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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A118404 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118407) are equal; also equals the matrix inverse of triangle A118400.

Original entry on oeis.org

1, 1, -1, -1, 0, 1, -1, 1, -1, -1, 1, 0, 0, 2, 1, 1, -1, 0, -2, -3, -1, -1, 0, 1, 2, 5, 4, 1, -1, 1, -1, -3, -7, -9, -5, -1, 1, 0, 0, 4, 10, 16, 14, 6, 1, 1, -1, 0, -4, -14, -26, -30, -20, -7, -1, -1, 0, 1, 4, 18, 40, 56, 50, 27, 8, 1, -1, 1, -1, -5, -22, -58, -96, -106, -77, -35, -9, -1, 1, 0, 0, 6, 27, 80, 154, 202, 183, 112, 44, 10, 1, 1, -1, 0, -6, -33, -107, -234, -356, -385, -295, -156, -54, -11, -1, -1, 0, 1, 6, 39, 140, 341, 590, 741, 680, 451, 210, 65, 12, 1, -1, 1, -1, -7, -45, -179, -481, -931, -1331, -1421, -1131, -661, -275, -77, -13, -1, 1, 0, 0, 8, 52, 224, 660, 1412, 2262, 2752, 2552, 1792, 936, 352, 90, 14, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Comments

Appears to coincide with triangle (5.2) in Lee-Oh (2016), although there is no obvious connection! - N. J. A. Sloane, Dec 07 2016

Examples

			Triangle begins:
1;
1,-1;
-1, 0, 1;
-1, 1,-1,-1;
1, 0, 0, 2, 1;
1,-1, 0,-2,-3,-1;
-1, 0, 1, 2, 5, 4, 1;
-1, 1,-1,-3,-7,-9,-5,-1;
1, 0, 0, 4, 10, 16, 14, 6, 1;
1,-1, 0,-4,-14,-26,-30,-20,-7,-1;
-1, 0, 1, 4, 18, 40, 56, 50, 27, 8, 1;
-1, 1,-1,-5,-22,-58,-96,-106,-77,-35,-9,-1;
1, 0, 0, 6, 27, 80, 154, 202, 183, 112, 44, 10, 1;
1, -1, 0, -6, -33, -107, -234, -356, -385, -295, -156, -54, -11, -1;
-1, 0, 1, 6, 39, 140, 341, 590, 741, 680, 451, 210, 65, 12, 1;
-1, 1, -1, -7, -45, -179, -481, -931, -1331, -1421, -1131, -661, -275, -77, -13, -1;
1, 0, 0, 8, 52, 224, 660, 1412, 2262, 2752, 2552, 1792, 936, 352, 90, 14, 1;
1, -1, 0, -8, -60, -276, -884, -2072, -3674, -5014, -5304, -4344, -2728, -1288, -442, -104, -15, -1;
-1, 0, 1, 8, 68, 336, 1160, 2956, 5746, 8688, 10318, 9648, 7072, 4016, 1730, 546, 119, 16, 1; ...
The matrix square is A118407:
1;
0, 1;
-2, 0, 1;
2,-2, 0, 1;
0, 2,-2, 0, 1;
-2, 0, 2,-2, 0, 1;
4,-2, 0, 2,-2, 0, 1;
-6, 4,-2, 0, 2,-2, 0, 1;
4,-6, 4,-2, 0, 2,-2, 0, 1;
6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...
in which all columns are equal.

Links

Crossrefs

Cf. A118405 (row sums), A118406 (unsigned row sums), A118407 (matrix square), A118400 (matrix inverse).
Columns or diagonals (modulo offsets): A219977, A011848, A212342, A007598, A005581, A007910.

Programs

  • Mathematica
    T[n_, k_] := SeriesCoefficient[(-1)^k/((1+x^2)(1+x)^(k-1)), {x, 0, n-k}];
Table[T[n, k], {n, 0, 16}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018 *)
  • PARI
    {T(n,k)=polcoeff(polcoeff((1+x)^2/(1+x^2)/(1+x+x*y +x*O(x^n)),n,x)+y*O(y^k),k,y)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

Formula

G.f.: A(x,y) = (1+x)^2 / ( (1+x^2) * (1+x + x*y) ).
G.f. of column k: (-1)^k / ( (1+x^2) * (1+x)^(k-1) ) for k>=0.

A118406 Unsigned row sums of triangle A118404.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 14, 28, 52, 104, 206, 412, 820, 1640, 3278, 6556, 13108, 26216, 52430, 104860, 209716, 419432, 838862, 1677724, 3355444, 6710888, 13421774, 26843548, 53687092, 107374184, 214748366, 429496732, 858993460, 1717986920, 3435973838
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Links

Crossrefs

Cf. A118404, A118405.

Programs

  • Mathematica
    CoefficientList[Series[(1-2x^2-5x^4)/(1-x^4)/(1-2x),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,0,1,-2},{1,2,2,4,4},40] (* Harvey P. Dale, Jul 29 2021 *)
  • PARI
    {a(n)=polcoeff((1 - 2*x^2 - 5*x^4)/(1-x^4)/(1-2*x+x*O(x^n)),n,x)}

Formula

G.f.: A(x) = (1 - 2*x^2 - 5*x^4)/(1-x^4)/(1-2*x).

A181586 a(0)=0; a(n+1) = 2*a(n) + period 4:repeat 0,1,-2,1.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 5, 8, 17, 34, 69, 136, 273, 546, 1093, 2184, 4369, 8738, 17477, 34952, 69905, 139810, 279621, 559240, 1118481, 2236962, 4473925, 8947848, 17895697, 35791394, 71582789, 143165576, 286331153, 572662306, 1145324613, 2290649224
Offset: 0

Views

Author

Paul Curtz, Jan 30 2011

Keywords

Comments

(**) See A115451, A139763, A139800, A139806, A139814.
a(n) + a(n+1) + a(n+2) + a(n+3) = 2^n.

Examples

			a(1)=2*a(0)+0=0, a(2)=2*a(1)+1=0+1=1, a(3)=2*a(2)-2=2-2=0, a(4)=2*a(3)+1=0+1=1, a(5)=2*a(4)+0=2+0=2, a(6)=2*a(5)+1=4+1=5.

Links

Crossrefs

Cf. A180343.

Programs

  • Maple
    a:= proc(n) option remember;
      `if`(n=0, 0, 2*a(n-1) +[0, 1, -2, 1][irem(n-1, 4)+1])
    end:
seq(a(n), n=0..40); # Alois P. Heinz, Jan 30 2011
  • Mathematica
    LinearRecurrence[{1, 1, 1, 2}, {0, 0, 1, 0}, 40] (* Jean-François Alcover, May 18 2018 *)

Formula

a(n) = a(n-4) + 2^(n-4).
a(n) = -a(n-2) + A078008(n).
a(n) = a(n-2) + A118405(n-2) unsigned.
a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) (**).
G.f.: x^2*(-1+x) / ( (2*x-1)*(1+x)*(x^2+1) ). - R. J. Mathar, Feb 06 2011
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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