A335903 Column 1 in the matrix of A279212 (whose indexing starts at 0).

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..35.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Formula

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.

From Colin Barker, Jun 29 2020: (Start)

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)

Example

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
number      antidiagonal     A279212
-------------------------------------
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.

Mathematica

a335903[1] = 2; a335903[2] = 6; a335903[n_] := (7n+9)*2^(n-4)
Map[a335903, Range[35]]  (* data  *)

Prog

(PARI) Vec(x*(1 - x)*(2 - x^2) / (1 - 2*x)^2 + O(x^30)) \\ Colin Barker, Jun 29 2020

Crossrefs

Cf. A011782, A233328, A279212.

Sequence in context: A018018 A030009 A061261 * A291414 A098790 A300344

Adjacent sequences: A335900 A335901 A335902 * A335904 A335905 A335906

Keyword

nonn,easy

Author

Hartmut F. W. Hoft, Jun 29 2020

Status

approved