A057079 -id:A057079 - cp's OEIS frontend

A112466 Riordan array ((1+2*x)/(1+x), x/(1+x)).

1, 1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 2, 0, -2, 1, 1, -3, 2, 2, -3, 1, -1, 4, -5, 0, 5, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, -1, 6, -14, 14, 0, -14, 14, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, -1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1, -1, 10, -44, 110, -165, 132, 0, -132, 165, -110, 44, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Inverse is A112465.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 07 2006; corrected by Philippe Deléham, Dec 11 2008
Equals A097808 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle starts
   1;
   1,  1;
  -1,  0,  1;
   1, -1, -1,  1;
  -1,  2,  0, -2,  1;
   1, -3,  2,  2, -3,  1;
  -1,  4, -5,  0,  5, -4,  1;
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1,  1;
  -2, -1,  1;
   2,  0, -1,  1;
  -2,  0,  0, -1,  1;
   2,  0,  0,  0, -1,  1;
  -2,  0,  0,  0,  0, -1,  1;
   2,  0,  0,  0,  0,  0, -1,  1; (End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

Cf. A000007, A008482, A037012, A057079, A097808, A112467.

Columns: A248157(n+2) (k=1), (-1)^n*A080956(n-2) (k=2), (-1)^(n-1)*A254749(n-2) (k=3).

A112466:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(n,k) - 2*Binomial(n-1,k)) >;
[A112466(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2025

seq(seq( (-1)^(n-k)*(2*binomial(n-1, k-1)-binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Feb 19 2020

{1}~Join~Table[(Binomial[n, n - k] - 2 Binomial[n - 1, n - k - 1])*(-1)^(n - k), {n, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 18 2020 *)

T(n,k) = (-1)^(n-k)*(binomial(n, n-k) - 2*binomial(n-1, n-k-1)); \\ Michel Marcus, Feb 19 2020

def A112466(n,k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n,k) - 2*binomial(n-1,k))
print(flatten([[A112466(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 30 2025

Number triangle: T(n,k) = (-1)^(n-k)*(C(n, n-k) - 2*C(n-1, n-k-1)), with T(0,0) = 1.
T(2*n, n) = 0 (main diagonal).
Sum_{k=0..n} T(n, k) = 0 + [n=0] + 2*[n=1] (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*Fibonacci(n-2) (diagonal sums).
Sum_{k=0..n} T(n,k)*x^k = (x+1)*(x-1)^(n-1), for n >= 1. - Philippe Deléham, Oct 03 2005
T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if n < 0 or if n < k, T(n,k) = T(n-1,k-1) - T(n-1,k) for n > 1. - Philippe Deléham, Nov 26 2006
G.f.: (1+2*x)/(1+x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Apr 30 2025: (Start)
T(2*n+1, 2*n+1-k) = T(2*n+1, k) (symmetric odd n rows).
T(2*n, 2*n-k) = (-1)*T(2*n, k) (antisymmetric even n rows).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) (signed row sums).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A057079(n+2) (signed diagonal sums). (End)

A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 12, 21, 9, 1, 24, 60, 45, 12, 1, 48, 156, 171, 78, 15, 1, 96, 384, 558, 372, 120, 18, 1, 192, 912, 1656, 1473, 690, 171, 21, 1, 384, 2112, 4608, 5160, 3225, 1152, 231, 24, 1, 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1, 1536, 10752

Offset: 0

Paul Barry, Feb 13 2006

Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.

Subtriangle of triangle given by (0, 3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

			Triangle begins
1,
3, 1,
6, 6, 1,
12, 21, 9, 1,
24, 60, 45, 12, 1,
48, 156, 171, 78, 15, 1
Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 12, 21, 9, 1
0, 24, 60, 45, 12, 1
0, 48, 156, 171, 78, 15, 1
... - _Philippe Deléham_, Jan 18 2012

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A003688, A003945.

With[{n = 10}, DeleteCases[#, 0] & /@ CoefficientList[Series[(1 + x)/(1 - (y + 2) x - y x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}
From Vladimir Kruchinin, Mar 17 2011: (Start)
T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.
T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)
G.f.: (1+x)/(1-(y+2)*x -y*x^2). - Philippe Deléham, Jan 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(2,0) = T(2,1) = 6, T(2,2) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 31 2013

A132357 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,4,14,41.

