A109437 -id:A109437 - cp's OEIS frontend

A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.

1, 2, 9, 32, 121, 450, 1681, 6272, 23409, 87362, 326041, 1216800, 4541161, 16947842, 63250209, 236052992, 880961761, 3287794050, 12270214441, 45793063712, 170902040409, 637815097922, 2380358351281, 8883618307200, 33154114877521, 123732841202882

Offset: 0

N. J. A. Sloane

Number of tilings of a box with sides 2 X 2 X n in R^3 by boxes of sides 2 X 1 X 1 (3-dimensional dominoes). - Frans J. Faase
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Also stacking bricks.
a(n)*(-1)^n = (1-T(n+1,-2))/3, n>=0, with Chebyshev's polynomials T(n,x) of the first kind, is the r=-2 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. - Wolfdieter Lang, Oct 18 2004
Partial sums of A217233. - Bruno Berselli, Oct 01 2012
The sequence is the case P1 = 2, P2 = -8, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

			G.f. = 1 + 2*x + 9*x^2 + 32*x^3 + 121*x^4 + 450*x^5 + ... - _Michael Somos_, Mar 17 2022

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 360.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Butler and S. Osborne, Counting tilings by taking walks, preprint, 2012; J. Combin. Math. Combin. Comput. 88 2014 17-25. - From _N. J. A. Sloane_, Dec 27 2012
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
D. Deford, Seating rearrangements on arbitrary graphs, preprint 2013; involve, Vol. 7 (2014), No. 6, 787-805.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
Charles H. Jepsen, Packing a box with bricks, Math. Mag. 64 (2) (1991) 92-97, Table 1
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (36).
Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.
László Németh, Tilings of (2 X 2 X n)-board with colored cubes and bricks, arXiv:1909.11729 [math.CO], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Radovan Potůček, The number of fillings a 2 X 2 X n prism with 1 X 1 X 2 prisms, Equations (2024) Vol. 3, 104-114.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to dominoes
Index entries for sequences related to bricks
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A002530, A004003, A006125, A217233 (first differences), A109437 (partial sums).
Column k=2 of A181206, A189650, A233308.
Cf. A100047.

a:=[1,2,9];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Nov 16 2018

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)/(1-3*x-3*x^2+x^3))); // G. C. Greubel, Nov 16 2018

CoefficientList[Series[(1 - x)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 15 2012 *)
RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == BitXor[1, a[n - 1]]^2/a[n - 2]}, a, {n, 30}] (* Jon Maiga, Nov 16 2018 *)
LinearRecurrence[{3,3,-1}, {1,2,9}, 30] (* G. C. Greubel, Nov 16 2018 *)
a[ n_] := (-1)^n * ChebyshevU[n, Sqrt[-1/2]]^2; (* Michael Somos, Mar 17 2022 *)

a(n)=(sqrt(3)+2)^(n+1) \/ 6 \\ Charles R Greathouse IV, Aug 18 2016

a(n)=([0,1,0; 0,0,1; -1,3,3]^n*[1;2;9])[1,1] \\ Charles R Greathouse IV, Aug 18 2016

Vec((1 - x) / ((1 + x)*(1 - 4*x + x^2)) + O(x^40)) \\ Colin Barker, Dec 16 2017

{a(n) = simplify((-1)^n * polchebyshev(n, 2, quadgen(-8)/2)^2)}; /* Michael Somos, Mar 17 2022 */

s=((1-x)/(1-3*x-3*x^2+x^3)).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018

G.f.: (1-x)/((1+x)*(1-4*x+x^2)) = (1-x)/(1-3*x-3*x^2+x^3). - Simon Plouffe in his 1992 dissertation; typo corrected by Vincenzo Librandi, Oct 15 2012
Nearest integer to (1/6)*(2+sqrt(3))^(n+1). - Don Knuth, Jul 15 1995
For n >= 4, a(n) = 3a(n-1) + 3a(n-2) - a(n-3). - Avi Peretz (njk(AT)netvision.net.il), Mar 30 2001
For n >= 3, a(n) = 4a(n-1) - a(n-2) + 2*(-1)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 14 2001
From Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 11 2001: The values are a(1) = 2 * 1^2, a(2) = 3^2, a(3) = 2 * 4^2, a(4) = 11^2, a(5) = 2 * 15^2, ... and in general for odd n a(n) is twice a square, for even n a(n) is a square. If we define b(n) by b(n) = sqrt(a(n)) for even n, b(n) = sqrt(a(n)/2) for odd n then apart from the first 2 elements b(n) is A002530(n+1).
a(n) + a(n+1) = A001835(n+2). - R. J. Mathar, Dec 06 2013
From Peter Bala, Apr 03 2014: (Start)
a(n) = |U(n,i/sqrt(2))|^2 where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n-1) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 2; 1, 1] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = (2*(-1)^n + (2-sqrt(3))^(1+n) + (2+sqrt(3))^(1+n)) / 6. - Colin Barker, Dec 16 2017
a(n) = (1 XOR a(n-1))^2/a(n-2). - Jon Maiga, Nov 16 2018
a(n) = a(-2-n) for all n in Z. - Michael Somos, Mar 17 2022
INVERT transform of sequence p(n), n > 0, where p is the number of nonreducible tilings by height of 2 X 2 X n using dicubes; p is (2, 5, 4, 4, 4, 4...). - Nicolas Bělohoubek, Jun 04 2024

