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A335903
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Column 1 in the matrix of A279212 (whose indexing starts at 0).
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2
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2, 6, 15, 37, 88, 204, 464, 1040, 2304, 5056, 11008, 23808, 51200, 109568, 233472, 495616, 1048576, 2211840, 4653056, 9764864, 20447232, 42729472, 89128960, 185597952, 385875968, 801112064, 1660944384, 3439329280, 7113539584, 14696841216, 30333206528, 62545461248, 128849018880, 265214230528, 545460846592
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OFFSET
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1,1
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COMMENTS
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Indexing for this sequence starts at 1 since then the index is the same as the number of the antidiagonal in the matrix for A279212 in which a number in column 1 of A279212 occurs.
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LINKS
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FORMULA
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a(1) = 2, a(2) = 6, a(3) = 15, a(n) = 2 * a(n-1) + 7 * 2^(n-4), for n >= 4 (recursion for column 1 in the matrix of A279212).
a(1) = 2, a(2) = 6, a(n) = (7*n + 9) * 2^(n - 4), for n >= 3.
G.f.: x*(1 - x)*(2 - x^2) / (1 - 2*x)^2.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 4.
(End)
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EXAMPLE
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a(17) = a(A233328(2)) = 1048576 = 2^20 = T(16, 1) = T(21, 0) in terms of matrix T of A279212; 2^20 is in column 1 of the 17th antidiagonal and in column 0 of the 21st antidiagonal of the matrix of A279212.
A search for duplicates in A279212 through antidiagonal 2000 produced only pairs of powers of 2 in columns 0 and 1 of the matrix of A279212. Let k_0 and k_1 be the antidiagonals in columns 0 and 1, respectively, for the pair of the n-th duplicates. Since k_0 = 2 and k_1 = 1 for the duplicates of 2, the first pair in both columns, then k_0 = k_1 + 3*n - 2 for the n-th pair, n >=1.
Table of duplicates in column 1 of the matrix of A279212 (the values for k_0 are one larger than the exponents in the left column of the table below because column 0 is sequence A011782):
value of number of index in
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2^1 1 2
2^20 17 154
2^151 145 10586
2^1178 1169 683866
2^9373 9361 43818842
2^74912 74897 2804817754
2^599203 599185 179511631706
... ... ...
The central column of the table is A233328. The values for the first 4 antidiagonals were computed using sequence A279212, the ones larger than antidiagonal 2000 were determined by computing those n for which 7*n + 9 is a power of 2.
The right column is n*(n+1)/2 + 1, where n is the number in the central column.
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MATHEMATICA
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a335903[1] = 2; a335903[2] = 6; a335903[n_] := (7n+9)*2^(n-4)
Map[a335903, Range[35]] (* data *)
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PROG
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(PARI) Vec(x*(1 - x)*(2 - x^2) / (1 - 2*x)^2 + O(x^30)) \\ Colin Barker, Jun 29 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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