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 1-5 of 5 results.

A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

Original entry on oeis.org

0, 3, 2, 1, 2, 0, 2, 1, -2, 3, 0, -4, 4, -1, -6, 5, -2, -8, 6, -3, -10, 7, -4, -12, 8, -5, -14, 9, -6, -16, 10, -7, -18, 11, -8, -20, 12, -9, -22, 13, -10, -24, 14, -11, -26, 15, -12, -28, 16, -13, -30, 17, -14, -32, 18, -15, -34, 19, -16, -36, 20, -17, -38, 21, -18, -40
Offset: 0

Views

Author

Paul Curtz, Mar 09 2008

Keywords

Comments

The entries are the coefficients in a family of Jacobsthal recurrences: a(n)=k*a(n-1)+(3-k)*a(n-2)+(2-2k)*a(n-3).
Examples for k=0 are in A001045 and A113954. Examples for k=1 are A001045, A078008.
Examples for k=2 are A000975, A087288, A084639, A000012 and A001045.
Examples for k=3 are A045883, A059570. Examples for k=4 are A094705 and A015518.

Examples

			The triples (k,3-k,2-2k) are (0,3,2), (1,2,0), (2,1,-2), (3,0,-4),...

Links

Programs

  • Mathematica
    CoefficientList[Series[x*(3 + 2*x + x^2 - 4*x^3 - 4*x^4)/((x - 1)^2*(1 + x + x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)
Table[{n,3-n,2-2n},{n,0,30}]//Flatten (* or *) LinearRecurrence[ {0,0,2,0,0,-1},{0,3,2,1,2,0},100] (* Harvey P. Dale, Jun 23 2019 *)
  • PARI
    x='x+O('x^50); Vec(x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x +x^2 )^2)) \\ G. C. Greubel, Sep 28 2017

Formula

From R. J. Mathar, Feb 25 2009: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x+x^2)^2). (End)

Extensions

Edited by R. J. Mathar, Jun 28 2008

A103196 a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).

Original entry on oeis.org

1, 2, 3, 8, 13, 30, 55, 116, 225, 458, 907, 1824, 3637, 7286, 14559, 29132, 58249, 116514, 233011, 466040, 932061, 1864142, 3728263, 7456548, 14913073, 29826170, 59652315, 119304656, 238609285, 477218598, 954437167
Offset: 0

Views

Author

Creighton Dement, Mar 18 2005

Keywords

Comments

A floretion-generated sequence relating to the Jacobsthal sequence A001045 as well as to A095342 (Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1). (a(n)) may be seen as the result of a certain transform of the natural numbers (see program code).
Floretion Algebra Multiplication Program, FAMP Code: 4jesleftforseq[A*B] with A = + 'i + 'j + i' + j' + 'ii' + 'jj' + 'ij' + 'ji' + e and B = - .25'i + .25'j + .25'k + .25i' - .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' - .25'ki' - .25'kj' - .25e; 1vesforseq[A*B](n) = n, ForType: 1A.

Links

Crossrefs

Cf. A001045, A095342, A053088, A034299, A052953, A026644.

Programs

  • Mathematica
    Table[(2^(n+3)-(-1)^n (3n-1))/9,{n,0,30}] (* or *) LinearRecurrence[ {0,3,2},{1,2,3},40] (* Harvey P. Dale, Jul 09 2018 *)

Formula

G.f. (2x+1)/((1-2x)(x+1)^2); Superseeker results: a(n) + a(n+1) = A001045(n+3); a(n+1) - a(n) = A095342(n+1); a(n+2) - a(n+1) - a(n) = A053088(n+1) = A034299(n+1) - A034299(n); a(n) + 2a(n+1) + a(n+2) = 2^(n+3); a(n+2) - 2a(n+1) + a(n) = A053088(n+1) - A053088(n); a(n+2) - a(n) = A001045(n+4) - A001045(n+3) = A052953(n+3) - A052953(n+2) = A026644(n+2) - A026644(n+1);
a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)phi(phi(3^k))}; a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)(2*3^k/9+C(1, k)/3+4*C(0, k)/9)}; a(n)=sum{k=0..n+2, J(n-k+3)((-1)^(k+1)-2C(1, k)+4C(0, k))} where J(n)=A001045(n); a(n)=A113954(n+2). - Paul Barry, Nov 09 2005

A210870 Triangle of coefficients of polynomials u(n,x) jointly generated with A210871; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 6, 5, 5, 1, 5, 8, 12, 8, 8, 1, 6, 12, 17, 23, 13, 13, 1, 7, 15, 29, 33, 43, 21, 21, 1, 8, 20, 38, 64, 63, 79, 34, 34, 1, 9, 24, 56, 86, 136, 117, 143, 55, 55, 1, 10, 30, 70, 140, 187, 279, 214, 256, 89, 89, 1, 11, 35, 95, 180, 332
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

In row n, for n>1, the first two terms are 1 and n-1, and the last two are F(n) and F(n), where F = A000045 (Fibonacci numbers).
Row sums: A000975
Alternating row sums: A113954
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
1...1
1...2...2
1...3...3...3
1...4...6...5....5
1...5...8...12...8...8
First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2.

Crossrefs

Cf. A210871, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A210870 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210871 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]   (* A113954 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)

Formula

u(n,x)=u(n-1,x)+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.

A210871 Triangle of coefficients of polynomials v(n,x) jointly generated with A210870; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 7, 3, 8, 1, 4, 5, 15, 5, 13, 1, 3, 12, 10, 30, 8, 21, 1, 5, 9, 31, 20, 58, 13, 34, 1, 4, 18, 22, 73, 38, 109, 21, 55, 1, 6, 14, 54, 51, 162, 71, 201, 34, 89, 1, 5, 25, 40, 145, 111, 344, 130, 365, 55, 144, 1, 7, 20, 85, 105, 361, 233
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

Row n, for n>2, starts with 1 and A028242(n) and ends with F(n-1) and F(n+1), where F=A000045 (Fibonacci numbers).
Row sums: A001045
Alternating row sums: A077925
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
1...2
1...1...3
1...3...2....5
1...2...7....3....8
1...4...5....15...5...13
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

Crossrefs

Cf. A210870, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A210870 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210871 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A113954 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

Formula

u(n,x)=u(n-1,x)+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.

A139790 a(n) = (5*2^(n+2) - 3*n*2^n - 2*(-1)^n) / 18.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 7, -7, -57, -199, -569, -1479, -3641, -8647, -20025, -45511, -101945, -225735, -495161, -1077703, -2330169, -5009863, -10718777, -22835655, -48467513, -102527431, -216239673, -454848967, -954437177, -1998352839, -4175662649, -8709239239
Offset: 0

Views

Author

Paul Curtz, May 21 2008

Keywords

Comments

Binomial transform of 1,1,0,1,-2,3,-8,13,-30,... (see A113954 and A103196). - R. J. Mathar, Feb 11 2010

Links

Crossrefs

Cf. A136298, A136299.

Programs

  • Magma
    [(5*2^(n+2)-3*n*2^n-2*(-1)^n) / 18: n in [0..35]]; // Vincenzo Librandi, Aug 09 2011
  • Mathematica
    LinearRecurrence[{3,0,-4},{1,2,3},40] (* Harvey P. Dale, May 27 2018 *)

Formula

a(n+1) - 2*a(n) = -A001045(n).
G.f.: (1 - x - 3*x^2)/((1+x)*(1-2*x)^2).
a(n) = 3*a(n-1) - 4*a(n-3).

Extensions

Definition replaced with closed formula by R. J. Mathar, Feb 11 2010
Showing 1-5 of 5 results.

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