A054881 -id:A054881 - cp's OEIS frontend

A003683 a(n) = 2^(n-1)*(2^n - (-1)^n)/3.

0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376, 2932030308352, 11728125427712

Offset: 0

N. J. A. Sloane

a(n) = A001045(n) * A011782(n). - Paul Barry, May 20 2003
The sequence 1,2,12,... is the binomial transform of (1, 1, 9, 9, 81, 81, ...) = 2*3^n/3 + (-3)^n/3. - Paul Barry, Jul 17 2003
Form a graph whose adjacency matrix is the tensor product of that of C_3 and [1,1;1,1]. a(n) counts walks of length n between any pair of adjacent nodes. A054881(n) counts closed walks of length n at a node.
Arises in connection with merit factor of the GRS sequences - see Hoeholdt et al.
2*a(n) = the constant term of the reduction by x^2->x+2 of the polynomial p(n,x) = ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2); see A192382.  For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232. - Clark Kimberling, Jun 30 2011
Apparently a(n+1) is the number of 3D tilings of a 2 X 2 X n room with bricks of 1 X 2 X 2 shape. - R. J. Mathar, Dec 06 2013
The ratio a(n+1)/a(n) converges to 4 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. Hoeholdt, H. E. Jensen, and J. Justesen, Aperiodic correlations and the merit factor of a class of binary sequences, IEEE Trans. Inform. Theory, 13 (1985), 549-552
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 7.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Eric Weisstein's World of Mathematics, Octahedral Graph
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003674, A011782, A054881, A192232, A192382, A238801.

[2^(n-1)*(2^n - (-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

A003683:=n->2^(n-1)*(2^n - (-1)^n)/3; seq(A003683(n), n=0..50); # Wesley Ivan Hurt, Dec 06 2013

Table[2^(n-1) (2^n-(-1)^n)/3,{n,0,30}] (* or *) LinearRecurrence[{2,8},{0,1},30] (* Harvey P. Dale, Sep 15 2013 *)

PARI
```
a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n)/3)
```

a(n)=(2^n-(-1)^n)<<(n-1)/3 \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,2,-8) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

a(n) = A003674(n)/3.
a(n) = 2*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1. - Barry E. Williams, Jan 04 2000
G.f.: x/((1+2*x)*(1-4*x)).
a(n) = ((1+3)^n-(1-3)^n)/6. - Paul Barry, May 14 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*9^k. - Paul Barry, May 20 2003
E.g.f.: exp(x)*sinh(3*x)/3. - Paul Barry, Jul 09 2003
a(n+1) = 2^n*A001045(n+1). - R. J. Mathar, Jul 08 2009
a(n+1) = Sum_{k=0..n} A238801(n,k)*3^k. - Philippe Deléham, Mar 07 2014

Erroneous references to spanning trees in K_2 X P_n deleted by Frans Faase, Feb 07 2009

A246036 Expansion of (1+4x)/((1+2x)(1-4x)).

Original entry on oeis.org

1, 6, 20, 88, 336, 1376, 5440, 21888, 87296, 349696, 1397760, 5593088, 22368256, 89481216, 357908480, 1431666688, 5726601216, 22906535936, 91625881600, 366504050688, 1466015154176, 5864062713856, 23456246661120, 93824995033088, 375299963355136, 1501199886974976, 6004799480791040, 24019198057381888

Offset: 0

N. J. A. Sloane, Aug 21 2014

Also, fourth moments of Rudin-Shapiro polynomials (see Doche, Doche-Habsieger, Ekhad papers). - Doron Zeilberger, Apr 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Mathematics of computation 74.252 (2005): 1923-1935.
Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.
Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003683, A054881, A246037, A271494, A271495, A271496.

I:=[1,6]; [n le 2 select I[n] else 2*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 22 2014

CoefficientList[Series[(1+4x)/((1+2x)(1-4x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2014 *)

Vec((1+4*x)/((1+2*x)*(1-4*x)) + O(x^100)) \\ Colin Barker, Aug 22 2014

apply( A246036(n)=(4^(1+n)-(-2)^n)/3, [0..30]) \\ M. F. Hasler, Sep 18 2020

A246036= BinaryRecurrenceSequence(2,8,1,6)
[A246036(n) for n in range(41)] # G. C. Greubel, Mar 08 2023

a(n) = 2*a(n-1) + 8*a(n-2).
a(n) = (4^(1+n) - (-2)^n)/3. - Colin Barker, Aug 22 2014
a(n) = A054881(n+3)/8. - L. Edson Jeffery, Apr 22 2015
a(n) = A003683(n+2)/2 and the above formula follow from the explicit expression for a(n), cf. second formula. - M. F. Hasler, Sep 11 2020
a(n) = 2^n*A001045(n+2). - R. J. Mathar, Mar 08 2021

A054883 Number of walks of length n along the edges of a dodecahedron between two opposite vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 12, 84, 192, 882, 2220, 8448, 22704, 78078, 218988, 710892, 2048256, 6430794, 18837516, 58008216, 171619248, 522598230, 1555243404, 4705481220, 14051590080, 42357719586, 126740502252, 381253030704, 1142062255152, 3431411494062

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,10,-16,-25,30).

