A372313
Table read by antidiagonals: A(n,1) = 2n-1, and for k > 1, A(n,k) = A372289(A(n,k-1)+A(n,1)).
Original entry on oeis.org
1, 5, 3, 13, 13, 5, 29, 341, 21, 7, 61, 2773, 53, 29, 9, 125, 22229, 117, 149, 37, 11, 253, 177877, 245, 629, 93, 45, 13, 509, 1423061, 501, 2549, 205, 469, 53, 15, 1021, 11384533, 1013, 10229, 429, 15701, 133, 61, 17, 2045, 91076309, 2037, 40949, 877, 503125, 293, 309, 69, 19
Offset: 1
A(3,1) = 5. A(3,2) = A372289(5+5) = 21. A(3,3) = A372289(21+5) = 53. A(3,4) = A372289(53+5) = 117.
Table begins:
1, 5, 13, 29, 61, 125, 253, 509, ...
3, 13, 341, 2773, 22229, 177877, 1423061, 11384533, ...
5, 21, 53, 117, 245, 501, 1013, 2037, ...
7, 29, 149, 629, 2549, 10229, 40949, 163829, ...
9, 37, 93, 205, 429, 877, 1773, 3565, ...
11, 45, 469, 15701, 503125, 16100693, 515222869, 16487132501, ...
13, 53, 133, 293, 613, 1253, 2533, 5093, ...
15, 61, 309, 1301, 5269, 21141, 84629, 338581, ...
17, 69, 173, 381, 797, 1629, 3293, 6621, ...
19, 77, 3413, 27477, 219989, 1760085, 14080853, 112646997, ...
A371094
a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.
Original entry on oeis.org
1, 21, 7, 21, 13, 341, 19, 45, 25, 117, 31, 69, 37, 341, 43, 93, 49, 213, 55, 117, 61, 5461, 67, 141, 73, 309, 79, 165, 85, 725, 91, 189, 97, 405, 103, 213, 109, 1877, 115, 237, 121, 501, 127, 261, 133, 1109, 139, 285, 145, 597, 151, 309, 157, 5461, 163, 333, 169, 693, 175, 357, 181, 1493, 187, 381, 193, 789, 199
Offset: 0
For n=1, 3*n+1 = 4, "100" in binary, when we substitute 01's for the two trailing 0's, we obtain 21, "10101" in binary, therefore a(1) = 21.
For n=6, 3*6+1 = 19, "10011" in binary, and there are no trailing 0's, and no changes, therefore a(6) = 19.
For n=7, 3*7+1 = 22, "10110" in binary, with one trailing 0, which when replaced with 01 gives us 45, "101101" in binary, therefore a(7) = 45.
For n=229, there are e=4 trailing bit expansions 0 -> 01,
3n+1 = binary 101011 0 0 0 0
a(n) = binary 101011 01010101
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Array[#2*(2^#3) + ((4^#3) - 1)/3 & @@ {#1, #2, IntegerExponent[#2, 2]} & @@ {#, 3 #1 + 1} &, 67, 0] (* Michael De Vlieger, Apr 19 2024 *)
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A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
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def A371094(n): return ((m:=3*n+1)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024
Showing 1-2 of 2 results.
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