Original entry on oeis.org

1, 4, 14, 41, 122, 364, 1093, 3280, 9842, 29525, 88574, 265720, 797161, 2391484, 7174454, 21523361, 64570082, 193710244, 581130733, 1743392200, 5230176602, 15690529805, 47071589414, 141214768240, 423644304721

Offset: 0

Paul Curtz, Nov 24 2007

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

First differences of A132353.

Cf. A129339.

LinearRecurrence[{3,0,-1,3},{1,4,14,41},50] (* Paolo Xausa, Dec 05 2023 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 3,-1,0,3]^n*[1;4;14;41])[1,1] \\ Charles R Greathouse IV, Oct 08 2016

O.g.f.: -(1+x+2*x^2)/((3*x-1)*(x+1)*(x^2-x+1)) = -(3/2)/(3*x-1)+(1/3)*(x-2)/(x^2-x+1)+(1/ 6)/(x+1). - R. J. Mathar, Nov 28 2007
a(n) = (1/2)*3^(n+1) + (1/6)*(-1)^n - (2/3)*cos(Pi*n/3). Or, a(n) = (1/2)*3^(n+1) + (1/2)*[ -1; -1; 1; 1; 1; -1]. - Richard Choulet, Jan 02 2008
a(n+1) - 3a(n) = A132367(n+1). - Paul Curtz, Dec 02 2007
6*a(n) = (-1)^n +3^(n+2) -2*A057079(n+1). - R. J. Mathar, Oct 03 2021

A244422 Quasi-Riordan triangle ((2-z)/(1-z), -z^2/(1-z)). Row reversed monic Chebyshev T-polynomials without vanishing columns.

Original entry on oeis.org

2, 1, 0, 1, -2, 0, 1, -3, 0, 0, 1, -4, 2, 0, 0, 1, -5, 5, 0, 0, 0, 1, -6, 9, -2, 0, 0, 0, 1, -7, 14, -7, 0, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Wolfdieter Lang, Aug 08 2014

This is a signed version of the triangle A061896.
The coefficient table for the monic Chebyshev polynomials of the first kind R(n, x) = 2*T(n, x/2) is given in A127672. For the T-polynomials see A053120. The present table is obtained from the row reversed coefficient table A127672 by deleting all odd numbered columns which have only zeros, and appending in the rows numbered n >= 1 zeros in order to obtain a triangle. This becomes the quasi-Riordan triangle T = ((2-z)/(1-z), -z^2/(1-z)). This means that the o.g.f. of the row polynomials Rrev(n, x) := sqrt(x)^n*R(n, 1/sqrt(x)) = Sum_{k=0..n} T(n, k)*x^k have o.g.f. (2-z)/(1 - z + x*z^2) like for ordinary Riordan triangles. However this is not a Riordan triangle (or lower triangular infinite dimensional matrix) in the usual sense because it is not invertible. Therefore, this lower triangular matrix is not a member of the Riordan group.
The row sums give repeat(2,1,-1,-2,-1) which is A057079(n+1), n >= 0. The alternating row sums give the Lucas numbers A000032.

			The triangle T(n,k) begins:
  n\k  0   1   2     3    4     5  6   7  8  9 10 11
  0:   2
  1:   1   0
  2:   1  -2   0
  3:   1  -3   0     0
  4:   1  -4   2     0    0
  5:   1  -5   5     0    0     0
  6:   1  -6   9    -2    0     0  0
  7:   1  -7  14    -7    0     0  0   0
  8:   1  -8  20   -16    2     0  0   0  0
  9:   1  -9  27   -30    9     0  0   0  0  0
  10:  1 -10  35   -50   25    -2  0   0  0  0  0
  11:  1 -11  44   -77   55   -11  0   0  0  0  0  0
  ...
Rrev(3, x) = 1 - 3*x = sqrt(x)^3*R(3,1/sqrt(x)) = sqrt(x)^3*(-3/sqrt(x) + 1/sqrt(x)^3 ) = -3*x + 1.
Rrev(4, x) = 1 - 4*x + 2*x^2 = sqrt(x)^4*(2 - 4/sqrt(x)^2 + 1/sqrt(x)^4) = 2*x^2 - 4*x + 1.
Recurrence: T(4,1) = T(3, 1) - T(2, 0) = -3 -1 = -4.

