Cf. A006012, A084120, A180034, A001653, A000040, A015451, A180029, A180028, A123362.

A180028, A123362.

From Gary W. Adamson, Jul 22 2016: (Start)

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, ...). (End)

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 22 24 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

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, ...

(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

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

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

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editing

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approved

proposed

reviewed

editing

proposed

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)

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

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

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

generated from matrix the matrices P: = [(1,N; 1,5]^n, (N=1,2,3,...); then extract 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

reviewed

editing

proposed

reviewed