A213947
Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.
Original entry on oeis.org
1, 1, 2, 1, 4, 3, 1, 10, 6, 4, 1, 20, 21, 8, 5, 1, 42, 57, 28, 10, 6, 1, 84, 150, 88, 35, 12, 7, 1, 170, 390, 252, 110, 42, 14, 8, 1, 340, 990, 712, 335, 132, 49, 16, 9, 1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10
Offset: 1
First few rows of the triangle:
1;
1, 2;
1, 4, 3;
1, 10, 6, 4;
1, 20, 21, 8, 5;
1, 42, 57, 28, 10, 6;
1, 84, 150, 88, 35, 12, 7;
1, 170, 390, 252, 110, 42, 14, 8;
1, 340, 990, 712, 335, 132, 49, 16, 9;
1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10;
1, 1364, 6138, 5464, 2805, 1200, 469, 176, 63, 20, 11;
...
-
read("transforms") ;
A213947i := proc(n,k)
L := [seq(i,i=1..n),seq(0,i=0..k)] ;
INVERT(L) ;
op(k,%) ;
end proc:
A213947 := proc(n,k)
if k = 1 then
1;
else
A213947i(k,n)-A213947i(k-1,n) ;
end if;
end proc: # R. J. Mathar, Jun 30 2012
A014131
a(n) = 2*a(n-1) if n odd else 2*a(n-1) + 6.
Original entry on oeis.org
0, 6, 12, 30, 60, 126, 252, 510, 1020, 2046, 4092, 8190, 16380, 32766, 65532, 131070, 262140, 524286, 1048572, 2097150, 4194300, 8388606, 16777212, 33554430, 67108860, 134217726, 268435452, 536870910
Offset: 0
-
[2^(n+2)-3-(-1)^n: n in [0..30]]; // Vincenzo Librandi, Apr 03 2012
-
Table[2^(n+2)-3-(-1)^n,{n,0,40}] (* or *) CoefficientList[Series[6x/((1-2x)(1-x)(1+x)),{x,0,30}],x] (* Vincenzo Librandi, Apr 03 2012 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],2a,2a+6]}; NestList[nxt,{1,0},30][[;;,2]] (* or *) LinearRecurrence[ {2,1,-2},{0,6,12},30] (* Harvey P. Dale, Aug 26 2024 *)
A293014
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) for n > 4, where a(n)=0 for n < 4 and a(4) = 1.
Original entry on oeis.org
0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 11, 17, 28, 44, 72, 116, 189, 305, 494, 798, 1292, 2090, 3383, 5473, 8856, 14328, 23184, 37512, 60697, 98209, 158906, 257114, 416020, 673134, 1089155, 1762289, 2851444, 4613732, 7465176, 12078908
Offset: 0
-
LinearRecurrence[{1, 1, -1, 1, 1}, {0, 0, 0, 0, 1}, 40]
-
a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,1,-1,1,1]^n)[1,5] \\ Charles R Greathouse IV, Sep 28 2017
Comments