 7, 1, 16, 48, 56, 32, 9, 1, 32, 112, 160, 120, 50, 11, 1, 64, 256, 432, 400, 220, 72, 13, 1, 128, 576, 1120, 1232, 840, 364, 98, 15, 1, 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1, 512, 2816, 6912, 9984, 9408, 6048, 2688, 816, 162, 19, 1

Offset: 0

Philippe Deléham, Nov 13 2011

Riordan array ((1-x)/(1-2x),x/(1-2x)).
Product A097805*A007318 as infinite lower triangular arrays.
Product A193723*A130595 as infinite lower triangular arrays.
T(n,k) is the number of ways to place n unlabeled objects into any number of labeled bins (with at least one object in each bin) and then designate k of the bins. - Geoffrey Critzer, Nov 18 2012
Apparently, rows of this array are unsigned diagonals of A028297. - Tom Copeland, Oct 11 2014
Unsigned A118800, so my conjecture above is true. - Tom Copeland, Nov 14 2016

			Triangle begins:
   1
   1,   1
   2,   3,   1
   4,   8,   5,   1
   8,  20,  18,   7,   1
  16,  48,  56,  32,   9,   1
  32, 112, 160, 120,  50,  11,   1

Cf. A118800 (signed version), A081277, A039991, A001333 (antidiagonal sums), A025192 (row sums); diagonals: A000012, A005408, A001105, A002492, A072819l; columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976.

Cf. A007318, A028297, A059260, A097805, A118801, A130595.

nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[(1-x)/(1-2x-y x) ,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 18 2012 *)

T(n,k) = 2*T(n-1,k)+T(n-1,k-1) with T(0,0)=T(1,0)=T(1,1)=1 and T(n,k)=0 for k<0 or for n

T(n,k) = A011782(n-k)*A135226(n,k) = 2^(n-k)*(binomial(n,k)+binomial(n-1,k-1))/2.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for n=-1,0,1,2,3,4,5,6,7,8,9,10 respectively.

G.f.: (1-x)/(1-(2+y)*x).

T(n,k) = Sum_j>=0 T(n-1-j,k-1)*2^j.

T = A007318*A059260, so the row polynomials of this entry are given umbrally by p_n(x) = (1 + q.(x))^n, where q_n(x) are the row polynomials of A059260 and (q.(x))^k = q_k(x). Consequently, the e.g.f. is exp[tp.(x)] = exp[t(1+q.(x))] = e^t exp(tq.(x)) = [1 + (x+1)e^((x+2)t)]/(x+2), and p_n(x) = (x+1)(x+2)^(n-1) for n > 0. - Tom Copeland, Nov 15 2016

T^(-1) = A130595*(padded A130595), differently signed A118801. Cf. A097805. - Tom Copeland, Nov 17 2016

The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)/(1 + 2*x) * (1 + 2*x)^n about 0. For example, for n = 4, (1 + x)/(1 + 2*x) * (1 + 2*x)^4 = (8*x^4 + 20*x*3 + 18*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 24 2018

A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1.

Original entry on oeis.org

1, 1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924

Offset: 0

Barry E. Williams, Jan 24 2000

First differences of A025192, also second differences of A000244.

			1 + x + 4*x^2 + 12*x^3 + 36*x^4 + 108*x^5 + 324*x^6 + 972*x^7 + 2916*x^8 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
P. Ribenhoim, The Little Book of Big Primes, Springer-Verlag, N.Y., 1991, p. 53.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A025192, A000244, A003462.