Cf. A054881, A054882, A054884, A054885.

[Round((5 +3^n +4*(-2)^n -3*(1+(-1)^n)*5^(n/2))/20): n in [0..30]]; // G. C. Greubel, Feb 07 2023

LinearRecurrence[{2,10,-16,-25,30},{0,0,0,0,0,6},30] (* Harvey P. Dale, Nov 13 2021 *)

concat([0,0,0,0,0], Vec(-6*x^5/((x-1)*(2*x+1)*(3*x-1)*(5*x^2-1)) + O(x^100))) \\ Colin Barker, Dec 21 2014

def A054883(n): return (5 +3^n +4*(-2)^n -3*(1+(-1)^n)*5^(n/2))/20 -int(n==0)/5
[A054883(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

G.f.: (1/20)*(-4 + 5/(1-t) + 1/(1-3*t) + 4/(1+2*t) - 6/(1-5*t^2)).
a(n) = (5 +3^n +(-1)^n*2^(n+2) -3*(1+(-1)^n)*sqrt(5)^n)/20 for n>0.
G.f.: 6*x^5/((1-x)*(1+2*x)*(1-3*x)*(1-5*x^2)). - Colin Barker, Dec 21 2014
E.g.f.: (1/20)*(4*exp(-2*x) + 5*exp(x) + exp(3*x) - 6*cosh(sqrt(5)*x) - 4). - G. C. Greubel, Feb 07 2023

A054882 Closed walks of length n along the edges of a dodecahedron based at a vertex.

Original entry on oeis.org

1, 0, 3, 0, 15, 6, 87, 84, 567, 882, 4095, 8448, 32079, 78078, 265863, 710892, 2282631, 6430794, 20009391, 58008216, 177478623, 522598230, 1584540279, 4705481220, 14198074455, 42357719586, 127472924127, 381253030704

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,10,-16,-25,30).

Cf. A054881, A054883, A054884, A054885.

[Ceiling((5+3^n+(-1)^n*2^(n+2)+3*(1+(-1)^n)*Sqrt(5)^n)/20): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011

LinearRecurrence[{2,10,-16,-25,30}, {1,0,3,0,15,6}, 41] (* G. C. Greubel, Feb 07 2023 *)

def A054882(n): return (5+3^n+4*(-2)^n+3*(1+(-1)^n)*5^(n/2)+4*0^n)/20
[A054882(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

G.f.: (1/20)*(4 + 5/(1-x) + 1/(1-3*x) + 4/(1+2*x) + 6/(1-5*x^2)).
G.f.: (1 - 2*x - 7*x^2 + 10*x^3 + 10*x^4 - 6*x^5)/((1-x)*(1+2*x)*(1-3*x)*(1-5*x^2)).
a(n) = (5 + 3^n + (-1)^n*2^(n+2) + 3*(1+(-1)^n)*sqrt(5)^n + 4*0^n)/20.
E.g.f.: (1/20)*(4 + 4*exp(-2*x) + 5*exp(x) + exp(3*x) + 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

A054884 Number of closed walks of length n along the edges of an icosahedron based at a vertex.

Original entry on oeis.org

1, 0, 5, 10, 65, 260, 1365, 6510, 32865, 162760, 815365, 4069010, 20352865, 101725260, 508665365, 2543131510, 12715852865, 63578287760, 317892415365, 1589457194010, 7947290852865, 39736429850260

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,10,-20,-25).

Cf. A030517, A030518, A054881, A054882, A054883, A054885.

[Floor((5^n+(-1)^n*5+3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011

LinearRecurrence[{4,10,-20,-25},{1,0,5,10},30] (* Harvey P. Dale, May 02 2022 *)

a(n) = if(n%2, 5^n-5, 5^n+5+6*5^(n/2))/12; \\ François Marques, Jul 11 2021

def A054884(n): return (5^n + 5*(-1)^n + 3*(1 + (-1)^n)*5^(n/2))/12
[A054884(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) + 6/(1-5*t^2)).
a(n) = (5^n + (-1)^n*5 + 3*(1 + (-1)^n)*sqrt(5)^n)/12.
a(n+1) = 5 * A030517(n) for n > 0.
a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4). - François Marques, Jul 10 2021
E.g.f.: (1/12)*(5*exp(-x) + exp(5*x) + 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

A054885 Number of walks of length n along the edges of an icosahedron between two opposite vertices.