Wolfdieter Lang, First rows of the triangle.

Cf. A053120, A127672, A054977, -A000027, A000096, -A005581, A005582, -A005583, A005584, A057079, A000032.

T(n,k) = [x^k] Rrev(n, x), k=0, 1, ..., n, with the row polynomials Rrev(n, x) = sqrt(x)^n*R(n,1/sqrt(x)), with R(n, x) given in A127672 (monic Chebyshev polynomials of the first kind).
O.g.f. row polynomials Rrev(n,x) = Sum_{k=0..n} T(n,k)*x^k: (2-z)/(1 - z + x*z^2) (quasi-Riordan).
O.g.f. for column number k entries with leading zeros: ((2-x)/(1-x))*(-x^2/(1-x))^k, k > = 0. See A054977, -A000027, A000096, -A005581, A005582, -A005583, A005584.
Recurrence: T(n,k) = T(n-1, k) - T(n-2, k-1), n >= k >= 1, T(n,k) = 0 if n < k, T(0,0) = 2, T(n,0) = 1 if n>=1, (Compare with A061896).
For n >= 1 the entries without trailing zeros are given by T(n,k) = (-1)^k*(n/(n-k))*binomial(n-k,k) where k=0..floor(n/2).

A359380 Numbers that are neither multiples of 3 nor of the form 4u+2.

Original entry on oeis.org

1, 4, 5, 7, 8, 11, 13, 16, 17, 19, 20, 23, 25, 28, 29, 31, 32, 35, 37, 40, 41, 43, 44, 47, 49, 52, 53, 55, 56, 59, 61, 64, 65, 67, 68, 71, 73, 76, 77, 79, 80, 83, 85, 88, 89, 91, 92, 95, 97, 100, 101, 103, 104, 107, 109, 112, 113, 115, 116, 119, 121, 124, 125, 127, 128, 131, 133, 136, 137, 139, 140

Offset: 1

Antti Karttunen, Dec 31 2022

Numbers that are congruent to {1, 4, 5, 7, 8, 11} mod 12. - Amiram Eldar, Jan 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Cf. A187074 (characteristic function).
Cf. also A359375.
Cf. A010892, A057079.

Select[Range[140], Mod[#, 3] > 0 && Mod[#, 4] != 2 &] (* Amiram Eldar, Dec 31 2022 *)

PARI
```
isA359380(n) = A187074(n);
```

Sum_{n>=1} (-1)^(n+1)/a(n) = (1-1/(4*sqrt(3)))*Pi/3. - Amiram Eldar, Jan 24 2023
From Stefano Spezia, Jul 10 2024: (Start)
G.f.: x*(1 + 2*x - 2*x^2 + 2*x^3 + x^4)/((1 - x)^2*(1 + x^3)).
a(n) = (6*n - 3 - A057079(n+1) + 2*A010892(n+2)/3)/3 for n > 0. (End)

A130707 a(n+3) = 3(a(n+2) - a(n+1)) + 2a(n).

Original entry on oeis.org

1, 2, 2, 2, 4, 10, 22, 44, 86, 170, 340, 682, 1366, 2732, 5462, 10922, 21844, 43690, 87382, 174764, 349526, 699050, 1398100, 2796202, 5592406, 11184812, 22369622, 44739242, 89478484, 178956970, 357913942, 715827884, 1431655766, 2863311530

Offset: 0

Paul Curtz, Jul 01 2007

nonn

Binomial transform of period-3 sequence with period 1 1 -1.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

a:=proc(n) options operator, arrow: (1/3)*2^n+(4/3)*(-1)^n*cos((1/3)*(2*n+1)*Pi) end proc: seq(a(n), n = 0 .. 33); # Emeric Deutsch, Jul 27 2007

RecurrenceTable[{a[0]==1,a[1]==a[2]==2,a[n]==3(a[n-1]-a[n-2])+2a[n-3]},a,{n,40}] (* or *) LinearRecurrence[{3,-3,2},{1,2,2},40] (* Harvey P. Dale, Jan 18 2015 *)

a(n) = 2^n/3 + 4*(-1)^n*(1/3)*cos((2n+1)*Pi/3). - Emeric Deutsch, Jul 27 2007
From R. J. Mathar, Nov 18 2007: (Start)
G.f.: (-1+x+x^2)/(2*x-1)/(x^2-x+1).
a(n) = (2*A057079(n) + 2^n)/3. (End)