CoefficientList[Series[(1 - x)^2/(1 - 3 x), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)

{a(n) = local(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-4*k + 9) * A[k-1] + 3 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n) = 4*3^(n-2); n >= 2; a(0) = 1; a(1) = 1.
G.f.: (1-x)^2/(1-3*x).
G.f.: 1/(1-sum(j>=1, (2*j-1)*x^j )). - Joerg Arndt, Jul 06 2011
a(n) = 3*a(n-1)+(-1)^n*C(2, 2-n).
a(n) = A003946(n-1), n>0. - R. J. Mathar, Oct 13 2008
a(n) = (-4*n + 9) * a(n-1) + 3 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
a(n) = Sum_{k, 0<=k<=n} A201780(n,k). - Philippe Deléham, Dec 05 2011

New name from Joerg Arndt, Jul 06 2011

A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

From G. C. Greubel, Jan 18 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
  (p, q) = (1, 1) : 2*A007318(n,k).
  (p, q) = (2, 2) : 2*A038208(n,k).
  (p, q) = (3, 2) : A154692(n,k).
  (p, q) = (3, 3) : 2*A038221(n,k). (End)

			Triangle begins as:
     2;
     3,    3;
     5,    8,     5;
     9,   18,    18,     9;
    17,   40,    48,    40,    17;
    33,   90,   120,   120,    90,    33;
    65,  204,   300,   320,   300,   204,    65;
   129,  462,   756,   840,   840,   756,   462,   129;
   257, 1040,  1904,  2240,  2240,  2240,  1904,  1040,   257;
   513, 2322,  4752,  6048,  6048,  6048,  6048,  4752,  2322,  513;
  1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.
Cf. A215149.
Sums include: A008776 (row), A010673 (alternating sign row).
Columns k: A000051 (k=0).
Main diagonal: A059304.

A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
[A154690(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # R. J. Mathar, Jan 13 2011

T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten

from sage.all import *
def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).
Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).
From G. C. Greubel, Jan 18 2025: (Start)
T(n, n-k) = T(n, k) (symmetry).
T(n, 1) = n + A215149(n), n >= 1.
T(2*n-1, n) = 3*A069720(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)

A061922 Xcatalans - produced as a self-convolved sequence like Catalan numbers (A000108) but use carryless GF(2)[ X ] polynomial multiplication.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 421, 1382, 4478, 15580, 54114, 181676, 650484, 2289320, 8028901, 28045302, 103229014, 372640460, 1336511110, 4882492452, 17534836812, 63692926552, 234287550818, 868236370364, 3281589811404

Offset: 0

Antti Karttunen, May 15 2001

Shifts one place left when Xmult-convolved (XMULTCONV) with itself.

For Xmult, see A048720 (Xmult table) or A048631 (Xfactorials). Other self-convolved sequences: A000108, A007460 - A007464, A025192.

Xcatalans(30); Xcatalans := proc(upto_n) local a,i,k; a := [1]; for i from 1 to upto_n do a := [ op(a), add(Xmult(a[k],a[i-k+1]), k=1..i)]; od; RETURN(a); end;
XMULTCONV := proc(a,b) local c,i,k,n; n := min( nops(a), nops(b) ); c := []; for i from 0 to n-1 do c := [ op(c), add(Xmult(a[k+1],b[i-k+1]), k=0..i)]; od; RETURN(c); end;

A080342 Number of weighings required to identify a single bad coin out of n coins, using a two-pan balance.

Original entry on oeis.org

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

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Mar 19 2003

It is known that there is exactly one bad coin, which is heavier than the others. No weights are used in the weighings.
0 appears once, 1 twice, 2 6 times, 3 18 times, 4 54 times, ... which is the same as the number of base-3 numbers of length n; see A007089. - Jonathan Vos Post, Apr 20 2011
Records appear at positions 3^n+1 (=A034472(n)). - Robert G. Wilson v, Aug 06 2012
The "Heavy Marble" section of "Brainteaser Problems" in the Mongan et al. reference describes the n = 8 case in detail and then derives the general formula given below. Of course this sequence applies also when the single, unlike object is lighter than all the others. If the unlike object is only known to have a different weight (that is, to be lighter than all the others or heavier than all the others), use A064099. - Rick L. Shepherd, Sep 05 2013
If it is unknown whether the bad coin is heavier or lighter, then the minimum number of weighings is A029837(n) and the number of coins that must be used in the first weighing is A004526(n), for n > 2. - Ivan N. Ianakiev, Apr 13 2017

			a(1) = 0 since no weighings are needed - the coin is bad. a(2) = 1 since one weighing is needed.