Original entry on oeis.org

0, 0, 0, 10, 40, 260, 1240, 6510, 32240, 162760, 812240, 4069010, 20337240, 101725260, 508587240, 2543131510, 12715462240, 63578287760, 317890462240, 1589457194010, 7947281087240, 39736429850260

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,10,-20,-25).

Cf. A030517, A030518, A054881, A054882, A054883, A054884.

[Floor((5^n+(-1)^n*5-3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011

LinearRecurrence[{4,10,-20,-25}, {0,0,0,10}, 41] (* G. C. Greubel, Feb 07 2023 *)

a(n) = if(n%2, 5^n-5, 5^n+5-6*5^(n/2))/12; \\ François Marques, Jul 11 2021

def A054885(n): return (5^n +5*(-1)^n -3*(1+(-1)^n)*5^(n/2))/12
[A054885(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) - 6/(1-5*t^2)).
a(n) = (5^n + 5*(-1)^n - 3*(1 + (-1)^n)*sqrt(5)^n)/12.
a(n+1) = 5 * A030518(n) for n > 0.
a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4). - François Marques, Jul 10 2021
E.g.f.: (1/12)*(5*exp(-x) + exp(5*x) - 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

A100284 Expansion of (1-4x-x^2)/((1-x)(1-4x-5x^2)).

Original entry on oeis.org

1, 1, 5, 21, 105, 521, 2605, 13021, 65105, 325521, 1627605, 8138021, 40690105, 203450521, 1017252605, 5086263021, 25431315105, 127156575521, 635782877605, 3178914388021, 15894571940105, 79472859700521, 397364298502605

Offset: 0

Paul Barry, Nov 11 2004

Binomial transform of A054881.

Binomial transform of A179607. - Johannes W. Meijer, Aug 01 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,1,-5)

Cf. A054881, A100285, A100286, A179607.

[(5^n +2*(-1)^n +3)/6: n in [0..40]]; // G. C. Greubel, Feb 06 2023

CoefficientList[Series[(1-4x-x^2)/((1-x)(1-4x-5x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{5,1,-5},{1,1,5},30] (* Harvey P. Dale, Apr 01 2013 *)

def A100284(n): return (1/6)*(5^n +1 +4*((n+1)%2))
[A100284(n) for n in range(41)] # G. C. Greubel, Feb 06 2023

a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3).
a(n) = (1/6)*(3 + 5^n + 2*(-1)^n).
E.g.f.: (1/6)*(exp(5*x) + 3*exp(x) + 2*exp(-x)). - G. C. Greubel, Feb 06 2023

A092807 Expansion of (1-6x+4x^2)/((1-2x)(1-6*x)).

Original entry on oeis.org

1, 2, 8, 40, 224, 1312, 7808, 46720, 280064, 1679872, 10078208, 60467200, 362799104, 2176786432, 13060702208, 78364180480, 470185017344, 2821109972992, 16926659575808, 101559956930560, 609359740534784

Offset: 0

Paul Barry, Mar 06 2004

Second binomial transform of A054881 (closed walks at a vertex of an octahedron) With interpolated zeros, counts closed walks of length n at a vertex of the edge-vertex incidence graph of K_4 associated with the edges of K_4.

This also gives the number of noncrossing, nonnesting, 2-colored permutations on {1, 2, ..., n}. - Lily Yen, Apr 22 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013 and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Cf. A054881, A074601, A092803, A124302.

[1] cat [6^(n-1) + 2^(n-1): n in [1..40]]; // G. C. Greubel, Jan 04 2023

CoefficientList[Series[(1-6x+4x^2)/((1-2x)(1-6x)),{x,0,40}],x] (* or *) LinearRecurrence[{8,-12},{1,2,8},41] (* Harvey P. Dale, Aug 23 2011 *)

[(6^n + 3*2^n + 2*0^n)/6 for n in range(41)] # G. C. Greubel, Jan 04 2023

G.f.: (1-6*x+4*x^2)/((1-2*x)*(1-6*x)).
a(n) = (6^n + 3*2^n + 2*0^n)/6.
a(n) = A074601(n-1), n>0. - R. J. Mathar, Sep 08 2008
a(0)=1, a(1)=2, a(2)=8, a(n) = 8*a(n-1)-12*a(n-2). - Harvey P. Dale, Aug 23 2011
a(n) = A124302(n)*2^n. - Philippe Deléham, Nov 01 2011
E.g.f.: (1/6)*( 1 + 3*exp(2*x) + exp(6*x) ). - G. C. Greubel, Jan 04 2023

A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

Original entry on oeis.org

1, 10, 220, 3080, 52976, 818720, 13333440, 211474560, 3398520576, 54257082880, 869067996160, 13897453373440, 222420341682176, 3558236809994240, 56935698394234880, 910939899548958720, 14575288593717067776, 233202615903456460800

Offset: 0

Ralf Steiner, May 16 2020

Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).
These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.
For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:
x(n) = A084175(n+2).
y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).
     = 2*A192382(n+1) = 4*A003683(n+1).
z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).
     = A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).
     = A000302(n+1) + A139818(n+1).
u(n) = A000079(n+1) = 2^(n+1).
v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.
For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).