More terms from Emeric Deutsch, Jul 27 2007

A131132 a(n) = a(n-1) + a(n-2) + 1 if n is a multiple of 6, otherwise a(n) = a(n-1) + a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 22, 36, 58, 94, 152, 247, 399, 646, 1045, 1691, 2736, 4428, 7164, 11592, 18756, 30348, 49104, 79453, 128557, 208010, 336567, 544577, 881144, 1425722, 2306866, 3732588, 6039454, 9772042, 15811496, 25583539, 41395035, 66978574, 108373609

Offset: 0

N. J. A. Sloane, May 25 2008

nonn

Also: convolution of A000045 with the period-6 sequence (0,0,0,0,0,0, 1,...). - R. J. Mathar, May 30 2008
Sequences of the form s(0)=a, s(1)= b, s(n) = s(n-1) + s(n-2) + k if n mod m = p, else s(n) = s(n-1) + s(n-2) have a form s(n) = fibonacci(n-1)*a + fibonacci(n)*b + P(x)*k. a(n) is the P(x) sequence for m=6. s(n) = fib(n)*a + fib(n-1)*b + a(n-6-p)*k. - Gary Detlefs, Dec 05 2010
a(n) is the number of compositions of n where the order of the 2 and the 3 does not matter. - Gregory L. Simay, May 18 2017

			Since 5 is not a multiple of 6, a(5) = a(4) + a(3) = 5 + 3 = 8. Since 6 is a multiple of 6, a(6) = a(5) + a(4) + 1 = 8 + 5 + 1 = 14. - _Michael B. Porter_, Jul 26 2016

H. Matsui et al., Problem B-1035, Fibonacci Quarterly, Vol. 45, Number 2; 2007; p. 182.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,1,-1,-1).

Cf. A052952, A004695, A080239, A124502, A066983.

A131132:=proc(n) option remember; local t1; if n <= 2 then RETURN(1); fi: if n mod 6 = 1 then t1:=1 else t1:=0; fi: procname(n-1)+procname(n-2)+t1; end; [seq(A131132(n), n=1..100)]; # N. J. A. Sloane, May 25 2008; Typo corrected by R. J. Mathar, May 30 2008

Print[Table[Round[(1 + Sqrt[5])/8 Fibonacci[n + 3]], {n, 0, 50}]] ;
Print[RecurrenceTable[{c[n] == c[n - 1] + c[n - 2] + c[n - 6] - c[n - 7] - c[n - 8], c[0] == 1, c[1] == 1, c[2] == 2, c[3] == 3, c[4] == 5, c[5] == 8, c[6] == 14, c[7] == 22}, c, {n, 0, 50}]] ;  (* John M. Campbell, Jul 07 2016 *)
LinearRecurrence[{1, 1, 0, 0, 0, 1, -1, -1}, {1, 1, 2, 3, 5, 8, 14, 22}, 40] (* Vincenzo Librandi, Jul 07 2016 *)

O.g.f.: 1/((1-x^6)(1 - x - x^2)). - R. J. Mathar, May 30 2008
a(n) = ((-1)^n-1)/6 + A099837(n+3)/12 + A000045(n+4)/4 + A057079(n)/12. - R. J. Mathar, Dec 07 2010
a(n) = floor(A066983(n+4)/3). - Gary Detlefs, Dec 19 2010
a(n) = round((1 + sqrt(5))/8 A000045(n+3)). - John M. Campbell, Jul 06 2016
a(n) = (number of compositions of n consisting of only 1 or 2 or 6) - (number of compositions with only 7 or ((1 or 2) and 7)) - (number of compositions with only 8 or ((1 or 2) and 8)). The "or" is inclusive. - Gregory L. Simay, May 25 2017

More specific name from R. J. Mathar, Dec 09 2009

A131562 a(n)= -3a(n-1) -3a(n-2)-2a(n-3), a(0)=1, a(1)=-2, a(2)=2.

Original entry on oeis.org

1, -2, 2, -2, 4, -10, 22, -44, 86, -170, 340, -682, 1366, -2732, 5462, -10922, 21844, -43690, 87382, -174764, 349526, -699050, 1398100, -2796202, 5592406, -11184812, 22369622, -44739242, 89478484, -178956970, 357913942, -715827884, 1431655766, -2863311530

Offset: 0

Paul Curtz, Aug 27 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-2).

