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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A134658 Triangle read by rows, giving coefficients of extended Jacobsthal recurrence.

Original entry on oeis.org

3, 1, 2, 2, -1, 2, 3, -3, 1, 2, 4, -6, 4, -1, 2, 5, -10, 10, -5, 1, 2, 6, -15, 20, -15, 6, -1, 2, 7, -21, 35, -35, 21, -7, 1, 2, 8, -28, 56, -70, 56, -28, 8, -1, 2, 9, -36, 84, -126, 126, -84, 36, -9, 1, 2, 10, -45, 120, -210, 252, -210, 120, -45, 10, -1, 2
Offset: 0

Views

Author

Paul Curtz, Feb 01 2008

Keywords

Comments

Sequence identical to half its p-th differences from the second term.
This sequence is the second of a family after A135356.
This triangle looks like a Pascal's triangle without first column, and with signs and with additional right diagonal consisting of 2's. - Michel Marcus, Apr 07 2019

Examples

			Triangle begins
3;                : A000244 = 1, 3, 9, 27, ... is the main sequence
1, 2;             : A001045 = 0, 1, 1, 3, ... is the main sequence
2, -1, 2;         : 0, 0, (A007910 = 1, 2, 3, ... ) is the main sequence
3, -3, 1, 2;      : 0, 0, 0, 1, 3, 6, 10, 17, ... is the main sequence
4, -6, 4, -1, 2;  : A134987 = 0, 0, 0, 0, 1, ... is the main sequence
...
See signatures of linear recurrence of corresponding sequences.

Crossrefs

Cf. A000244, A001045, A007910, A134977 (sum of antidiagonals), A134987, A135356.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = Which[n == 0, 3, k == n, 2, k == 0, n, k == n-1, (-1)^k, True, T[n-1, k] - T[n-1, k-1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 06 2019 *)

Formula

Every row sums to 3. - Jean-François Alcover, Apr 04 2019 (further to a remark e-mailed by Paul Curtz).

Extensions

In agreement with author, T(0, 0) = 3 and offset 0 by Michel Marcus, Apr 06 2019

A135541 a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3), with a(0) = 2, a(1) = 2.

Original entry on oeis.org

0, 2, 7, 12, 21, 44, 91, 180, 357, 716, 1435, 2868, 5733, 11468, 22939, 45876, 91749, 183500, 367003, 734004, 1468005, 2936012, 5872027, 11744052, 23488101, 46976204, 93952411, 187904820, 375809637, 751619276, 1503238555, 3006477108
Offset: 0

Views

Author

Paul Curtz, Feb 22 2008

Keywords

Links

Crossrefs

Cf. A007909, A007910, A010712, A016029, A134658.

Programs

  • Magma
    I:=[0, 2, 7]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 17 2012
  • Mathematica
    LinearRecurrence[{2,-1,2},{0,2,7},40] (* Vincenzo Librandi, Jun 17 2012 *)

Formula

From R. J. Mathar, Feb 23 2008: (Start)
O.g.f.: -7/(5*(2x-1)) - (4x+7)/(5*(x^2+1)).
a(n) = (7*2^n - (-1)^floor(n/2)*A010712(n+1))/5. (End)
E.g.f.: (1/5)*(7*cosh(2*x) + 7*sinh(2*x) - 7*cos(x) - 4*sin(x)). - G. C. Greubel, Oct 18 2016

Extensions

More terms from R. J. Mathar, Feb 23 2008

A161204 a(0)=2. a(n+1) = 2*a(n) + period 4: repeat -5,1,3,1.

Original entry on oeis.org

2, -1, -1, 1, 3, 1, 3, 9, 19, 33, 67, 137, 275, 545, 1091, 2185, 4371, 8737, 17475, 34953, 69907, 139809, 279619, 559241, 1118483, 2236961, 4473923, 8947849, 17895699, 35791393, 71582787, 143165577, 286331155, 572662305, 1145324611, 2290649225, 4581298451, 9162596897
Offset: 0

Views

Author

Paul Curtz, Jan 20 2011

Keywords

Links

Programs

  • Magma
    I:=[2, -1, -1, 1]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 17 2012
  • Maple
    A000034 := proc(n) if type(n,'even') then 1 ; else 2 ; end if; end proc:
A161204 := proc(n) 4*(-1)^floor((n+1)/2)*A000034(n+1)/5+2^n/15+(-1)^n/3 ; end proc: # R. J. Mathar, Jan 26 2011
  • Mathematica
    CoefficientList[Series[(-2+3*x-x^3+2*x^2)/((2*x-1)*(1+x)*(1+x^2)),{x,0,40}],x] (* Vincenzo Librandi, Jun 17 2012 *)
LinearRecurrence[{1,1,1,2},{2,-1,-1,1},40] (* Harvey P. Dale, Dec 01 2019 *)

Formula

First differences of A180343(n).
G.f.: ( -2 + 3*x - x^3 + 2*x^2 ) / ( (2*x-1)*(1+x)*(1+x^2) ). - R. J. Mathar, Jan 26 2011
a(n) = 4*(-1)^floor((n+1)/2)*A000034(n+1)/5 + 2^n/15 + (-1)^n/3. - R. J. Mathar, Jan 26 2011
a(n) = a(n-4) + 2^(n-4).
a(n) = a(n-2) + (-3,2,4,0,0,8,16,24,=sixth differences of A007910(n-1) = 0,0,1,2,3,6,13 or fifth differences of A007909(n); also -3,2,4,8*A007910(n-1)).
a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4). - Vincenzo Librandi, Jun 17 2012
Previous Showing 11-13 of 13 results.

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