J. Mongan, N. Suojanen, and E. Giguère, Programming Interviews Exposed: Secrets to Landing Your Next Job, 2nd Edition, Wiley Publishing, Inc., 2007, pp. 169-172.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000
R. K. Guy, R. J. Nowakowski, Coin-weighing problems, Am. Math. Monthly 102 (2) (1995) 164-167.
B. Manvel, Counterfeit coin problems, Math. Mag. 50 (2) (1977) 90-92, theorem 1.

Cf. A000244, A007089, A025192, A064235.
Cf. also A004526, A029837, A064099.
Cf. A081604.

import Data.List (transpose)
a080342 n = genericIndex a080342_list (n - 1)
a080342_list = 0 : zs where
   zs = 1 : 1 : (map (+ 1) $ concat $ transpose [zs, zs, zs])
-- Reinhard Zumkeller, Sep 02 2015

f[n_] := Floor[ Log[3, n]] - Floor[2^-FractionalPart[ Log[3, n]]] + 1; Array[f, 105] (* Robert G. Wilson v, Aug 05 2012 *)

a(n) = ceil(log(n)/log(3)) \\ Rick L. Shepherd, Sep 05 2013

a(n) = floor(L) - floor(2^(-f(L))) + 1, where L = log_3(n) and f() = fractional part.
a(n) = ceiling(log_3(n)). - Rick L. Shepherd, Sep 05 2013
A064235(n) = 3 ^ a(n). - Reinhard Zumkeller, Sep 02 2015

A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

Original entry on oeis.org

1, 0, 3, 2, 19, 30, 143, 322, 1179, 3110, 10183, 28842, 89939, 262990, 802623, 2380562, 7196299, 21479670, 64657463, 193535482, 581480259, 1742693150, 5231574703, 15687733602, 47077181819, 141203583430, 423666674343

Offset: 0

Paul Barry, May 11 2004

For n > 0, 6*a(n) is the number of 3-colorings of the prism of size 2 X n (i.e., C_2 X C_n).More generally, the number of k-colorings of the prism of size 2 X n is given by (k^2 - 3*k + 3)^n + (k - 1) * ((3 - k)^n + (1 - k)^n) + k^2 - 3*k + 1 (chromatic polynomial). - Sela Fried, Oct 07 2023

Andrew Howroyd, Table of n, a(n) for n = 0..1000
N. L. Biggs, R. M. Damerell, and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Cf. A002001, A025192, A094555, A094556, A328778.

LinearRecurrence[{2, 5, -6}, {1, 0, 3, 2}, 30] (* Greg Dresden, Jun 19 2021 *)

a(n) = if(n==0, 1, (1 + 3^n + 2*(-2)^n)/6) \\ Andrew Howroyd, Jun 14 2021

G.f.: (1 - 2*x - 2*x^2 + 2*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 1/6 + 3^n/6 + (-2)^n/3 for n > 0.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
E.g.f.: exp(-2*x)*(1 + exp(2*x))*(2 + exp(3*x))/6. - Stefano Spezia, Sep 26 2023

A099856 Expansion of (1+3x)/(1-3x).

Original entry on oeis.org

1, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974, 45753584909922

Offset: 0

Paul Barry, Oct 28 2004

A099858 gives a Chebyshev transform. Binomial transform is A083420.

Hankel transform is 1, -18, 0, 0, 0, 0, 0, 0, 0, ... - Philippe Deléham, Dec 13 2011

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

Cf. A025192, A083420, A093561, A099858.

