A065447
Concatenation of 1, 00, 111, 0000, ..., n 1's (if n is odd) or n 0's (if n is even).
Original entry on oeis.org
1, 100, 100111, 1001110000, 100111000011111, 100111000011111000000, 1001110000111110000001111111, 100111000011111000000111111100000000, 100111000011111000000111111100000000111111111, 1001110000111110000001111111000000001111111110000000000
Offset: 1
a(2) = 100, the concatenation of one 1, two 0's.
a(3) = 100111, the concatenation of one 1, two 0's, three 1's.
a(4) = 1001110000, the concatenation of one 1, two 0's, three 1's, four 0's.
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a:= n-> parse(cat((irem(i,2)$i)$i=1..n)):
seq(a(n), n=1..10); # Alois P. Heinz, Mar 08 2024
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FoldList[Join, {1}, Map[ConstantArray[Mod[#, 2], #] &, Range[2, 10]]] (* Peter J. C. Moses, Mar 08 2024 *)
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{ m=10; for (n=1, 44, if (n==1, a=1, m*=10; a*=m; if (n%2, a+=(m - 1)/9)); write("b065447.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 19 2009
A065761
Concatenation of increasing number of alternating digits in base 2, starting with 0.
Original entry on oeis.org
0, 3, 24, 399, 12768, 817215, 104603520, 26778501375, 13710592704000, 14039646928897023, 28753196910381103104, 117773094544920998318079, 964797190511992818221703168, 15807237169348490333744384720895
Offset: 1
a(5) = 12768 is formed by appending 0 five times (00000) to a(4) in base 2: (0)11000111100000.
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baseI(x, b)= { local(d, e=0, f=1); while (x>0, d=x-10*(x\10); x\=10; e+=d*f; f*=b); return(e) } { c=1; for (n=1, 50, if (n==1, a=0; b=0, c=c*10 + 1; if (n%2, d=0, d=c); b=b*10^n + d; a=baseI(b, 2)); write("b065761.txt", n, " ",a) ) } \\ Harry J. Smith, Oct 30 2009
A371033
a(n) is the integer whose binary expansion starts with 1 and such that the runs of identical bits have lengths n, n-1, n-2, ..., 3, 2, 1.
Original entry on oeis.org
1, 6, 57, 966, 31801, 2065350, 266370105, 68453106630, 35115918982201, 35993681099981766, 73750982613738224697, 302157703921043555451846, 2475577920866839506242796601, 40562343629382474008388259775430, 1329187433441286490429798672020569145
Offset: 1
Representations as binary words (as in A371032) have decreasing runlengths:
1: 1
6: 110
57: 111001
966: 1111000110 (runlengths 4,3,2,1)
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a:= n-> Bits[Join]([seq((1-(n-i) mod 2)$i, i=1..n)]):
seq(a(n), n=1..15); # Alois P. Heinz, Jul 09 2024
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Map[FromDigits[#, 2] &, Table[Flatten[Map[ConstantArray[Mod[#, 2], n + 1 - #] &, Range[n]]], {n, 16}]] (* Peter J. C. Moses, Mar 08 2024 *)
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def A371033(n):
c = 0
for i in range(n):
c <<= n-i
if i&1^1:
c += (1<Chai Wah Wu, Mar 18 2024
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