A214992 Power ceiling-floor sequence of (golden ratio)^4.

Original entry on oeis.org

7, 47, 323, 2213, 15169, 103969, 712615, 4884335, 33477731, 229459781, 1572740737, 10779725377, 73885336903, 506417632943, 3471038093699, 23790849022949, 163064905066945, 1117663486445665, 7660579500052711

Offset: 0

Clark Kimberling, Nov 08 2012, Jan 24 2013

Let f = floor and c = ceiling.  For x > 1, define four sequences as functions of x, as follows:
p1(0) = f(x), p1(n) = f(x*p1(n-1));
p2(0) = f(x), p2(n) = c(x*p2(n-1)) if n is odd and p2(n) = f(x*p1(n-1)) if n is even;
p3(0) = c(x), p3(n) = f(x*p3(n-1)) if n is odd and p3(n) = c(x*p3(n-1)) if n is even;
p4(0) = c(x), p4(n) = c(x*p4(n-1)).
The present sequence is given by a(n) = p3(n).
Following the terminology at A214986, call the four sequences power floor, power floor-ceiling, power ceiling-floor, and power ceiling sequences.  In the table below, a sequence is identified with an A-numbered sequence if they appear to agree except possibly for initial terms.  Notation: S(t)=sqrt(t), r = (1+S(5))/2 = golden ratio, and Limit = limit of p3(n)/p2(n).
x ......p1..... p2..... p3..... p4.......Limit
r.......A000045 A000045 A000045 A000045..r
r^2.....A001519 A001654 A061646 A001906..-1+S(5)
r^3.....A024551 A001076 A015448 A049652..-1+S(5)
r^4.....A049685 A157335 A214992 A004187..-19+9*S(5)
r^5.....A214993 A049666 A015457 A214994...(-9+5*S(5))/2
r^6.....A007805 A156085 A214995 A049660..-151+68*S(5)
1+S(2)..A024537 A000129 A001333 A048739..S(2)
2+S(2)..A007052 A214996 A214997 A007070..(1+S(2))/2
1+S(3)..A057960 A002605 A028859 A077846..(1+S(3))/2
2+S(3)..A001835 A109437 A214998 A001353..-4+3*S(3)
S(5)....A214999 A215091 A218982 A218983..1.26879683...
2+S(5)..A024551 A001076 A015448 A049652..-1+S(5)
2+S(6)..A218984 A090017 A123347 A218985..S(3/2)
2+S(7)..A218986 A015530 A126473 A218987..(1+S(7))/3
2+S(8)..A218988 A057087 A086347 A218989..(1+S(2))/2
3+S(8)..A001653 A084158 A218990 A001109..-13+10*S(2)
3+S(10).A218991 A005668 A015451 A218992..-2+S(10)
...
Properties of p1, p2, p3, p4:
(1) If x > 2, the terms of p2 and p3 interlace: p2(0) < p3(0) < p2(1) < p3(1) < p2(2) < p3(2)... Also, p1(n) <= p2(n) <= p3(n) <= p4(n) <= p1(n+1) for all x>0 and n>=0.
(2) If x > 2, the limits L(x) = limit(p/x^n) exist for the four functions p(x), and L1(x) <= L2(x) <= L3(x) <= L4 (x). See the Mathematica programs for plots of the four functions; one of them also occurs in the Odlyzko and Wilf article, along with a discussion of the special case x = 3/2.
(3) Suppose that x = u + sqrt(v) where v is a nonsquare positive integer.  If u = f(x) or u = c(x), then p1, p2, p3, p4 are linear recurrence sequences.  Is this true for sequences p1, p2, p3, p4 obtained from x = (u + sqrt(v))^q for every positive integer q?
(4) Suppose that x is a Pisot-Vijayaraghavan number. Must p1, p2, p3, p4 then be linearly recurrent?  If x is also a quadratic irrational b + c*sqrt(d), must the four limits L(x) be in the field Q(sqrt(d))?
(5) The Odlyzko and Wilf article (page 239) raises three interesting questions about the power ceiling function; it appears that they remain open.

			a(0) = ceiling(r) = 7, where r = ((1+sqrt(5))/2)^4 = 6.8...; a(1) = floor(7*r) = 47; a(2) = ceiling(47) = 323.

