|
| |
|
|
A246534
|
|
a(n) = Sum_{k=1..n} 2^(T(k)-1), where T(k)=k(k+1)/2 = A000217(k).
|
|
24
|
|
|
|
0, 1, 5, 37, 549, 16933, 1065509, 135283237, 34495021605, 17626681066021, 18032025190548005, 36911520172609651237, 151152638972001256489509, 1238091191924352276155613733, 20283647694843594776223406899749, 664634281540152780046679753547072037
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Similar to A181388, this occurs as binary encoding of a straight n-omino lying on the y-axis, when the grid points of the first quadrant (N x N, N={0,1,2,...}) are given the weight 2^k, with k=0, 1,2, 3,4,5, ... filled in by antidiagonals.
Numbers k such that the k-th composition in standard order (row k of A066099) is a reversed initial interval. - Gus Wiseman, Apr 02 2020
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Label the cells of an infinite square matrix with 0,1,2,3,... along antidiagonals:
0 1 3 6 10 ...
2 4 7 ...
5 8 ...
9 ...
....
Now any subset of these cells can be represented by the sum of 2 raised to the power written in the given cells. In particular, the subset consisting of the first cell in the first 1, 2, 3, ... rows is represented by 2^0, 2^0+2^2, 2^0+2^2+2^5, ...
|
|
|
MATHEMATICA
|
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[0, 1000], normQ[stc[#]]&&Greater@@stc[#]&] (* Gus Wiseman, Apr 02 2020 *)
|
|
|
PROG
|
(PARI) t=0; vector(20, n, t+=2^(n*(n+1)/2-1)) \\ yields the vector starting with a[1]=1
(PARI) t=0; vector(20, n, if(n>1, t+=2^(n*(n-1)/2-1))) \\ yields the vector starting with 0
(Python)
a = 0
for n in range(1, 17): print(a, end =', '); a += 1<<(n-1)*(n+2)//2 # Ya-Ping Lu, Jan 23 2024
|
|
|
CROSSREFS
|
The version for prime (rather than binary) indices is A002110.
The non-strict generalization is A114994.
The non-reversed version is A164894.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