Cf. A130707.

LinearRecurrence[{-3,-3,-2},{1,-2,2},40] (* Harvey P. Dale, Jan 11 2017 *)

|v(n)| = 2^n+A130772(n); 2*|v(n)|-|v(n+1)|= 2*A057079(n), where v(n)=a(n+1)-a(n) are first differences.
O.g.f.: (1+x-x^2)/((1+2*x)*(1+x+x^2)). a(n)=(-1)^n*A130707(n). - R. J. Mathar, Jul 07 2008
Binomial transform yields A130151 without the first two terms. - R. J. Mathar, Jul 07 2008

Edited by R. J. Mathar, Jul 07 2008

A132868 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,3,7,20.

Original entry on oeis.org

1, 3, 7, 20, 60, 182, 547, 1641, 4921, 14762, 44286, 132860, 398581, 1195743, 3587227, 10761680, 32285040, 96855122, 290565367, 871696101, 2615088301, 7845264902, 23535794706, 70607384120, 211822152361, 635466457083, 1906399371247, 5719198113740

Offset: 0

Paul Curtz, Nov 22 2007

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

Cf. A129339, A132951.

LinearRecurrence[{3,0,-1,3},{1,3,7,20},50] (* Harvey P. Dale, Jan 21 2012 *)

4*a(n) = 3^(n+1) + A132951(n).
O.g.f.: (-1+2*x^2)/((3*x-1)*(x+1)*(x^2-x+1)) = -(3/4)/(3*x-1)-(1/12)/(x+1)+(1/3)*(x+1)/(x^2-x+1). - R. J. Mathar, Nov 28 2007
a(n) = (1/12)*(3^(n+2) - 4*cos((n+1)*Pi/3) + cos((n+1)*Pi) + 4*sqrt(3) * sin(((n+1)*Pi)/3) + I*sin((n+1)*Pi)). - Harvey P. Dale, Jan 21 2012
12*a(n) = -(-1)^n +3^(n+2) +4*A057079(n). - R. J. Mathar, Oct 03 2021

A143100 A007318 * A143098.

Original entry on oeis.org

1, 3, 4, 6, 13, 30, 64, 129, 256, 510, 1021, 2046, 4096, 8193, 16384, 32766, 65533, 131070, 262144, 524289, 1048576, 2097150, 4194301, 8388606, 16777216, 33554433, 67108864, 134217726, 268435453, 536870910, 1073741824, 2147483649, 4294967296, 8589934590

Offset: 1

Gary W. Adamson, Jul 24 2008

nonn

			a(4) = 6 = (1, 3, 3, 1) dot (1, 2, -1, 2) = (1 + 6 - 3 + 2).

Nathaniel Johnston, Table of n, a(n) for n = 1..500

Cf. A143097, A143098, A143099.

A143098 := proc(n) if(n=1)then return 1: elif(n mod 3 = 0)then return -1: else return 2: fi: end: A143100 := proc(n) return add(binomial(n-1,k-1)*A143098(k),k=1..n): end: seq(A143100(n),n=1..34); # Nathaniel Johnston, Apr 30 2011

Binomial transform of A143098: (1, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, ...).
From R. J. Mathar, Jul 31 2008: (Start)
G.f.: (3x^3 - 2x^2 - x + 1)*x/((x^2-x+1)*(2x-1)*(x-1)).
a(n) = -1 + 2^(n-1) + A057079(n-1). (End)

cp's OEIS Frontend

A112466 Riordan array ((1+2*x)/(1+x), x/(1+x)).

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A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

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A132357 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,4,14,41.

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A244422 Quasi-Riordan triangle ((2-z)/(1-z), -z^2/(1-z)). Row reversed monic Chebyshev T-polynomials without vanishing columns.

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A359380 Numbers that are neither multiples of 3 nor of the form 4u+2.

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A130707 a(n+3) = 3*(a(n+2) - a(n+1)) + 2*a(n).

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A131132 a(n) = a(n-1) + a(n-2) + 1 if n is a multiple of 6, otherwise a(n) = a(n-1) + a(n-2).

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A132357 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,4,14,41.

A130707 a(n+3) = 3(a(n+2) - a(n+1)) + 2a(n).

A132868 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,3,7,20.