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A154626 a(n) = 2^n*A001519(n). 8

%I

%S 1,2,8,40,208,1088,5696,29824,156160,817664,4281344,22417408,

%T 117379072,614604800,3218112512,16850255872,88229085184,461973487616,

%U 2418924584960,12665653559296,66318223015936,347246723858432,1818207451086848,9520257811087360

%N a(n) = 2^n*A001519(n).

%C Hankel transform of 1,1,3,11,45,.... (see A026375). Binomial transform of A015448.

%C A production matrix for the sequence is M =

%C 1, 1, 0, 0, 0,...

%C 1, 0, 5, 0, 0,...

%C 1, 0, 0, 5, 0,...

%C 1, 0, 0, 0, 5,...

%C ...Take powers of M, extracting the upper left terms; getting

%C the sequence starting (1, 1, 2, 8, 40, 208,...).

%C - _Gary W. Adamson_, Jul 22 2016

%C The sequence is N=5 in an infinite set of INVERT transforms of powers of N prefaced with a "1". (1, 2, 8, 40,...) is the INVERT transform of (1, 1, 5, 25, 125,...). The first six of such sequences are shown in A006012 (N=3). - _Gary W. Adamson_, Jul 24 2016

%C From _Gary W. Adamson_, Jul 27 2016: (Start)

%C The sequence is the first in an infinite set in which we perform the operation for matrix M (Cf. Jul 22 2016), but change the left border successively from (1, 1, 1, 1,...) then to (1, 2, 2, 2...), then (1, 3, 3, 3,...)...; generally (1, N, N, N,...). Extracting the upper left terms of each matrix operation, we obtain the infinite set beginning:

%C N=1 (A154626): 1, 2, 8, 40, 208, 1088,...

%C N=2 (A084120): 1, 3, 15, 81, 441, 1403,...

%C N=3 (A180034): 1, 4, 22, 124, 700, 3952,...

%C N=4 (A001653): 1, 5, 29, 169, 985, 5741,...

%C N=5 (A000040): 1, 6, 36, 216, 1296, 7776,...

%C N=6 (A015451): 1, 7, 43, 265, 1633, 10063,...

%C N=7 (A180029): 1, 8, 50, 316, 1996, 12608,...

%C N=8 (A180028): 1, 9, 57, 369, 1285, 15417,...

%C N=9 (.......): 1, 10, 64, 424, 2800, 18496,...

%C N=10 (A123361): 1, 11, 71, 481, 3241, 21851,...

%C N=11 (.......): 1, 12, 78, 540, 3708, 25488,...

%C ... Each of the sequences begins (1, (N+1), (7*N + 1),

%C (40*N + (N-1)^2),... (End)

%C The set of infinite sequences shown (Cf. comment of Jul 27 2016), can be

%C generated from the matrices P = [(1,N; 1,5]^n, (N=1,2,3,...) by extracting the upper left terms. Example: N=6 sequence (A015451): (1, 7, 43, 265,...) can be generated from the matrix P = [(1,6); (1,5)]^n. - _Gary W. Adamson_, Jul 28 2016

%H Karl V. Keller, Jr., <a href="/A154626/b154626.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).

%F G.f.: (1 - 4*x) / (1 - 6*x + 4*x^2).

%F a(n) = A084326(n+1)-4*A084326(n). - _R. J. Mathar_, Jul 19 2012

%F From _Colin Barker_, Sep 22 2017: (Start)

%F a(n) = (((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n)) / (2*sqrt(5)).

%F a(n) = 6*a(n-1) - 4*a(n-2) for n>1.

%F (End)

%t LinearRecurrence[{6, -4}, {1, 2}, 30] (* _Vincenzo Librandi_, May 15 2015 *)

%o (MAGMA) [n le 2 select (n) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, May 15 2015

%o (PARI) Vec((1-4*x) / (1-6*x+4*x^2) + O(x^30)) \\ _Colin Barker_, Sep 22 2017

%Y Cf. A006012, A084120, A180034, A001653, A000040, A015451, A180029,

%Y A180028, A123362.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jan 13 2009

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)