A105476 -id:A105476 - cp's OEIS frontend

A384633 Expansion of (1+x-2x^2-2x^3) / (1-6x^2-4x^3+2*x^4).

1, 1, 4, 8, 26, 62, 180, 460, 1276, 3356, 9136, 24320, 65688, 175752, 473136, 1268624, 3410448, 9152784, 24590912, 66021248, 177335712, 476185568, 1278917440, 3434413760, 9223575488, 24769781184, 66521273088, 178644161536, 479759612288, 1288410499200

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 0 in the following graph:
     2
    / \
 0-1---3
    \ /
     4.
Also, for n>=1, the number of walks of length n-1 starting at vertex 1 in the same graph.

			a(3)=8 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, 0-1-3-2, 0-1-3-4, 0-1-4-1, 0-1-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,6,4,-2).

Cf. A384634 (vertices 2, 4), A384635 (vertex 3), A005824 (missing edge {1,3}), A105476 (missing edge {1,2}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+x-2*x^2-2*x^3) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]

A105963 Expansion of (1+4x)/(1-x-3x^2).

Original entry on oeis.org

1, 5, 8, 23, 47, 116, 257, 605, 1376, 3191, 7319, 16892, 38849, 89525, 206072, 474647, 1092863, 2516804, 5795393, 13345805, 30731984, 70769399, 162965351, 375273548, 864169601, 1989990245, 4582499048, 10552469783, 24299966927

Offset: 0

Creighton Dement, Apr 28 2005

Inversion of the periodic sequence with initial period (1,4,-1,-4). Sequence appears to have the property: for m > n, if s divides both a(n) and a(m) then s also divides a(2*m-n). E.g., 23 divides both a(3) = 23 and a(25) = 1989990245; 23 also divides a(2*25-3) = a(47) = 185518234185384428 = (2)^2*(23)*(131)*(15393149202239).

Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[.5'k + .5k' + 2'kk' + 2e]

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

Cf. A006130, A105476, A274977.

a:=[1,5];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020

I:=[ 1,5]; [n le 2 select I[n] else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013

seq(coeff(series((1+4*x)/(1-x-3*x^2), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Jan 15 2020

CoefficientList[Series[(1+4x)/(1-x-3x^2), {x,0,40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
Table[Round[3^((n-1)/2)*(Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] + 4*Fibonacci[n, 1/Sqrt[3]] )], {n,0,40}] (* G. C. Greubel, Jan 15 2020 *)

Vec((1+4*x)/(1-x-3*x^2)+O(x^40)) \\ Charles R Greathouse IV, Sep 27 2012

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+4*x)/(1-x-3*x^2) ).list()
A077952_list(30) # G. C. Greubel, Jan 15 2020

a(n) = A006130(n) + 4*A006130(n-1) = A006130(n+1) + A006130(n-1). - R. J. Mathar, Dec 12 2009
From Colin Barker, May 01 2019: (Start)
a(n) = (2^(-1-n)*((1-sqrt(13))^n*(-9+sqrt(13)) + (1+sqrt(13))^n*(9+sqrt(13)))) / sqrt(13).
a(n) = a(n-1) + 3*a(n-2) for n > 1. (End)
a(n) = 3^((n-1)/2)*( sqrt(3)*Fibonacci(n+1, 1/sqrt(3)) + 4*Fibonacci(n, 1/sqrt(3)) ). - G. C. Greubel, Jan 15 2020

A131406 3A128174 - 2A000012(signed).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Gary W. Adamson, Jul 07 2007

Row sums = A032766, congruent to {0,1} mod 3: (1, 3, 4, 6, 7, 9, 10,...).

Sequence array for the expansion of (1+2x)/(1-x^2). A105476 is an eigensequence. [From Paul Barry, Nov 03 2010]

			First few rows of the triangle are:
1;
2, 1;
1, 2, 1;
2, 1, 2, 1;
1, 2, 1, 2, 1;
2, 1, 2, 1, 2, 1;
...

Cf. A128174, A000012, A032766.

T[n_, k_] := Mod[n-k, 2]+1; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2016 *)

3*A128174 - 2*A000012(signed + - + 1 by columns). (1, 2, 1, 2, 1,...) in every column.

