A005032 -id:A005032 - cp's OEIS frontend

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A199109 a(n) = (7*3^n + 1)/2.

Original entry on oeis.org

4, 11, 32, 95, 284, 851, 2552, 7655, 22964, 68891, 206672, 620015, 1860044, 5580131, 16740392, 50221175, 150663524, 451990571, 1355971712, 4067915135, 12203745404, 36611236211, 109833708632, 329501125895, 988503377684, 2965510133051, 8896530399152, 26689591197455

Offset: 0

Vincenzo Librandi, Nov 03 2011

Also the number of (not necessarily maximal) cliques in the (n+2)-Mycielski graph. - Eric W. Weisstein, Nov 29 2017

			Ternary....................Decimal
11...............................4
102.............................11
1012............................32
10112...........................95
101112.........................284
1011112........................851
10111112......................2552
101111112.....................7655
1011111112...................22964, etc.
- _Philippe Deléham_, Feb 16 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Mycielski Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005032 (first differences), A199110, A237930.

Magma
```
[(7*3^n+1)/2 : n in [0..30]];
```

Table[(7 3^n + 1)/2, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
(7 3^Range[0, 20] + 1)/2 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{4, -3}, {11, 32}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(4 - 5 x)/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=7*3^n\2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1) - 1.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (4-5*x)/((1-x)*(1-3*x)). - Bruno Berselli, Nov 03 2011
a(n) = A000244(n+1) + A003462(n) + 1 = A237930(n) + 1. - Philippe Deléham, Feb 16 2014
From Elmo R. Oliveira, Apr 02 2025: (Start)
E.g.f.: exp(x)*(7*exp(2*x) + 1)/2.
a(n) = A199110(n)/2. (End)

A082541 a(n) = (73^n - 40^n)/3.

Original entry on oeis.org

1, 7, 21, 63, 189, 567, 1701, 5103, 15309, 45927, 137781, 413343, 1240029, 3720087, 11160261, 33480783, 100442349, 301327047, 903981141, 2711943423, 8135830269, 24407490807, 73222472421, 219667417263, 659002251789, 1977006755367

Offset: 0

Paul Barry, May 02 2003

Binomial transform of A083495.

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

Cf. A083497.

[(7*3^n-4*0^n)/3: n in [0..30]]; // Vincenzo Librandi, Sep 15 2011

Join[{1},NestList[3#&,7,30]] (* Harvey P. Dale, May 07 2019 *)

a(n)=(7*3^n-4*0^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+4*x)/(1-3*x).
E.g.f.: (7*exp(3*x) - 4*exp(0))/3.
a(n) = A005032(n-1), n > 0. - R. J. Mathar, Sep 17 2008

A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.

Original entry on oeis.org

10, 100, 2890, 96100, 3237610, 109202500, 3683712490, 124263300100, 4191798484810, 141402777864100, 4769968258260490, 160906295771812900, 5427884341892493610, 183099910962324064900, 6176546013641762558890, 208354665265158340802500, 7028469704892605715408010

Offset: 0

Kival Ngaokrajang, Oct 29 2014

The denominators are conjectured to be A005032.
Refer to comments and links of A240926. Consider a unit circle with a chord of length sqrt(84)/5. This has been chosen such that the larger sagitta has length 7/5. The input, besides the unit circle C, is the circle C_0 with radius R_0 = 7/10, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n).
If one considers the curvature of touching circles inscribed in the smaller segment (sagitta length 3/5), the rational sequence would be A249458/A169634. See an illustration given in the link.
For the proof and the formula for the rational curvatures of the circles in the larger segment see a comment under A249862. C_n = (5/7)*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Nov 07 2014

Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Sagitta.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (37,-111,27).

Cf. A005032, A049310, A078986, A097315, A169364, A240926, A247335, A247512, A248834, A249458, A249862.

