A321002
a(0)=3; thereafter a(n) = 20*6^(n-1)-2^(n-1).
Original entry on oeis.org
3, 19, 118, 716, 4312, 25904, 155488, 933056, 5598592, 33592064, 201553408, 1209322496, 7255939072, 43535642624, 261213872128, 1567283265536, 9403699658752, 56422198083584, 338533188763648, 2031199133106176, 12187194799685632, 73123168800210944, 438739012805459968, 2632434076841148416
Offset: 0
Essentially the first differences of
A321003.
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Vec((1 - x)*(3 - 2*x) / ((1 - 2*x)*(1 - 6*x)) + O(x^25)) \\ Colin Barker, Nov 02 2018
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a(n) = if (n, 20*6^(n-1)-2^(n-1), 3); \\ Michel Marcus, Nov 02 2018
A385178
Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.
Original entry on oeis.org
0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171
Offset: 0
Triangle begins:
0;
1, 1;
3, 4, 5;
7, 10, 14, 19;
15, 22, 32, 46, 65;
31, 46, 68, 100, 146, 211;
63, 94, 140, 208, 308, 454, 665;
127, 190, 284, 424, 632, 940, 1394, 2059;
255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305;
511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
...
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/* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025
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T:= proc(n,k) option remember;
`if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
end:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Jun 24 2025
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t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)
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