A003683 -id:A003683 - cp's OEIS frontend

A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, 1398784, 5591040, 22372352, 89473024, 357924864, 1431633920, 5726666752, 22906404864, 91626143744, 366503526400, 1466016202752, 5864060616704, 23456250855424, 93824986644480

Offset: 1

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1.
  p(1, x) = 2*x -> 2*x.
  p(2, x) = 2 + x + 3*x^2 -> 8 + 4*x.
  p(3, x) = 8*x + 4*x^2 + 4*x^3 -> 16 + 24*x.
  p(4, x) = 4 + 4*x + 21*x^2 + 10*x^3 + 5*x^4 -> 96 + 80*x.
From these, read A083086 = (1, 0, 9, 16, 96, ...) and A192382 =(0, 2, 4, 24, 80, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003683, A083086, A162517, A192232.

[(4^(n-1) - (-2)^(n-1))/3: n in [1..40]]; // G. C. Greubel, Feb 19 2023

seq(4^n*(1-(-1/2)^n)/3, n=0..24); # Peter Luschny, Oct 02 2019

q[x_]:= x+2; d= Sqrt[x+2];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2 d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n, 6}]
reductionRules= {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x*q[x]^((y- 1)/2)};
t = Table[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,30}];
Table[Coefficient[Part[t,n], x, 0], {n,30}] (* abs value of A083086 *)
Table[Coefficient[Part[t,n], x, 1], {n,30}] (* 2*A003683 *)
Table[Coefficient[Part[t,n]/2, x, 1], {n,30}] (* A003683 *)
LinearRecurrence[{2,8}, {0,2}, 40] (* G. C. Greubel, Feb 19 2023 *)

[(4^(n-1) - (-2)^(n-1))/3 for n in range(1,41)] # G. C. Greubel, Feb 19 2023

Conjectures from Colin Barker, May 12 2014: (Start)
a(n) = 2^(n-2)*(2*(-1)^n + 2^n)/3 = 2*A003683(n-1).
a(n) = 2*a(n-1) + 8*a(n-2).
G.f.: 2*x^2 / ((1+2*x)*(1-4*x)). (End).
a(n) = 4^n*(1 - (-1/2)^n)/3. - Peter Luschny, Oct 02 2019
E.g.f: (1/3)*(2 + exp(2*x))*(sinh(x))^2. - G. C. Greubel, Feb 19 2023

A124860 A Jacobsthal-Pascal triangle.

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 5, 15, 15, 5, 11, 44, 66, 44, 11, 21, 105, 210, 210, 105, 21, 43, 258, 645, 860, 645, 258, 43, 85, 595, 1785, 2975, 2975, 1785, 595, 85, 171, 1368, 4788, 9576, 11970, 9576, 4788, 1368, 171, 341, 3069, 12276, 28644, 42966, 42966, 28644, 12276, 3069, 341

Offset: 0

Paul Barry, Nov 10 2006

Triangle T(n, k) read by rows given by [1, 2, -2, 0, 0, 0, ...] DELTA [1, 2, -2, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 11 2006

			Triangle begins
   1;
   1,   1;
   3,   6,   3;
   5,  15,  15,   5;
  11,  44,  66,  44,  11;
  21, 105, 210, 210, 105,  21;
  43, 258, 645, 860, 645, 258, 43;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001045, A003683 (row sums), A016095, A084938, A124862 (diagonal sums), A193449.

A124860:= func< n,k | Binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3 >;
[A124860(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

A := proc(n,k) ## n >= 0 and k = 0 .. n
    ((-1)^n+2^(n+1))/3*binomial(n, k)
end proc: # Yu-Sheng Chang, Jan 15 2020

jacobPascal[n_, k_]:= Binomial[n, k]*(2^(n+1) -(-1)^(n+1))/3; ColumnForm[Table[jacobPascal[n, k], {n,0,12}, {k,0,n}], Center] (* Alonso del Arte, Jan 16 2020 *)

def A124860(n,k): return binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3
flatten([[A124860(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

G.f.: 1/(1 - x*(1+y) - 2*x^2*(1+y)^2).
T(n, k) = J(n+1) * C(n, k), where J(n) = A001045(n).
T(n, 0) = T(n, n) = A001045(n+1).
T(2*n, n) = A124862(n).
Sum_{k=0..n} T(n, k) = A003683(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A124861(n).
T(n, k) = T(n-1, k-1) + T(n-1, k) + 2*T(n-2, k-2) + 4*T(n-2, k-1) + 2*T(n-2, k), T(0, 0) = 1, T(n, k) = 0 if k < 0 or if k > n . - Philippe Deléham, Nov 11 2006
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k + 1 + 2*x*(1+y))*x*(1 + y)/((2*k + 2 + 2*x*(1+y))*x*(1+y) + 1/T(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
From G. C. Greubel, Feb 17 2023: (Start)
T(n, n-k) = T(n, k).
T(n, 1) = A193449(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A003674 a(n) = 2^(n-1)*(2^n - (-1)^n).