I:=[10,100,2890]; [n le 3 select I[n] else 37*Self(n-1) - 111*Self(n-2) + 27*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

LinearRecurrence[{37, -111, 27},{10, 100, 2890},16] (* Ray Chandler, Aug 11 2015 *)
CoefficientList[Series[10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

{
r=0.7;dn=7;print1(round(dn/r),", ");r1=r;
for (n=1,40,
     if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
     ac=sqrt(ab^2-r^2);
     if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
     b=acos(r/ab)-z;
     r=r*(1-cos(b))/(1+cos(b)); dn=dn*3;
     print1(round(dn/r),", ");
)
}

x='x+O('x^30); Vec(10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x))) \\ G. C. Greubel, Dec 20 2017

Empirical g.f.: -10*(30*x^2-27*x+1) /((3*x - 1)*(9*x^2-34*x+1)). - Colin Barker, Oct 29 2014
From Wolfdieter Lang, Nov 07 2014: (Start)
a(n) = 5*(A249862(n) + 3^n) = 5*3^n*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249862 for the proof.
O.g.f.: 5*((1 - 17*x)/(1 - 34*x + 9*x^2) + 1/(1-3*x)) = 10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)) proving the conjecture of Colin Barker above. (End)
E.g.f.: 5*exp(3*x)*(1 + exp(14*x)*cosh(2*sqrt(70)*x)). - Stefano Spezia, Mar 24 2023

Edited. Name and comment small changes, keyword easy added. - Wolfdieter Lang, Nov 07 2014

a(16) from Stefano Spezia, Mar 24 2023

A108765 Expansion of g.f. (1 - x + x^2)/((1-3x)(x-1)^2).

Original entry on oeis.org

1, 4, 14, 45, 139, 422, 1272, 3823, 11477, 34440, 103330, 310001, 930015, 2790058, 8370188, 25110579, 75331753, 225995276, 677985846, 2033957557, 6101872691, 18305618094, 54916854304, 164750562935, 494251688829, 1482755066512

Offset: 0

Creighton Dement, Jun 24 2005

Superseeker suggests a(n+2) - 2*a(n+1) + a(n) = 7*3^n = A005032(n).
Inverse binomial transform gives match with first differences of A026622.
Floretion Algebra Multiplication Program, FAMP Code: kbasefor[(- 'j + 'k - 'ii' - 'ij' - 'ik')], vesfor = A000004, Fortype: 1A, Roktype (leftfactor) is set to:Y[sqa.Findk()] = Y[sqa.Findk()] + Math.signum(Y[sqa.Findk()])*p (internal program code)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A005032, A026622.

s=1;lst={s};Do[s+=(s+(n+=s));AppendTo[lst, s], {n, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 11 2008 *)
CoefficientList[Series[(1-x+x^2)/((1-3x)(x-1)^2),{x,0,40}],x] (* or *) LinearRecurrence[{5,-7,3},{1,4,14},40] (* Harvey P. Dale, Dec 11 2012 *)

From Rolf Pleisch, Feb 10 2008: (Start)
a(0) = 1; a(n) = 3*a(n-1) + n.
a(n) = (7*3^n - 2*n - 3)/4. (End)
a(0)=1, a(1)=4, a(2)=14, a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Harvey P. Dale, Dec 11 2012

A249458 The numerators of curvatures of touching circles inscribed in a special way in the smaller segment of unit circle divided by a chord of length sqrt(84)/5.

Original entry on oeis.org

10, 100, 1690, 36100, 835210, 19802500, 472931290, 11318832100, 271066588810, 6492762648100, 155527144782490, 3725543446072900, 89243180863948810, 2137770243127864900, 51209104645650371290, 1226685938180259902500

Offset: 0

Kival Ngaokrajang, Oct 29 2014

The denominators are conjectured to be A169634.
Refer to comments and links of A240926. Consider a unit circle with a chord of length sqrt(84)/5. This has been chosen such that the smaller sagitta has length 3/5. The input, besides the circle C, is the circle C_0 with radius R_0 = 3/10, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n). If one considers the curvature of touching circles inscribed in the larger segment (sagitta length 7/5), the sequence would be A249457/A005032. See an illustration given in the link.
For the proof and the formula for the rational curvatures of the circles in the smaller segment see a comment under A249864. C_n = (5/(3*7))*(7*S(n, 26/7) - 13*S(n-1, 26/7) + 7), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Nov 08 2014

Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Sagitta.
Index entries for linear recurrences with constant coefficients, signature (33,-231,343).
Index entries for sequences related to Chebyshev polynomials.

Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248834, A169634, A249457, A049310, A249863, A249864.

I:=[10, 100, 1690]; [n le 3 select I[n] else 33*Self(n-1) - 231*Self(n-2) + 343*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

LinearRecurrence[{33, -231, 343},{10, 100, 1690},16] (* Ray Chandler, Aug 11 2015 *)
CoefficientList[Series[10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x)), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

{
r=0.3;dn=3;print1(round(dn/r),", ");r1=r;
for (n=1,40,
     if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
     ac=sqrt(ab^2-r^2);
     if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
     b=acos(r/ab)-z;
     r=r*(1-cos(b))/(1+cos(b)); dn=dn*7;
     print1(round(dn/r),", ");
)
}

x='x+O('x^30); Vec(10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x))) \\ G. C. Greubel, Dec 20 2017

Empirical g.f.: -10*(70*x^2-23*x+1) / ((7*x-1)*(49*x^2-26*x+1)). - Colin Barker, Oct 29 2014
From Wolfdieter Lang, Nov 09 2014 (Start)
a(n) = 5*(A249864(n) + 7^n) = (5*7^n)*(S(n, 26/7) - (13/7)*S(n-1, 26/7) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249864 for the proof.
O.g.f.: 5*((1 - 13*x)/(1 - 26*x + (7*x)^2) + 1/(1-7*x)) = 10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x)) proving the conjecture of Colin Barker above. (End)

Edited. In name and comment small changes, keyword easy and crossrefs added. - Wolfdieter Lang, Nov 08 2014

A141495 a(n) = 3*a(n-1) for n>2; a(0)=1, a(1)=3, a(2)=7.

Original entry on oeis.org

1, 3, 7, 21, 63, 189, 567, 1701, 5103, 15309, 45927, 137781, 413343, 1240029, 3720087, 11160261, 33480783, 100442349, 301327047, 903981141, 2711943423, 8135830269, 24407490807, 73222472421, 219667417263, 659002251789

Offset: 0

Roger L. Bagula, Aug 10 2008

A sequence of the form: a(0)=1, a(1)=prime(m), a(2)=prime(m+2), a(n)=a(1)*a(n-1).

a(n) is divisible by 7 for n>1. - Colin Barker, Jan 09 2014

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

Essentially the same as A005032 and A084215. For other examples see A141496, etc.

a[0] = 1; a[1] = 3; a[2] = 7; a[n_] := a[n] = a[1]*a[n - 1]; Table[a[n], {n, 0, 30}]
Join[{1,3},NestList[3#&,7,30]] (* Harvey P. Dale, Aug 05 2024 *)

a(n) = A082541(n-1), n>1. - R. J. Mathar, Aug 27 2008

a(n) = 7*3^(n-2) for n>1. a(n)=3*a(n-1) for n>2. G.f.: (2*x^2-1) / (3*x-1). - Colin Barker, Jan 09 2014

Edited by N. J. A. Sloane, Aug 16 2008

A171382 a(n) = (22^n+7(-1)^n)/3.

Original entry on oeis.org

3, -1, 5, 3, 13, 19, 45, 83, 173, 339, 685, 1363, 2733, 5459, 10925, 21843, 43693, 87379, 174765, 349523, 699053, 1398099, 2796205, 5592403, 11184813, 22369619, 44739245, 89478483, 178956973, 357913939, 715827885, 1431655763, 2863311533

Offset: 0

Klaus Brockhaus, Dec 07 2009

a(n) = A155980(n+2).
a(n) = A135351(n+3)-A135351(n+2).
Second binomial transform of a signed version of A005032 preceded by 3.
Inverse binomial transform of A008776 preceded by 3.

