The beam and the double-pan balance

Put the beam of a double-pan balance between any two successive terms of the hereunder sequence D.

D = 1,2,3,3,4,5,6,3,7,5,8,7,9,9,1,8,10,11,12,3,13,11,14,7,…

There are always k terms to the left of the beam with sum t and k terms to the right

of the beam with sum 2t.

Examples (the beam is the ampersand &)

t=1.  D=1 & 2  (sum 2 on the right pan = 2t)

t=3.  D=1,2 & 3,3  (sum 6 on the right pan = 2t)

t=6.  D=1,2,3 & 3,4,5  (sum 12 on the right pan = 2t)

t=9.  D=1,2,3,3 & 4,5,6,3  (sum 18 on the right pan = 2t)

t=13. D=1,2,3,3,4 & 5,6,3,7,5  (sum 26 on the right pan = 2t)

t=18. D=1,2,3,3,4,5 & 6,3,7,5,8,7  (sum 36 on the right pan = 2t)

t=24. D=1,2,3,3,4,5,6 & 3,7,5,8,7,9,9  (sum 48 on the right pan = 2t)

t=27. D=1,2,3,3,4,5,6,3 & 7,5,8,7,9,9,1,8  (sum 54 on the right pan = 2t)

…

There are lots of sequences with this property — we decided to extend S at each step with a pair of (underlined) integers such that:

— the first element of the pair is the smallest one not yet present in S;

— if this is impossible to achieve, the first element is 1 (as above, in the last line of the table, t=27).

Carole D. was quick to compute a few more terms of D:

D = 1, 2, 3, 3, 4, 5, 6, 3, 7, 5, 8, 7, 9, 9, 1, 8, 10, 11, 12, 3, 13, 11, 14, 7, 15, 12, 16, 11, 1, 2, 17, 7, 18, 12, 19, 14, 20, 16, 1, 8, 21, 18, 22, 11, 23, 19, 1, 20, 24, 21, 25, 11, 26, 22, 27, 6, 1, 2, 1, 5, 28, 23, 1, 20, 29, 25, 30, 6, 31, 26, 32, 10, 33, 27, 34, 14, 1, 2, 1, 23, 35, 28, 36, 18, 37, 29, 1, 32, 38, 31, 39, 18, 1, 2, 40, 20, 41, 31, 42, 21, 43, 32, 1, 32, 44, 34, 45, 21, 46, 35, 1, 17, 1, 2, 1, 5, 1, 2, 1, 14, 47, 37, 48, 21, 1, 2, 49, 11, 50, 37, 51, 24, 52, 38, 1, 17, 53, 40, 54, 24, 55, 41, 1, 29, 56, 43, 57, 24, 58, 44, 1, 41, 1, 2, 1, 5, 1, 2, 59, 10, 60, 45, 61, 23, 62, 46, 1, 53, 63, 48, 64, 23, 1, 2, 65, 31, 66, 48, 67, 26, 68, 49, 1, 53, 1, 2, 1, 5, 69, 51, 1, 59, 70, 53, 71, 22, 72, 54, 1, 62, 73, 56, 74, 22, 1, 2, 75, 21, 76, 56, 77, 25, 78, 57, 1, 62, 79, 59, 80, 25, 1, 2, 1, 50, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 81, 60, 82, 29, 83, 61, 1, 62, 1, 2,1, 5, 84, 63, 1, 32, 85, 65, 86, 25, 87, 66, 1, 71, 88, 68, 89, 25, 1, 2, 1, 50, 90, 69, 91, 29, 92, 70, 1, 71, 93, 72, 94, 29, 1, 2, 1, 86, 95, 73, 96, 33, 97, 74, 1, 71, 98, 76, 99, 33, 1, 2, 100, 23, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 101, 76, 1, 29, 102, 78, 103, 32, 104, 79, 1, 68, 105, 81, 106, 32, 1, 2, 107, 52, 108, 81, 109, 35, 110, 82, 1, 68, 1, 2, 1, 5, 111, 84, 1, 92, 112, 86, 113, 31, 114, 87, 1, 77, 115, 89, 116, 31, 1, 2, 117, 42, 1, 2, 1, 5, 1, 2, 1, 14, 118, 89, 119, 34, 1, 2, 120, 57, 121, 89, 122, 37, 123, 90, 1, 65, 124, 92, 125, 37, 1, 2, 126, 60, 127, 92, 128, 40, 129, 93, 1, 65, 1, 2, 1, 5, 130, 95, 1, 62, 131, 97, 132, 36, 133, 98, 1, 74, 134, 100, 135, 36, 1, 2, 136, 50, 137, 100, 138, 39, 139, 101, 1, 74, 1, 2, 1, 5, 1, 2, 140, 10, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 122, 141, 102, 142, 38, 143, 103, 1, 86, 144, 105, 145, 38, 1, 2, 146, 40, 1, 2, 1, 5, ...