Clark Kimberling, Table of n, a(n) for n = 0..250
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33, 235-240, 1991.
Index entries for linear recurrences with constant coefficients, signature (6,6,-1).
Index entries for sequences related to the Josephus Problem.

Cf. A001622, A214986, A049685, A157335, A004187.

(* Program 1.  A214992 and related sequences *)
x = GoldenRatio^4; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A049685 *)
Table[p2[n], {n, 0, z}]  (* A157335 *)
Table[p3[n], {n, 0, z}]  (* A214992 *)
Table[p4[n], {n, 0, z}]  (* A004187 *)
Table[p4[n] - p1[n], {n, 0, z}]  (* A004187 *)
Table[p3[n] - p2[n], {n, 0, z}]  (* A098305 *)
(* Program 2.  Plot of power floor and power ceiling functions, p1(x) and p4(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p1[x_, 0] := f[x]; p1[x_, n_] := f[x*p1[x, n - 1]];
p4[x_, 0] := c[x]; p4[x_, n_] := c[x*p4[x, n - 1]];
Plot[Evaluate[{p1[x, 10]/x^10, p4[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]
(* Program 3. Plot of power floor-ceiling and power ceiling-floor functions, p2(x) and p3(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p2[x_, 0] := f[x]; p3[x_, 0] := c[x];
p2[x_, n_] := If[Mod[n, 2] == 1, c[x*p2[x, n - 1]], f[x*p2[x, n - 1]]]
p3[x_, n_] := If[Mod[n, 2] == 1, f[x*p3[x, n - 1]], c[x*p3[x, n - 1]]]
Plot[Evaluate[{p2[x, 10]/x^10, p3[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]

a(n) = floor(r*a(n-1)) if n is odd and a(n) = ceiling(r*a(n-1)) if n is even, where a(0) = ceiling(r), r = (golden ratio)^4 = (7 + sqrt(5))/2.
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3).
G.f.: (7 + 5*x - x^2)/((1 + x)*(1 - 7*x + x^2)).
a(n) = (10*(-2)^n+(10+3*sqrt(5))*(7-3*sqrt(5))^(n+2)+(10-3*sqrt(5))*(7+3*sqrt(5))^(n+2))/(90*2^n). - Bruno Berselli, Nov 14 2012
a(n) = 7*A157335(n) + 5*A157335(n-1) - A157335(n-2). - R. J. Mathar, Feb 05 2020
E.g.f.: exp(-x)*(5 + 2*exp(9*x/2)*(155*cosh(3*sqrt(5)*x/2) + 69*sqrt(5)*sinh(3*sqrt(5)*x/2)))/45. - Stefano Spezia, Oct 28 2024

A011916 a(n) = ((b(n)-1)+sqrt(3b(n)^2-4b(n)+1))/2, where b(n) is A011922(n).

Original entry on oeis.org

0, 3, 44, 615, 8568, 119339, 1662180, 23151183, 322454384, 4491210195, 62554488348, 871271626679, 12135248285160, 169022204365563, 2354175612832724, 32789436375292575, 456697933641263328, 6360981634602394019

Offset: 0

Mario Velucchi (mathchess(AT)velucchi.it)

Integers k such that k^2 = Sum_{i=1..x} (k+i) for some value of x. 3 is a term because 3^2=9 and 4+5=9; 44 is a term because 44^2=1936 and the sum of (45,46,47,...,76) = 1936. - Gil Broussard, Dec 23 2008
Also the index of the first of two consecutive octagonal numbers whose sum is equal to the sum of two consecutive squares. - Colin Barker, Dec 20 2014
Also the index of a triangular number included in A239071. - Ivan Neretin, May 31 2015

Mario Velucchi, "Seeing couples" in Recreational and Educational Computing, to appear 1997. [apparently never materialized, Colin Barker, Dec 23 2014]

Colin Barker, Table of n, a(n) for n = 0..874
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A011918, A109437.

