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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A189800 a(n) = 6*a(n-1) + 8*a(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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55
Offset: 0

Views

Author

Philippe Deléham, Dec 18 2006

Keywords

Comments

Rising and falling diagonals are A008999, A124400.
Subtriangle of triangle given by (1, 1, -1, 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, Feb 17 2012
Jointly generated with A209130 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

Examples

			Triangle begins:
  1;
  2, 1;
  3, 4, 2;
  4, 10, 10, 3;
  5, 20, 31, 20, 5;
  6, 35, 76, 78, 40, 8;
  7, 56, 161, 232, 184, 76, 13;
  8, 84, 308, 582, 636, 406, 142, 21;
  9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
  10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  2, 1, 0
  3, 4, 2, 0
  4, 10, 10, 3, 0
  5, 20, 31, 20, 5, 0
  6, 35, 76, 78, 40, 8, 0
  7, 56, 161, 232, 184, 76, 13, 0

Crossrefs

Cf. A209130.
Cf. A008999, A124400, A084938, A209130.
Cf. A254006, A000012, A000027, A000244, A015530, A015544.
Cf. A001629, A000045, A000292, A051747.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
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[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)
(* Clark Kimberling, Mar 05 2012 *)

Formula

Sum_{k=0..n} x^k*T(n,k) = A254006(n), A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for x = -2, -1, 0, 1, 2, 3 respectively.
T(n,n-1) = 2*A001629(n+1) for n>=1.
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n,0) = n+1.
T(n,1) = A000292(n) for n>=1.
T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1.
G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012
Showing 1-2 of 2 results.

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