G. C. Greubel, Table of n, a(n) for n = 0..825
V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
H. Lee Price, The Pythagorean Tree: A New Species, arXiv:0809.4324 [math.HO], 2008-2011
R. Steiner, Spezielle Folge primitiver pythagoräischer Dreiecke, researchgate.net, 2020
Index entries for linear recurrences with constant coefficients, signature (10,120,-320,-1024).

Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # G. C. Greubel, Feb 18 2023

a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.
G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - Colin Barker, Jun 11 2020
E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - G. C. Greubel, Feb 18 2023

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

Original entry on oeis.org

4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249

Offset: 0

Paul Curtz, Mar 29 2022

Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
    4,  -3,   5,  -1,   9,   7, 25, ...
   -7,   8,  -6,  10,  -2,  18, 14,  50, ...
   15, -14,  16, -12,  20,  -4, 36,  28, 100, ...
  -29,  30, -28,  32, -24,  40, -8,  72,  56, 200, ...
   59, -58,  60, -56,  64, -48, 80, -16, 144, 112, 400, ...
  ...
The diagonals are given by D(n,n+k) = a(k)*2^n.
D(n,1) = -(-1)^n* A340627(n).
a(n)   - a(n) =  0,  0,  0,  0,   0, ...       (trivially)
a(n+1) + a(n) =  1,  2,  4,  8,  16, ... = 2^n (by definition)
a(n+2) - a(n) =  1,  2,  4,  8,  16, ... = 2^n
a(n+3) + a(n) =  3,  6, 12, 24,  48, ... = 2^n*3
a(n+4) - a(n) =  5, 10, 20, 40,  80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
     (...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
Main diagonal: A192382(n).
First upper diagonal: A054881(n+1).
First subdiagonal: A003683(n+1).
Second subdiagonal: A246036(n).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
   4,   3,   5,    1,    9,   -7,   25,   -39, ...  = c(n)
  -1,   2,  -4,    8,  -16,   32,  -64,   128, ...  = -A122803(n)
   3,  -6,  12,  -24,   48,  -96,  192,  -384, ...  =
  -9,  18, -36,   72, -144,  288, -576,  1152, ...
  27, -54, 108, -216,  432, -864, 1728, -3456, ...
...
The first subdiagonal is -A000400(n). The second is A169604(n).

Index entries for linear recurrences with constant coefficients, signature (1,2).

If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
Essentially the same as A115335.
Cf. A000079, A002450, A340627.
Cf. A020714, A175805.
Cf. A045883, A192382, A003683, A246036.
Cf. A054881, A000400, A169604, A122803.
Cf. A132805, A082311, A082365.
Cf. A024495, A132804, A163868.

a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
    (11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022

LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)

a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

abs(a(n)) = A115335(n-1) for n >= 1.
a(3*n) - (-1)^n*4 = A132805(n).
a(3*n+1) + (-1)^n*4 = A082311(n).
a(3*n+2) - (-1)^n*4 = A082365(n).
From Thomas Scheuerle, Mar 29 2022: (Start)
G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
a(n) = (11*(-1)^n + 2^n)/3.
a(n + 2*m) = a(n) + A002450(m)*2^n.
a(2*n) = A192382(n+1) + (-1)^n*a(n).
a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
a(n) = A001045(n) + 4*(-1)^n.
a(n+1) = 2*a(n) -11*(-1)^n.
a(n+2) = a(n) + 2^n.
a(n+4) = a(n) + A020714(n).
a(n+6) = a(n) + A175805(n).
a(2*n) = A163868(n).
a(2*n+1) = (2^(2*n+1) - 11)/3.

Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022

Edited by M. F. Hasler, Apr 26 2022.

cp's OEIS Frontend

A003683 a(n) = 2^(n-1)*(2^n - (-1)^n)/3.

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A246036 Expansion of (1+4*x)/((1+2*x)*(1-4*x)).

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A054883 Number of walks of length n along the edges of a dodecahedron between two opposite vertices.

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A054882 Closed walks of length n along the edges of a dodecahedron based at a vertex.

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A054884 Number of closed walks of length n along the edges of an icosahedron based at a vertex.

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A054885 Number of walks of length n along the edges of an icosahedron between two opposite vertices.

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A100284 Expansion of (1-4*x-x^2)/((1-x)*(1-4*x-5*x^2)).

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A100284 Expansion of (1-4x-x^2)/((1-x)(1-4x-5x^2)).

A092807 Expansion of (1-6x+4x^2)/((1-2x)(1-6*x)).