CoefficientList[Series[(1+3x)/(1-3x),{x,0,30}],x] (* or *) Join[{1}, NestList[3#&,6,30]] (* Harvey P. Dale, Nov 08 2011 *)

Vec((1+3*x)/(1-3*x) + O(x^40)) \\ Michel Marcus, Dec 11 2015

a(n) = 2*3^n - 0^n.
a(n) = A025192(n+1), n > 0. - R. J. Mathar, Sep 02 2008
a(n) = Sum_{k=0..n} A093561(n,k)*2^k. - Philippe Deléham, Dec 13 2011
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: 2*exp(3*x) - 1.
a(n) = 3*a(n-1) for n > 1. (End)

a(26)-a(28) from Elmo R. Oliveira, Aug 23 2024

A052948 Expansion of g.f.: (1-2x)/(1-3x+2*x^3).

Original entry on oeis.org

1, 1, 3, 7, 19, 51, 139, 379, 1035, 2827, 7723, 21099, 57643, 157483, 430251, 1175467, 3211435, 8773803, 23970475, 65488555, 178918059, 488813227, 1335462571, 3648551595, 9968028331, 27233159851, 74402376363, 203271072427, 555346897579, 1517235940011, 4145165675179

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 3, s(n) = 3.

In general, a(n,m,j,k) = (2/m)*Sum_{r=1..m-1} sin(j*r*Pi/m)*sin(k*r*Pi/m)*(1+2*cos(Pi*r/m))^n is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = j, s(n) = k. - Herbert Kociemba, Jun 02 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Denis Chebikin and Richard Ehrenborg, The f-vector of the descent polytope, arXiv:0812.1249 [math.CO], 2008-2010; Disc. Comput. Geom., 45 (2011), 410-424.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1007
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Alina F. Y. Zhao, Bijective proofs for some results on the descent polytope, Australasian Journal of Combinatorics, Volume 65(1) (2016), Pages 45-52.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Cf. A026150, A077846.

a:=[1,1,3];; for n in [4..30] do a[n]:=3*a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Oct 21 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-3*x+2*x^3) )); // G. C. Greubel, Oct 21 2019

spec := [S,{S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z),Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((1-2*x)/(1-3*x+2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 21 2019

CoefficientList[Series[(1-2x)/(1-3x+2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,0,-2},{1,1,3},30] (* Harvey P. Dale, Aug 22 2012 *)

PARI
```
Vec((1-2*x)/(1-3*x+2*x^3)+O(x^30))
```

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,1,2,2, lambda n: -1); [next(it) for i in range(0,29)] # Zerinvary Lajos, Jul 09 2008

a(n) = 2*a(n-1) + 2*a(n-2) - 1.
a(n) = Sum_{alpha=RootOf(1-3*z+2*z^3)} alpha^(-n)/3.
a(n) = (1 + (1+sqrt(3))^n + (1-sqrt(3))^n)/3. Binomial transform of A025192 (with interpolated zeros). - Paul Barry, Sep 16 2003
a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)^2 * (1 + 2*cos(Pi*k/6))^n. - Herbert Kociemba, Jun 02 2004
a(0)=1, a(1)=1, a(2)=3, a(n) = 3*a(n-1) - 2*a(n-3). - Harvey P. Dale, Aug 22 2012
a(n) = A077846(n) - 2*A077846(n-1). - R. J. Mathar, Feb 27 2019
E.g.f.: exp(x)*(1 + 2*cosh(sqrt(3)*x))/3. - Stefano Spezia, Mar 02 2024

More terms from James Sellers, Jun 06 2000

Definition revised by N. J. A. Sloane, Feb 24 2011

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cp's OEIS Frontend

A106516 A Pascal-like triangle based on 3^n.

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A189315 Expansion of g.f. 5*(1-3*x+x^2)/(1-5*x+5*x^2).

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A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1.

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A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

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A061922 Xcatalans - produced as a self-convolved sequence like Catalan numbers (A000108) but use carryless GF(2)[ X ] polynomial multiplication.

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Maple

A080342 Number of weighings required to identify a single bad coin out of n coins, using a two-pan balance.

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A189315 Expansion of g.f. 5(1-3x+x^2)/(1-5x+5x^2).

A099856 Expansion of (1+3x)/(1-3x).

A052948 Expansion of g.f.: (1-2x)/(1-3x+2*x^3).