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.

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A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

Programs

  • Magma
    [n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012
  • Mathematica
    a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)
  • PARI
    A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009
  • PARI
    x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018
  • Sage
    [lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

Formula

a(n) = 5*a(n-1) + 8*a(n-2).
G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

Extensions

More precise definition by M. F. Hasler, Mar 06 2009

A193727 Mirror of the triangle A193726.

Original entry on oeis.org

1, 2, 1, 10, 9, 2, 50, 65, 28, 4, 250, 425, 270, 76, 8, 1250, 2625, 2200, 920, 192, 16, 6250, 15625, 16250, 9000, 2800, 464, 32, 31250, 90625, 112500, 77500, 32000, 7920, 1088, 64, 156250, 515625, 743750, 612500, 315000, 103600, 21280, 2496, 128
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

This triangle is obtained by reversing the rows of the triangle A193726.
Triangle T(n,k), read by rows, given by (2,3,0,0,0,0,0,0,0,...) DELTA (1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
     1;
     2,    1;
    10,    9,    2;
    50,   65,   28,   4;
   250,  425,  270,  76,   8;
  1250, 2625, 2200, 920, 192; 16;

Links

Crossrefs

Cf. A011782, A015535, A020699, A081040, A084938.
Cf. A107839, A133494, A169634, A193722, A193726.

Programs

  • Magma
    function T(n, k) // T = A193727
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 5*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023
  • Mathematica
    (* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 5*T[n-1, k] + 2*T[n-1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)
  • SageMath
    def T(n, k): # T = A193727
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 5*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

Formula

T(n,k) = A193726(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 5*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-3*x-x*y)/(1-5*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A020699(n).
T(n, 1) = A081040(n-1).
T(n, n) = A011782(n).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = 2*A015535(n) + A015535(n-1) + (1/2)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 2*A107839(n-1) - A107839(n-2) + (1/2)*[n=0]. (End)

A233020 Number of n X 2 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, vertically, diagonally or antidiagonally, and top left element zero.

Original entry on oeis.org

3, 15, 81, 435, 2337, 12555, 67449, 362355, 1946673, 10458075, 56183721, 301834755, 1621541217, 8711375595, 46799960409, 251422553235, 1350712686993, 7256408541435, 38983468081161, 209430157488675, 1125117723605697
Offset: 1

Views

Author

R. H. Hardin, Dec 03 2013

Keywords

Examples

			Some solutions for n=5:
..0..2....0..0....0..0....0..1....0..0....0..0....0..1....0..2....0..0....0..0
..2..2....1..1....2..2....1..1....1..1....0..2....0..1....2..0....1..1....0..0
..3..2....1..1....2..3....0..0....0..0....2..2....0..1....2..0....1..1....1..1
..3..2....1..3....3..3....2..2....1..1....2..3....0..1....0..2....3..3....3..1
..2..3....3..3....3..2....0..0....3..1....2..3....0..1....0..2....2..3....3..1

Links

Crossrefs

Column 2 of A233026.

Formula

Empirical: a(n) = 5*a(n-1) + 2*a(n-2).
Conjectures from Colin Barker, Oct 06 2018: (Start)
G.f.: 3*x / (1 - 5*x - 2*x^2).
a(n) = sqrt(3/11)*(((5+sqrt(33))/2)^n - ((5-sqrt(33))/2)^n) = 3*A015535(n).
(End)

A201002 Primes in the Lucas U(5,-2) sequence.

Original entry on oeis.org

5, 22483, 450237562331, 12994489360387, 260223602644254517259, 7510419207086038206643, 3615797716913419563139283609508083, 2089820990182844019889959238674869772826787
Offset: 1

Views

Author

R. J. Mathar, Jan 08 2013

Keywords

Comments

The Lucas U(5,-2) sequence is A015535. This sequence here contains U(5,-2,n) at n = 2, 7, 17, 19, 29, 31, 47, 59, ...

Links

Crossrefs

Cf. A015535.

A106835 Expansion of 2*x^2*(-2+9*x+3*x^2)/((2*x^2+5*x-1)*(2*x^2-5*x+1)).

Original entry on oeis.org

0, 4, 22, 114, 590, 3066, 15998, 83786, 440270, 2320314, 12260382, 64931114, 344562670, 1831630106, 9751275838, 51981730186, 277413656590, 1481919831674, 7922862005342, 42388551182314, 226923616315950, 1215450062928346
Offset: 1

Views

Author

Roger L. Bagula, May 30 2005

Keywords

Links

Programs

  • Mathematica
    M = {{0, 0, 0, 2}, {1, 5, 0, 0}, {0, 2, 0, 0}, {0, 0, 1, 5}};
v[1] = {0, 1, 1, 2};
v[n_] := v[n] = M.v[n - 1];
a = Table[v[n][[1]], {n, 1, 50}]
LinearRecurrence[{10,-25,0,4},{0,4,22,114},30] (* Harvey P. Dale, Jul 05 2025 *)

Formula

From R. J. Mathar_, Apr 07 2009: (Start)
G.f.: 2*x^2*(-2+9*x+3*x^2)/((2*x^2+5*x-1)*(2*x^2-5*x+1)).
a(n)=(-7*A107839(n)+33*A107839(n-1)+A015535(n+1)-3*A015535(n))/4.
a(n) = 10*a(n-1)-25*a(n-2)+4*a(n-4). (End)

Extensions

Edited by Associate Editors of the OEIS, Apr 05 2009
Meaningful name from Joerg Arndt, Dec 26 2022

A321045 a(n) is the value of the first entry in the matrix A^n where A = [{1,2,3}, {4,5,6}, {7,8,9}].

Original entry on oeis.org

1, 1, 30, 468, 7560, 121824, 1963440, 31644432, 510008400, 8219725776, 132476037840, 2135095631568, 34411003154640, 554596768687824, 8938349587100880, 144057985642894032, 2321760077211226320, 37419444899740487376, 603083354885909384400, 9719800331483969538768
Offset: 0

Views

Author

Peter James Foreman, Oct 26 2018

Keywords

Links

Programs

  • PARI
    a(n) = ([1,2,3; 4,5,6; 7,8,9]^n)[1,1]; \\ Michel Marcus, Oct 26 2018

Formula

a(n) = (1/(132*2^n)) * ((55-7*sqrt(33))*(15+3*sqrt(33))^n + (55+7*sqrt(33))*(15-3*sqrt(33))^n).
G.f.: (3*x^2 + 14*x - 1)/(18*x^2 + 15*x - 1).
3^n | a(n+1). - R. J. Mathar, Jan 09 2020
Let b(n)=3^n*A015535(n) = 1,15,243,3915,.. (n>=0). Then 6*a(n) = 5*b(n)-69*b(n-1), n>0. - R. J. Mathar, Aug 19 2022
Previous Showing 11-16 of 16 results.

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