Triangle T(n,k)=if(k<=n,(3-(-1)^(n-k))/2). [From Paul Barry, Nov 03 2010]

A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 8, 8, 5, 1, 5, 12, 18, 15, 8, 1, 6, 18, 32, 39, 28, 13, 1, 7, 24, 53, 77, 80, 51, 21, 1, 8, 32, 80, 142, 176, 160, 92, 34, 1, 9, 40, 116, 234, 352, 384, 312, 164, 55, 1, 10, 50, 160, 370, 632, 830, 812, 598, 290, 89, 1, 11, 60, 215

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 2: 1,2,3,4,5,6,7,8,...
Row sums: A006138.
Alternating row sums: signed Fibonacci numbers.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

			First five rows:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 4, 3;
  1, 4, 8, 8, 5;
First three polynomials u(n,x):
  1
  1 + x
  1 + 2x + 2x^2.
From _Philippe Deléham_, Mar 28 2012: (Start)
(1, 0, 0, -1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1;
  1, 0;
  1, 1, 0;
  1, 2, 2, 0;
  1, 3, 4, 3, 0;
  1, 4, 8, 8, 5, 0; (End)

Cf. A210790, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210789 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210790 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A006138 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A105476 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 28 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1+x-y*x-y*x^2-y^2*x^2)/(1-y*x-y*x^2-x^2-y^2*x^2).
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 5, 5, 1, 4, 10, 10, 8, 1, 6, 14, 24, 20, 13, 1, 6, 21, 38, 52, 38, 21, 1, 8, 27, 65, 96, 109, 71, 34, 1, 8, 36, 92, 176, 224, 220, 130, 55, 1, 10, 44, 136, 280, 446, 500, 434, 235, 89, 1, 10, 55, 180, 440, 772, 1066, 1074, 839, 420, 144, 1

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: 2,2,4,4,6,6,8,8,...
Row sums: A105476.
Alternating row sums: signed Fibonacci numbers.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

			First five rows:
  1;
  1,  2;
  1,  2,  3;
  1,  4,  5,  5;
  1,  4, 10, 10,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 2x + 3x^2.
From _Philippe Deléham_, Mar 28 2012: (Start)
(1, 0, -1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  3,  0;
  1,  4,  5,  5,  0;
  1,  4, 10, 10,  8,  0; (End)

Cf. A210789, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210789 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210790 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A006138 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A105476 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 28 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1+x-y*x-y^2*x^2)/(1-y*x-x^2-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A384634 Expansion of (1+2x+x^2) / (1-6x^2-4x^3+2x^4).

Original entry on oeis.org

1, 2, 7, 16, 48, 120, 338, 880, 2412, 6392, 17316, 46240, 124640, 333920, 898168, 2409600, 6475408, 17382432, 46694512, 125377024, 336745984, 904275328, 2428594976, 6521881856, 17515179200, 47037120384, 126321412672, 339239675392, 911046599168, 2446649462272

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 2 in the following graph:
     2
    / \
 0-1---3
    \ /
     4.

			a(2)=7 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,6,4,-2).

Cf. A384633 (vertices 0, 1), A384635 (vertex 3), A005824 (missing edge {1,3}), A105476 (missing edge {1,2}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]

A384635 Expansion of (1+3x+2x^2) / (1-6x^2-4x^3+2*x^4).

Original entry on oeis.org

1, 3, 8, 22, 58, 158, 420, 1136, 3036, 8180, 21920, 58952, 158168, 425032, 1140976, 3064960, 8229648, 22103600, 59355776, 159410272, 428089760, 1149677536, 3087468096, 8291603712, 22267339200, 59800139584, 160595513856, 431286986880, 1158238963072

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 3 in the following graph:
     2
    / \
 0-1---3
    \ /
     4.

			a(2)=8 because we have the walks 3-1-0, 3-1-2, 3-1-3. 3-1-4, 3-2-1, 3-2-3, 3-4-1, 3-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,6,4,-2).

Cf. A384633 (vertices 0, 1), A384634 (vertices 2, 4), A005824 (missing edge {1,3}), A105476 (missing edge {1,2}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]

A140168 Triangle read by rows, iterates of X * [1,0,0,0,...]; where X = an infinite bidiagonal matrix with [2, -1, 2, -1, 2, ...] in the main diagonal, [1, 1, 1, ...] in the subdiagonal and rest zeros.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 8, 3, 3, 1, 16, 5, 9, 2, 1, 32, 11, 23, 7, 4, 1, 64, 21, 57, 16, 15, 3, 1, 128, 43, 135, 41, 46, 12, 5, 1, 256, 85, 313, 94, 133, 34, 22, 4, 1, 512, 171, 711, 219, 360, 99, 78, 18, 6, 1, 1024, 341, 1593, 492, 939, 261, 255, 60, 30, 5, 1, 2048, 683, 3527, 1101

Offset: 0

Gary W. Adamson, May 10 2008

Row sums = A105476 starting (1, 3, 6, 15, 33, 78, 177, ...).

