A360800 Numbers Sum_{i=1..2r+1} 2^k(i) such that k(1) is even and, for r > 0 and i < 2r+1, the difference k(i+1)-k(i) is > 0 and odd.

1, 4, 7, 16, 19, 25, 28, 31, 64, 67, 73, 76, 79, 97, 100, 103, 112, 115, 121, 124, 127, 256, 259, 265, 268, 271, 289, 292, 295, 304, 307, 313, 316, 319, 385, 388, 391, 400, 403, 409, 412, 415, 448, 451, 457, 460, 463, 481, 484, 487, 496, 499, 505, 508, 511, 1024

Offset: 1

Author

Gerhard Kirchner, Feb 24 2023

Keywords

• nonn

Comments

This is a subsequence of A360799. Another description of the terms: in the base-2 representation, the number of ones is odd and all zeros are grouped in blocks of even length.

That is why the terms less than 2^(2j+1) describe start profiles for tiling a (2j+1) X m wall with 1 X 2 dominos, see examples and A360799.

Examples

			A 5 X m wall is tiled bottom-up with dominos, start profiles:
            _        _            _ _ _    _     _ _    _ _ _ _ _
    ___ ___| |   ___| |___    ___| | | |  | |___| | |  | | | | | |
   |___|___|_|  |___|_|___|  |___|_|_|_|  |_|___|_|_|  |_|_|_|_|_|
    0 0 0 0 1    0 0 1 0 0    0 0 1 1 1    1 0 0 1 1    1 1 1 1 1
    1 = a(1)     4 = a(2)     7 = a(3)     19 = a(5)    31 = a(7)
    also the mirror images of 1 (16), 19 (25) and 7 (28).
		

Crossrefs

Cf. A001835, A003775, A028469, A028471, A180963, A187616, A360799.

Programs

Maxima

block(kmax: 100, a:[],
 oddsum(y):= block(su1:0, su2:0, pold:0, ok: true,
  while y>0 and ok do(p:mod(y,2), y:(y-p)/2,
   if p=1 then(if pold=0 and su2=1 then ok:false, su1:1-su1, su2:0)
   elseif p=0 then su2:1-su2, pold:p), return(is(ok and su1=1))),
for k from 1 thru kmax do if oddsum(k) then a:append(a,[k]),a);