cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295
Offset: 0

Views

Author

Philippe Deléham, Nov 28 2008

Keywords

Comments

Unsigned version of A152185.
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
Inverse binomial transform of A015449 (without the first leading 1).
(End)

Links

Crossrefs

Cf. A015523, A072263, A072264, A179606, A197189.

Programs

  • Mathematica
    LinearRecurrence[{3,5},{1,5},40] (* Harvey P. Dale, May 03 2013 *)

Formula

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010
G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A207636 Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.

Original entry on oeis.org

1, 3, 2, 6, 7, 2, 12, 20, 11, 2, 24, 52, 42, 15, 2, 48, 128, 136, 72, 19, 2, 96, 304, 400, 280, 110, 23, 2, 192, 704, 1104, 960, 500, 156, 27, 2, 384, 1600, 2912, 3024, 1960, 812, 210, 31, 2, 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2, 1536, 7936
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

Comments

As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

Examples

			First five rows:
   1;
   3,  2;
   6,  7,  2;
  12, 20, 11,  2;
  24, 52, 42, 15,  2;

Crossrefs

Cf. A207635.
Cf. A084938, A003945, A040000.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207635 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207636 *)

Formula

u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 3, T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x+y*x)/(1-2*x-y*x).
Sum_{k=0..n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (End)

A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 10, 5, 1, 10, 22, 23, 11, 1, 15, 40, 65, 60, 21, 1, 21, 65, 145, 195, 137, 43, 1, 28, 98, 280, 490, 518, 322, 85, 1, 36, 140, 490, 1050, 1484, 1372, 723, 171, 1, 45, 192, 798, 2016, 3570, 4368, 3447, 1624, 341, 1, 55, 255, 1230, 3570
Offset: 1

Views

Author

Clark Kimberling, Feb 29 2012

Keywords

Comments

Alternating row sums: 1,0,1,0,1,0,1,0,...

Examples

			First five rows:
1
1...1
1...3....3
1...6....10...5
1...10...22...23...11
First five polynomials u(n,x):
1
1 + x
1 + 3x + 3x^2
1 + 6x + 10x^2 + 5x^3
1 + 10x + 22x^2 + 23x^3 + 11x^4

Crossrefs

Cf. A208525.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]  (* A208524 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208525 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
Table[u[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)
Table[v[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)

Formula

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208525 Triangle of coefficients of polynomials v(n,x) jointly generated with A208524; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 7, 5, 4, 12, 18, 11, 5, 18, 42, 49, 21, 6, 25, 80, 135, 116, 43, 7, 33, 135, 295, 381, 279, 85, 8, 42, 210, 560, 966, 1050, 638, 171, 9, 52, 308, 966, 2086, 2996, 2724, 1453, 341, 10, 63, 432, 1554, 4032, 7182, 8688, 6921, 3240, 683, 11, 75
Offset: 1

Views

Author

Clark Kimberling, Feb 29 2012

Keywords

Comments

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...

Examples

			First five rows:
1
2...3
3...7....5
4...12...18...11
5...18...42...49...21
First five polynomials v(n,x):
1
2 + 3x
3 + 7x + 5x^2
4 + 12x + 18x^2 + 11x^3
5 + 18x + 42x^2 + 49x^3 + 21x^4

Crossrefs

Cf. A208524.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]  (* A208524 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208525 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
Table[u[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)
Table[v[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)

Formula

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Showing 1-4 of 4 results.

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