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A155980 (First differences of A135351), A135351 ((2^n+3-7*(-1)^n+3*0^n)/6), A005032 (7*3^n), A008776 (2*3^n).

Magma
```
[ (2*2^n+7*(-1)^n)/3: n in [0..32] ];
```

Nest[Append[#,Last[#]+2#[[-2]]]&,{3,-1},40]  (* Harvey P. Dale, Apr 07 2011 *)

a(n) = a(n-1)+2*a(n-2) for n > 1; a(0) = 3, a(1) = -1.
a(n) = 2^n-a(n-1) for n > 0; a(0) = 3.
G.f.: (3-4*x)/((1+x)*(1-2*x)).

A270472 Expansion of g.f. (1-2x)/(1-9x).

Original entry on oeis.org

1, 7, 63, 567, 5103, 45927, 413343, 3720087, 33480783, 301327047, 2711943423, 24407490807, 219667417263, 1977006755367, 17793060798303, 160137547184727, 1441237924662543, 12971141321962887, 116740271897665983, 1050662447078993847, 9455962023710944623, 85103658213398501607

Offset: 0

Colin Barker, Mar 17 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A005032, A187709 (partial sums).

Cf. A055275: (1-x)/(1-9*x); A092810: (1-3*x)/(1-9*x).

CoefficientList[Series[(1 - 2 x)/(1 - 9 x), {x, 0, 20}], x] (* Michael De Vlieger, Mar 18 2016 *)

PARI
```
Vec((1-2*x)/(1-9*x) + O(x^30))
```

G.f.: (1-2*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 7*9^(n-1) for n>0.
a(n) = A005032(2*n-2). - R. J. Mathar, Jan 28 2025
E.g.f.: (7*exp(9*x) + 2)/9. - Elmo R. Oliveira, Mar 25 2025

A199110 a(n) = 7*3^n + 1.

Original entry on oeis.org

8, 22, 64, 190, 568, 1702, 5104, 15310, 45928, 137782, 413344, 1240030, 3720088, 11160262, 33480784, 100442350, 301327048, 903981142, 2711943424, 8135830270, 24407490808, 73222472422, 219667417264, 659002251790, 1977006755368, 5931020266102, 17793060798304, 53379182394910

Offset: 0

Vincenzo Librandi, Nov 03 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A005032, A199109.

Magma
```
[7*3^n+1: n in [0..30]];
```

7*3^Range[0, 30] + 1 (* Paolo Xausa, Jan 28 2025 *)

def a(n): return 7*3**n + 1
print([a(n) for n in range(26)]) # Michael S. Branicky, Aug 22 2021

a(n) = 3*a(n-1) - 2 = A005032(n) + 1.
a(n) = 4*a(n-1) - 3*a(n-2).
From Bruno Berselli, Nov 03 2011: (Start)
G.f.: 2*(4-5*x)/((1-x)*(1-3*x)).
a(n) = 2*A199109(n). (End)
E.g.f.: exp(x)*(1 + 7*exp(2*x)). - Elmo R. Oliveira, Apr 02 2025

cp's OEIS Frontend

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

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A199109 a(n) = (7*3^n + 1)/2.

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A082541 a(n) = (7*3^n - 4*0^n)/3.

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A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.

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Crossrefs

Programs

Magma

Mathematica

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A108765 Expansion of g.f. (1 - x + x^2)/((1-3*x)*(x-1)^2).

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Comments

Links

Crossrefs

Programs

Mathematica

Formula

A249458 The numerators of curvatures of touching circles inscribed in a special way in the smaller segment of unit circle divided by a chord of length sqrt(84)/5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A141495 a(n) = 3*a(n-1) for n>2; a(0)=1, a(1)=3, a(2)=7.

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A082541 a(n) = (73^n - 40^n)/3.

A108765 Expansion of g.f. (1 - x + x^2)/((1-3x)(x-1)^2).

A171382 a(n) = (22^n+7(-1)^n)/3.

A270472 Expansion of g.f. (1-2x)/(1-9x).