			First few rows of the triangle:
    1;
    2,   1;
    4,   1,   1;
    8,   3,   3,   1;
   16,   5,   9,   2,   1;
   32,  11,  23,   7,   4,  1;
   64,  21,  57,  16,  15,  3,  1;
  128,  43, 135,  41,  46, 12,  5,  1;
  256,  85, 313,  94, 133, 34, 22,  4, 1;
  512, 171, 711, 219, 360, 99, 78, 18, 6, 1;
  ...

Cf. A105476.

Triangle read by rows, iterates of X * [1,0,0,0,...]; where X = an infinite bidiagonal matrix with [2, -1, 2, -1, 2, ...] in the main diagonal, [1, 1, 1, ...] in the subdiagonal and rest zeros.

A238988 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, 0, 2, 1, -1, -1, 1, 2, 1, 0, -1, -2, 1, 3, 1, 1, 0, -4, -2, 3, 3, 1, 1, 1, -2, -4, -2, 3, 4, 1, 0, 1, 3, -2, -9, -2, 6, 4, 1, -1, 0, 6, 3, -9, -9, 0, 6, 5, 1, -1, -1, 3, 6, 3, -9, -15, 0, 10, 5, 1, 0, -1, -4, 3, 18, 3, -24, -15, 5, 10, 6, 1

Offset: 0

Philippe Deléham, Mar 07 2014

T(n,0) = T(n+1,1) = A010892(n), T(n+2,2) = T(n+3,3) = A099254(n), T(n+4,4) = T(n+5,5) = A128504(n).

Triangle T(n,k) = A101950(n - floor((k+1)/2),floor(k/2)).

			Triangle begins:
1;
1, 1;
0, 1, 1;
-1, 0, 2, 1;
-1, -1, 1, 2, 1;
0, -1, -2, 1, 3, 1;
1, 0, -4, -2, 3, 3, 1;
1, 1, -2, -4, -2, 3, 4, 1;
0, 1, 3, -2, -9, -2, 6, 4, 1;
-1, 0, 6, 3, -9, -9, 0, 6, 5, 1;
-1, -1, 3, 6, 3, -9, -15, 0, 10, 5, 1;
0, -1, -4, 3, 18, 3, -24, -15, 5, 10, 6, 1;
1, 0, -8, -4, 18, 18, -6, -24, -20, 5, 15, 6, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A101950

nmax=11; Flatten[CoefficientList[Series[CoefficientList[Series[(1 + x*y)/(1 - x + x^2 - x^2*y^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 14 2017 *)

G.f.: (1 + x*y)/(1 - x + x^2 - x^2*y^2).
T(n,k) = T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A010892(n), A040000(n), A105476(n+1) for x = -1, 0, 1, 2 respectively.

A384614 Expansion of (1+x+x^2) / (1-x-3*x^2).

Original entry on oeis.org

1, 2, 6, 12, 30, 66, 156, 354, 822, 1884, 4350, 10002, 23052, 53058, 122214, 281388, 648030, 1492194, 3436284, 7912866, 18221718, 41960316, 96625470, 222506418, 512382828, 1179902082, 2717050566, 6256756812, 14407908510, 33178178946, 76401904476, 175936441314

Offset: 0

Sean A. Irvine, Jun 04 2025

Number of walks of length n starting at vertex 2 in the following graph:
  0-1
    |\
    | 2
    |/
  4-3.

			a(2)=6 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-3-1, 2-3-2, 2-3-4.

Sean A. Irvine, Walks on Graphs.

Cf. A105476 (vertices 0, 1), A382683 (missing edge {4,3}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|1|0>, <0|1|0|1|0>, <0|1|1|0|1>, <0|0|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+x+x^2) / (1-x-3*x^2), {x, 0, 32}], x]

cp's OEIS Frontend

A384633 Expansion of (1+x-2*x^2-2*x^3) / (1-6*x^2-4*x^3+2*x^4).

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A105963 Expansion of (1+4*x)/(1-x-3*x^2).

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A131406 3*A128174 - 2*A000012(signed).

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A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section.

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A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.

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A384634 Expansion of (1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4).

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A384635 Expansion of (1+3*x+2*x^2) / (1-6*x^2-4*x^3+2*x^4).

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A140168 Triangle read by rows, iterates of X * [1,0,0,0,...]; where X = an infinite bidiagonal matrix with [2, -1, 2, -1, 2, ...] in the main diagonal, [1, 1, 1, ...] in the subdiagonal and rest zeros.

A384633 Expansion of (1+x-2x^2-2x^3) / (1-6x^2-4x^3+2*x^4).

A105963 Expansion of (1+4x)/(1-x-3x^2).

A131406 3A128174 - 2A000012(signed).

A384634 Expansion of (1+2x+x^2) / (1-6x^2-4x^3+2x^4).

A384635 Expansion of (1+3x+2x^2) / (1-6x^2-4x^3+2*x^4).