Here is the sequence T where the weight on the right pan is always 3 times the weight on the left pan (thank you again, Carole!-)

T = 1, 3, 2, 10, 4, 4, 5, 35, 6, 10, 7, 9, 8, 12, 11, 129, 13, 11, 14, 26, 15, 13, 16, 20, 17, 15, 18, 30, 19, 25, 21, 495, 22, 30, 23, 21, 24, 32, 27, 77, 28, 32, 29, 23, 31, 33, 34, 46, 36, 32, 37, 23, 38, 34, 39, 81, 40, 36, 41, 59, 42, 42, 43, 1937, 44, 44, 45, 75, 47, 45, 48, 36, 49, 47, 50, 78, 51, 57, 52, 256, 53, 59, 54, 74, 55, 61, 56, 36, 58, 66, 60, 72, 62, 74, 63, 121, 64, 80, 65, 63, 67, 81, 68, 24, 69, 83, 70, 66, 71, 85, 73, 251, 76, 84, 79, 65, 82, 82, 86, 150, 87, 81, 88, 80, 89, 83, 90, 7658, 91, 85, 92, 84, 93, 87, 94, 206, 95, 93, 96, 84, 97, 95, 98, 46, 99, 97, 100, 88, 101, 99, 102, 210, 103, 101, 104, 124, 105, 103, 106, 918, 107, 105, 108, 128, 109, 107, 110, 186, 111, 109, 112, 132, 113, 111, 114, 30, 115, 117, 116, 148, 118, 122, 119, 169, 120, 128, 123, 173, 125, 127, 126, 358, 130, 126, 131, 189, 133, 127, 134, 118, 135, 133, 136, 188, 137, 135, 1, 95, 138, 138, 139, 193, 140, 140, 141, 123, 142, 142, 143, 197, 144, 148, 145, 859, 146, 158, 147, 189, 149, 167, 151, 109, 152, 176, 153, 175, 154, 190, 155, 445, 156, 192, 157, 167, 159, 193, 160, 160, 161, 195, 162, 170, 163, 197, 164, 30468, 165, 199, 166, 174, 168, 200, 171, 165, 172, 200, 177, 171, 178, 198, 179, 645, 180, 200, 181, 191, 182, 202, 183, 153, 184, 204, 185, 195, 187, 205, 1, 183, 194, 202, 196, 192, 201, 199, 203, 149, 207, 197, 208, 188, 209, 199, 211, 629, 212, 200, 213, 191, 214, 202, 215, 281, 216, 204, 217, 195, 218, 206, 219, 3453, 220, 208, 221, 199, 222, 210, 223, 289, 224, 212, 225, 203, 226, 214, 227, 517, 228, 216, 229, 207, 230, 218, 231, 297, 232, 220, 233, 211, 234, 222, 1, 119, 235, 225, 236, 232, 237, 227, 238, 354, 239, 233, 240, 248, 241, 235, 242, 434, 243, 237, 244, 268, 245, 247, 246, 446, 249, 251, 250, 258, 252, 252, 253, 1179, 254, 266, 255, 249, 257, 267, 259, 497, 260, 272, 261, 247, 262, 274, 263, 209, 264, 276, 265, 267, 269, 275, 270, 482, 271, 277, 273, 267, 1, 3, 278, 102, 279, 273, 280, 272, 282, 274, 283, 489, 284, 276, 285, 275, 286, 278, 287, 205, 288, 280, 290, 278, 291, 281, 292, 496, 293, 283, 294, 298, 295, 285, 296, 3140, 299, 285, 300, 332, 301, 287, 302, 454, 303, 293, 304, 364, 305, 299, 306, 130, 307, 301, 308, 396, 309, 303, 310, 390, 311, 305, 312, 448, 313, 307, 314, 1466, 315, 309, 316, 452, 317, 311, 318, 350, 319, 317, 320, 452, 321, 319, 322, 318, 323, 321, 324, 456, ...