Original entry on oeis.org

0, 3, 6, 36, 120, 528, 2016, 8256, 32640, 131328, 523776, 2098176, 8386560, 33558528, 134209536, 536887296, 2147450880, 8590000128, 34359607296, 137439215616, 549755289600, 2199024304128, 8796090925056

Offset: 0

N. J. A. Sloane

M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003683 (one-third), A062510, A071930.

[(4^n -(-2)^n)/2: n in [0..40]]; // G. C. Greubel, Feb 17 2023

Table[(4^n-(-2)^n)/2, {n,0,40}] (* G. C. Greubel, Feb 17 2023 *)

PARI
```
a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n))
```

[(4^n-(-2)^n)/2 for n in range(41)] # G. C. Greubel, Feb 17 2023

G.f.: 3*x/((1+2*x)*(1-4*x)).
a(n) = 3*A003683(n).
Given the 2 X 2 matrix M = [1,3; 3,1], a(n) = term (1,2) in M^n, n>0. - Gary W. Adamson, Aug 06 2010
From G. C. Greubel, Feb 17 2023: (Start)
a(n) = 2*a(n-1) + 8*a(n-2).
a(n) = 3*2^(n-1)*A001045(n).
a(n) = 2^(n-1)*A062510(n).
a(n) = (1/2)*A071930(n+1).
E.g.f.: (1/2)*(exp(4*x) - exp(-2*x)). (End)

A096977 a(n) = 4a(n-1) + 3a(n-2) - 14a(n-3) + 8a(n-4).

Original entry on oeis.org

0, 1, 2, 11, 36, 157, 598, 2447, 9672, 38913, 155194, 621683, 2484908, 9943269, 39765790, 159077719, 636281744, 2545185225, 10180624386, 40722730555, 162890456180, 651562756781, 2606249162982, 10425000380191, 41699994064216

Offset: 0

Paul Barry, Jul 17 2004

Original name was: A Jacobsthal summation.

The convolution of A024000 and A003683. Inverse binomial transform is A055275, with interpolated zeros.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,3,-14,8).

Cf. A001654.

[4*4^n/27-4*(-2)^n/27+n/9: n in [0..30]]; // Vincenzo Librandi, Jul 01 2011

LinearRecurrence[{4,3,-14,8},{0,1,2,11},30] (* Harvey P. Dale, Jul 01 2015 *)

a(n)=(4*4^n-4*(-2)^n+3*n)/27 \\ Charles R Greathouse IV, Jul 01 2011

G.f.: x*(1-2*x)/((1-x)^2*(1+2*x)*(1-4*x)).
a(n) = 4*4^n/27 - 4*(-2)^n/27 + n/9.
a(n) = Sum_{k=0..n} A001045(k)^2.
a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 5, 8, 5, 0, 1, 2, 6, 12, 16, 6, 0, 1, 2, 7, 16, 29, 32, 7, 0, 1, 2, 8, 20, 44, 70, 64, 8, 0, 1, 2, 9, 24, 61, 120, 169, 128, 9, 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10, 0, 1, 2, 11, 32, 101, 256, 547, 896, 985, 512, 11

Offset: 0

Ralf Stephan, Oct 13 2004

			Square array, A(n, k), begins as:
  0, 1, 2,  3,  4,   5,    6,    7,     8, ... A001477;
  0, 1, 2,  4,  8,  16,   32,   64,   128, ... A000079;
  0, 1, 2,  5, 12,  29,   70,  169,   408, ... A000129;
  0, 1, 2,  6, 16,  44,  120,  328,   896, ... A002605;
  0, 1, 2,  7, 20,  61,  182,  547,  1640, ... A015518;
  0, 1, 2,  8, 24,  80,  256,  832,  2688, ... A063727;
  0, 1, 2,  9, 28, 101,  342, 1189,  4088, ... A002532;
  0, 1, 2, 10, 32, 124,  440, 1624,  5888, ... A083099;
  0, 1, 2, 11, 36, 149,  550, 2143,  8136, ... A015519;
  0, 1, 2, 12, 40, 176,  672, 2752, 10880, ... A003683;
  0, 1, 2, 13, 44, 205,  806, 3457, 14168, ... A002534;
  0, 1, 2, 14, 48, 236,  952, 4264, 18048, ... A083102;
  0, 1, 2, 15, 52, 269, 1110, 5179, 22568, ... A015520;
  0, 1, 2, 16, 56, 304, 1280, 6208, 27776, ... A091914;
Antidiagonal triangle, T(n, k), begins as:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  2,  3;
  0,  1,  2,  4,  4;
  0,  1,  2,  5,  8,  5;
  0,  1,  2,  6, 12, 16,   6;
  0,  1,  2,  7, 16, 29,  32,   7;
  0,  1,  2,  8, 20, 44,  70,  64,   8;
  0,  1,  2,  9, 24, 61, 120, 169, 128,   9;
  0,  1,  2, 10, 28, 80, 182, 328, 408, 256,  10;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

Rows m: A001477 (m=0), A000079 (m=1), A000129 (m=2), A002605 (m=3), A015518 (m=4), A063727 (m=5), A002532 (m=6), A083099 (m=7), A015519 (m=8), A003683 (m=9), A002534 (m=10), A083102 (m=11), A015520 (m=12), A091914 (m=13).
Columns q: A000004 (q=0), A000012 (q=1), A009056 (q=2), A008586 (q=3).
Main diagonal gives A357502.

