A246534 a(n) = Sum_{k=1..n} 2^(T(k)-1), where T(k)=k(k+1)/2 = A000217(k).

0, 1, 5, 37, 549, 16933, 1065509, 135283237, 34495021605, 17626681066021, 18032025190548005, 36911520172609651237, 151152638972001256489509, 1238091191924352276155613733, 20283647694843594776223406899749, 664634281540152780046679753547072037

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

Table of n, a(n) for n=0..15.

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.

Intersection of A333256 and A333217.

Partial sums of A036442.

Cf. A000120, A029931, A048793, A066099, A070939, A124766, A228351, A233564, A272919, A333218.

Sequence in context: A244820 A359796 A360933 * A095957 A354862 A121834

Adjacent sequences: A246531 A246532 A246533 * A246535 A246536 A246537

Keyword

nonn

Author

M. F. Hasler, Aug 28 2014

Status

approved