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A154626 a(n) = 2^n*A001519(n). 8
1, 2, 8, 40, 208, 1088, 5696, 29824, 156160, 817664, 4281344, 22417408, 117379072, 614604800, 3218112512, 16850255872, 88229085184, 461973487616, 2418924584960, 12665653559296, 66318223015936, 347246723858432, 1818207451086848, 9520257811087360 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

A production matrix for the sequence is M =

1, 1, 0, 0, 0,...

1, 0, 5, 0, 0,...

1, 0, 0, 5, 0,...

1, 0, 0, 0, 5,...

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

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

- Gary W. Adamson, Jul 22 2016

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

From Gary W. Adamson, Jul 27 2016: (Start)

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:

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

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

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

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

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

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

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

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

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

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

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

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

  (40*N + (N-1)^2),... (End)

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

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

LINKS

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-4).

FORMULA

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

a(n) = A084326(n+1)-4*A084326(n). - R. J. Mathar, Jul 19 2012

From Colin Barker, Sep 22 2017: (Start)

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

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

(End)

MATHEMATICA

LinearRecurrence[{6, -4}, {1, 2}, 30] (* Vincenzo Librandi, May 15 2015 *)

PROG

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

(PARI) Vec((1-4*x) / (1-6*x+4*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

CROSSREFS

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

    A180028, A123362.

Sequence in context: A071007 A027617 A187071 * A003305 A076625 A139415

Adjacent sequences:  A154623 A154624 A154625 * A154627 A154628 A154629

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jan 13 2009

STATUS

approved

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Last modified February 27 06:36 EST 2020. Contains 332299 sequences. (Running on oeis4.)