A228826 -id:A228826 - cp's OEIS frontend

A301287 Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).

1, 3, 6, 7, 8, 15, 18, 17, 20, 25, 28, 29, 30, 35, 40, 39, 40, 47, 50, 49, 52, 57, 60, 61, 62, 67, 72, 71, 72, 79, 82, 81, 84, 89, 92, 93, 94, 99, 104, 103, 104, 111, 114, 113, 116, 121, 124, 125, 126, 131, 136, 135, 136, 143, 146, 145, 148, 153, 156, 157, 158

Offset: 0

N. J. A. Sloane, Mar 23 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, bottom row, first tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 2 from the list of 20 2-uniform tilings.
Brian Galebach, Enlarged illustration of tiling, suitable for coloring (taken from the web site in the previous link)
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Reticular Chemistry Structure Resource (RCSR), The cph tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program for A301287
N. J. A. Sloane, Trunks and branches for determining coordination sequence, central view. Blue = trunks, red = branches, green = twigs.
N. J. A. Sloane, Trunks and branches, a different scan, truncated on right but otherwise shows quadrants I and IV in detail
N. J. A. Sloane, Trunks and branches in first quadrant, in full.
N. J. A. Sloane, Trunks and branches in fourth quadrant, in full.
N. J. A. Sloane, Details of proof (page 1)
N. J. A. Sloane, Details of proof (page 2)
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,-1).

Cf. A301289.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

Join[{1, 3, 6}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {7, 8, 15, 18, 17, 20}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

PARI
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See Links section.
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G.f. = -(2*x^8-2*x^7-x^6-4*x^5-2*x^4-2*x^3-4*x^2-2*x-1) / ((x^2+1)*(x^2+x+1)*(x-1)^2). N. J. A. Sloane, Mar 28 2018 (This is now a theorem. - N. J. A. Sloane, Apr 05 2018)
Equivalent conjecture: 3*a(n) = 8*n+2*A057078(n+1)+3*A228826(n+2). - R. J. Mathar, Mar 31 2018 (This is now a theorem. - N. J. A. Sloane, Apr 05 2018)
Theorem: G.f. = (1+2*x+4*x^2+2*x^3+2*x^4+4*x^5+1*x^6+2*x^7-2*x^8) / ((1-x)*(1+x^2)*(1-x^3)).
Proof. This follows by applying the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the links, and the details of the proof (by calculating the generating function) are on the next two scanned pages. - N. J. A. Sloane, Apr 05 2018

More terms from Rémy Sigrist, Mar 27 2018

A228825 Delayed continued fraction of e.

Original entry on oeis.org

2, 2, -1, -1, -1, -2, 2, -2, 1, 1, 1, 2, -2, 2, -2, 2, -1, -1, -1, -2, 2, -2, 2, -2, 2, -2, 1, 1, 1, 2, -2, 2, -2, 2, -2, 2, -2, 2, -1, -1, -1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 1, 1, 1, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -1, -1, -1, -2, 2, -2

Offset: 0

Clark Kimberling, Sep 04 2013

An algorithm for the (usual) continued fraction of r > 0 follows:  x(0) = r, a(n) = floor(x(n)), x(n+1) = 1/(x(n) - a(n)).
The accelerated continued fraction uses "round" instead of "floor" (cf. A133593, A133570, A228667), where round(x) is the integer nearest x.
The delayed continued fraction (DCF) uses "second nearest integer", so that all the terms are in {-2, -1, 1, 2}.  If s/t and u/v are consecutive convergents of a DCF, then |s*x-u*t| = 1.
Regarding DCF(e), after the initial (2,2), the strings (-1,-1,-1) and (1,1,1) alternate with odd-length strings of the forms (-2,2,...,-2) and (2,-2,...,2).  The string lengths form the sequence 2,3,3,3,5,3,7,3,9,3,11,3,13,3,...
Comparison of convergence rates is indicated by the following approximate values of x-e, where x is the 20th convergent: for delayed CF, x-e = 5.4x10^-7; for classical CF, x-e = 6.1x10^-16; for accelerated CF, x-e = -6.6x10^-27.  The convergents for accelerated CF are a proper subset of those for classical CF, which are a proper subset of those for delayed CF (which are sampled in Example).

			Convergents: 2, 5/2, 3, 8/3, 11/4, 30/11, 49/18, 68/25, 19/7, 87/32, 106/39, 299/110, 492/181,...

Cf. A133570, A228826

$MaxExtraPrecision = Infinity; x[0] = E; s[x_] := s[x] = If[FractionalPart[x] < 1/2, Ceiling[x], Floor[x]]; a[n_] := a[n] = s[Abs[x[n]]]*Sign[x[n]]; x[n_] := 1/(x[n - 1] - a[n - 1]); t = Table[a[n], {n, 0, 100}]

A264018 Number of (2+1) X (n+1) arrays of permutations of 0..n*3+2 with each element having index change +-(.,.) 0,0 1,2 or 2,2.