A099173:= func< n,k | (&+[n^j*Binomial(k,2*j+1): j in [0..Floor(k/2)]]) >;
[A099173(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

A[k_, n_]:= Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/(2 Sqrt[k])]; Table[A[k-n, n]//Simplify, {k, 0, 12}, {n, 0, k}]//Flatten (* Jean-François Alcover, Jan 21 2019 *)

A(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))

def A099173(n,k): return sum( n^j*binomial(k, 2*j+1) for j in range((k//2)+1) )
flatten([[A099173(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

A(n, k) = Sum_{i=0..floor(k/2)} n^i * C(k, 2*i+1) (array).
Recurrence: A(n, k) = 2*A(n, k-1) + (n-1)*A(n, k-2), with A(n, 0) = 0, A(n, 1) = 1.
T(n, k) = A(n-k, k) (antidiagonal triangle).
T(2*n, n) = A357502(n).
A(n, k) = ((1+sqrt(n))^k - (1-sqrt(n))^k)/(2*sqrt(n)). - Jean-François Alcover, Jan 21 2019

A083217 a(n) = (2*5^n + (-1)^n)/3.

Original entry on oeis.org

1, 3, 17, 83, 417, 2083, 10417, 52083, 260417, 1302083, 6510417, 32552083, 162760417, 813802083, 4069010417, 20345052083, 101725260417, 508626302083, 2543131510417, 12715657552083, 63578287760417, 317891438802083

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A003683 (without leading zero). Inverse binomial transform of A067411.

a(n) is the number of compositions of n when there are 3 types of 1 and 8 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A003683, A067411, A082412.

[(2*5^n +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Feb 17 2023

LinearRecurrence[{4,5},{1,3},30] (* Harvey P. Dale, Sep 18 2018 *)

from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1,3,4,5, lambda n: 0)
[next(it) for i in range(1,24)] # Zerinvary Lajos, Jul 03 2008

a(n) = (2*5^n + (-1)^n)/3.
G.f.: (1-x)/((1-5*x)*(1+x)).
E.g.f.: (2*exp(5*x) + exp(-x))/3
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*J(n-j+1) where J(n) = A001045(n). - Paul Barry, May 19 2006
a(0)=1, a(n) = 5*a(n-1) - 2 if n is odd, and a(n) = 5*a(n) + 2 if n is even. - Vincenzo Librandi, Nov 18 2010

A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

Offset: 1

Philippe Deléham, Aug 27 2005

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014
From Vladimir Shevelev, Feb 07 2014: (Start)
Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

			Starred terms in Pascal's triangle (A007318), read by rows:
1;
1*, 1;
1, 2*, 1;
1*, 3, 3*, 1;
1, 4*, 6, 4*, 1;
1*, 5, 10*, 10, 5*, 1;
1, 6*, 15, 20*, 15, 6*, 1;
1*, 7, 21*, 35, 35*, 21, 7*, 1;
1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
Triangle T(n,k) begins:
1;
2;
1,    3;
4,    4;
1,   10,  5;
6,   20,  6;
1,   21,  35,   7;
8,   56,  56,   8;
1,   36, 126,  84,  9;
10, 120, 252, 120, 10;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A109446.

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A099138 a(n) = 6^(n-1)*J(n), where J(n) = A001045(n).

Original entry on oeis.org

0, 1, 6, 108, 1080, 14256, 163296, 2006208, 23794560, 287214336, 3436494336, 41298398208, 495217981440, 5944792559616, 71324450021376, 855971764420608, 10271190988062720, 123257112966660096, 1479068428940476416

Offset: 0

Paul Barry, Sep 29 2004

In general k^(n-1)*J(n), where J(n) = A001045(n), is given by ((2*k)^n - (-k)^n)/(3*k) with g.f. x/((1+k*x)*(1-2*k*x)).

G. C. Greubel, Table of n, a(n) for n = 0..925
Index entries for linear recurrences with constant coefficients, signature (6,72).

Cf. A001045, A003683, A080424, A091903, A091904.

[(12^n - (-6)^n)/18: n in [0..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{6,72}, {0,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

[(12^n - (-6)^n)/18 for n in range(41)] # G. C. Greubel, Feb 18 2023

G.f.: x/((1+6*x)*(1-12*x)).
a(n) = 6^(n-1)*Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) * 2^k.
a(n) = (12^n - (-6)^n)/18.
a(n) = 6^(n-1)*A001045(n).
E.g.f.: (1/18)*(exp(12*x) - exp(-6*x)). - G. C. Greubel, Feb 18 2023

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