If we want to use the same idea to build the sequence U where the two pans have the same weight (no factor 2 or 3 to multiply the sum on the left-pan to get the sum on the right-pan), we have to put the beam after a(2) – NOT immediately after a(1) as above – and start weighting from there (indeed, 1+4=2+3)

U =  1, 4, 2, 3, 1, 3, 5, 1, 1, 1, 1, 5, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 7, 3, 8, 4, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 5, 1, 5, 10, 6, 1, 7, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 6, 1, 9, 1, 1, 1, 9, 13, 7, 1, 11, 1, 1, 1, 13, 1, 1, 1, 1, 14, 8, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 9, 1, 11, 1, 1, 16, 2, 1, 1, 1, 1, 1, 1, 17, 1, 18, 8, 1, 13, 1, 1, 19, 3, 1, 1, 1, 1, 1, 1, 20, 6, 1, 1, 1, 1, 1, 1, 1, 1, 21, 7, 1, 15, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 22, 8, 1, 17, 1, 1, 1, 21, 1, 1, 1, 1, 23, 9, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 10, 1, 1, 25, 11, 1, 15, 1, 1, 1, 25, 1, 1, 1, 1, 26, 12, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 13, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 14, 1, 13, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

We see that the 1s are quickly invading the sequence U.

Other variants imply to multiply the terms instead of adding them. See hereunder the beginning of the sequence DD where the terms on the right pan weight twice the terms on the left pan – when multiplied (instead of added):

DD = 1, 2, 4, 1, 8, 2, 1, 1, 16, 4, 1, 4, 1, 1, 1, 1, 32, 8, 1, 16, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 64, 16, 1, 64, 1, 1, 128, 2, 1, 1, 1, 1, 1, 1, 1, 256, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 512, 8, 1, 256, 1, 1, 1024, 4, 1, 1, 1, 1, 2048, 8, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4096, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8192, 32, 1, 64, 1, 1, 16384, 4, 1, 1, 1, 1, 32768, 32, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, ...

Again, an invasion of 1s – and the presence of nothing else but powers of 2.

The sequence TT shows the same idea with the right pan weighting three times the left pan (another invasion of 1s and the exclusive presence of powers of 3):

TT = 1, 3, 9, 1, 27, 3, 1, 1, 81, 9, 1, 9, 1, 1, 1, 1, 243, 27, 1, 81, 1, 1, 1, 81, 1, 1, 1, 1, 1, 1, 1, 1, 729, 81, 1, 729, 1, 1, 2187, 3, 1, 1, 1, 1, 1, 1, 1, 6561, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19683, 27, 1, 6561, 1, 1, 59049, 9, 1, 1, 1, 1, 177147, 27, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

The hereunder sequence UU shows the same idea where the two pans are balanced (indeed, 2 = 2, then 2*2=4*1 then 2*2*4=1*8*2, etc.):

UU = 2, 2, 4, 1, 8, 2, 1, 1, 16, 4, 1, 4, 1, 1, 1, 1, 32, 8, 1, 16, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 64, 16, 1, 64, 1, 1, 128, 2, 1, 1, 1, 1, 1, 1, 1, 256, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 512, 8, 1, 256, 1, 1, 1024, 4, 1, 1, 1, 1, 2048, 8, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4096, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8192, 32, 1, 64, 1, 1, 16384, 4, 1, 1, 1, 1, 32768, 32, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 65536, 64, 1, 64, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 131072, 128, 1, 256, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 262144, 256, 1, 1024, 1, 1, 1, 4096, 1, 1, 1, 1, ...

More variants to come soon (and submissions to the OEIS).