Original entry on oeis.org

1, 5, 25, 105, 441, 1869, 7921, 33553, 142129, 602069, 2550409, 10803705, 45765225, 193864605, 821223649, 3478759201, 14736260449, 62423800997, 264431464441, 1120149658761, 4745030099481, 20100270056685, 85146110326225

Offset: 1

R. H. Hardin, Nov 01 2015

nonn

			Some solutions for n=4:
..7..1..2..3..4....0.13..9..3..4....0..8..9..3..4....7..1..9..3..4
.12..6..0..8..9...12..6..7..8..2...12..6.14..1..2...12.13..0..8..2
.10.11..5.13.14...10.11..5..1.14...10.11..5.13..7...10.11..5..6.14

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 2 of A264017.

Empirical: a(n) = 4*a(n-1) + 4*a(n-3) + a(n-4).
Empirical g.f.: x*(1 + x + 5*x^2 + x^3) / ((1 + x^2)*(1 - 4*x - x^2)). - Colin Barker, Jan 03 2019
Empirical: 5*a(n) = 2*A228826(n) + A048876(n). - R. J. Mathar, Sep 09 2020

A322171 Expansion of x(3 + 5x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

Original entry on oeis.org

3, 11, 17, 19, 23, 31, 37, 39, 43, 51, 57, 59, 63, 71, 77, 79, 83, 91, 97, 99, 103, 111, 117, 119, 123, 131, 137, 139, 143, 151, 157, 159, 163, 171, 177, 179, 183, 191, 197, 199, 203, 211, 217, 219, 223, 231, 237, 239, 243, 251, 257, 259, 263, 271, 277, 279, 283, 291, 297, 299

Offset: 1

Mark A. Thomas, Nov 29 2018

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

Cf. A228826.

I:=[3,11,17,19]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Dec 06 2018

seq(coeff(series(x*(x^3+x^2+5*x+3)/((1-x)^2*(1+x^2)),x,n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 06 2018

CoefficientList[Series[(x^3 + x^2 + 5 x + 3)/((x - 1)^2 (x^2 + 1)), {x, 0, 50}], x] (* or *)
a[n_]:= (1/2) (10 n - (1 + 2 * I) (-I)^n - (1 - 2 I) I^n); Simplify[Array[a, 50]] (* Stefano Spezia, Nov 29 2018 *)
LinearRecurrence[{2, -2, 2, -1}, {3, 11, 17, 19}, 60] (* Vincenzo Librandi, Dec 06 2018 *)

Vec((3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)) + O(x^60)) \\ Andrew Howroyd, Nov 29 2018

a(n) = (1/2)*(10*n - (1+2*i)*(-i)^n - (1-2*i)*i^n), where i = sqrt(-1).
a(n) = 5*n - 2*sin(Pi*n/2) - cos(Pi*n/2).
a(n) = 5*n - A228826(n-1). - Andrew Howroyd, Nov 29 2018
G.f.: x*(x^3 + x^2 + 5*x + 3) / ((x - 1)^2 *(x^2 + 1)). - Vincenzo Librandi, Dec 06 2018

A356050 a(n) = 2*A135318(n+1) - A135318(n).

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 11, 14, 21, 26, 43, 54, 85, 106, 171, 214, 341, 426, 683, 854, 1365, 1706, 2731, 3414, 5461, 6826, 10923, 13654, 21845, 27306, 43691, 54614, 87381, 109226, 174763, 218454, 349525, 436906, 699051, 873814, 1398101, 1747626, 2796203, 3495254, 5592405

Offset: 0

Paul Curtz, Aug 19 2022

Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Cf. A001045, A084214, A135318, A230096.

Cf. A094958.

LinearRecurrence[{0, 1, 0, 2}, {1, 1, 3, 4}, 50] (* Amiram Eldar, Aug 19 2022 *)

a(n) = A135318(n) + A230096(n+1).
a(n) = a(n-8) + 5*A094958(n-5).
a(2*n) = A001045(n+2).
a(2*n+1) = A084214(n+1).
From Stefano Spezia, Aug 20 2022: (Start)
O.g.f.: (1 + x + 2*x^2 + 3*x^3)/((1 + x^2)*(1 - 2*x^2)).
E.g.f.: (8*cosh(sqrt(2)*x) - 2*cos(x) + 5*sqrt(2)*sinh(sqrt(2)*x) - 4*sin(x))/6. (End)
3*a(n) = A228826(n+1) +A094958(n+3). - R. J. Mathar, Jan 25 2023

cp's OEIS Frontend

A301287 Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).

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A228825 Delayed continued fraction of e.

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A264018 Number of (2+1) X (n+1) arrays of permutations of 0..n*3+2 with each element having index change +-(.,.) 0,0 1,2 or 2,2.

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A322171 Expansion of x*(3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

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A356050 a(n) = 2*A135318(n+1) - A135318(n).

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A322171 Expansion of x(3 + 5x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).