RecurrenceTable[{a[n] == 15 a[n - 1] - 15 a[n - 2] + a[n - 3], a[0] == 0, a[1] == 3, a[2] == 44}, a, {n, 0, 17}] (* Michael De Vlieger, Jul 02 2015 *)
LinearRecurrence[{15,-15,1},{0,3,44},30] (* Harvey P. Dale, Jul 26 2018 *)

{a(n) = if( n<0, n = -n; polcoeff( x*(1 - 3*x) / ((x-1) * (x^2 - 14*x + 1)) + x * O(x^n), n), polcoeff( x*(x - 3) / ((x-1) * (x^2 - 14*x + 1)) + x * O(x^n), n))} /* Michael Somos, Jul 27 2012 */

concat(0, Vec(x*(-3+x)/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, Dec 20 2014

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +15*a(n-1) -15*a(n-2) +a(n-3).
G.f.: x*(-3 + x) / ((x - 1)*(x^2 - 14*x + 1)). (End)
From Michael Somos, Jul 27 2012: (Start)
a(n) = A109437(2*n).
a(-1 - n) = -A109437(2*n + 1). (End)
a(n) = (A001353(n+1)^2 - A001075(n)^2)/4. - Richard R. Forberg, Aug 26 2013
a(n) = (-2-(7-4*sqrt(3))^n*(-1+sqrt(3))+(1+sqrt(3))*(7+4*sqrt(3))^n)/12. - Colin Barker, Mar 05 2016

More terms from R. J. Mathar, Apr 15 2010

Added a(0)=0, Michael Somos, Jul 27 2012

A217233 Expansion of (1-2x+x^2)/(1-3x-3*x^2+x^3).

Original entry on oeis.org

1, 1, 7, 23, 89, 329, 1231, 4591, 17137, 63953, 238679, 890759, 3324361, 12406681, 46302367, 172802783, 644908769, 2406832289, 8982420391, 33522849271, 125108976697, 466913057513, 1742543253359, 6503259955919, 24270496570321, 90578726325361

Offset: 0

Bruno Berselli, Sep 28 2012

Numbers with the property a(n)^2+a(n-1)^2 = 2*(a(n)-a(n-1)-(-1)^n)^2.

			a(3)=23, a(2)=7: 23^2+7^2 = 2*(23-7-(-1)^3)^2 = 578;
a(6)=1231, a(5)=329: 1231^2+329^2 = 2*(1231-329-(-1)^6)^2 = 1623602.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A109437 (1/(1-3*x-3*x^2+x^3)), A006253 ((1-x)/(1-3*x-3*x^2+x^3)).

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+x^2)/(1-3*x-3*x^2+x^3)));

CoefficientList[Series[(1 - 2 x + x^2)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 25}], x]

makelist(coeff(taylor((1-2*x+x^2)/(1-3*x-3*x^2+x^3), x, 0, n), x, n), n, 0, 25);

Vec((1-2*x+x^2)/(1-3*x-3*x^2+x^3)+O(x^26))

G.f.: (1-x)^2/((1+x)*(1-4*x+x^2)).
a(n) = (4*(-2)^n+(1-sqrt(3))^(2*n+1)+(1+sqrt(3))^(2*n+1))/(6*2^n).
a(n) = -a(-n-1) = 3*a(n-1)+3*a(n-2)-a(n-3) = 4*a(n-1)-a(n-2)+4*(-1)^n.
a(n)+a(n-1) = A052530(n) with a(-1)=-1.
a(n)-a(n-2) = A003699(n) with n>1.
Sum(a(i), i=0..n) = A006253(n).

A109438 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).

Original entry on oeis.org

1, 5, 18, 68, 253, 945, 3526, 13160, 49113, 183293, 684058, 2552940, 9527701, 35557865, 132703758, 495257168, 1848324913, 6898042485, 25743845026, 96077337620, 358565505453, 1338184684193, 4994173231318, 18638508241080

Offset: 0

Creighton Dement, Jun 28 2005

See A109437 for comments.

Floretion Algebra Multiplication Program, FAMP Code: (-1)^(n)jbasejfor[ + .5'ii' + .5'kk' + .5'ij' + .5'ji' + .5'jk' + .5'kj'] 1vesfor = (-1,-1,-1,-1,)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A109437, A001834, A102871.

LinearRecurrence[{3,3,-1},{1,5,18},30] (* Harvey P. Dale, Sep 07 2021 *)

Vec((1 + 2*x) / ((1 + x)*(1 - 4*x + x^2)) + O(x^30)) \\ Colin Barker, May 12 2019

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

a(n) = (-2*(-1)^n + (7-5*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(7+5*sqrt(3))) / 12. - Colin Barker, May 12 2019

A214998 Power ceiling-floor sequence of 2 + sqrt(3).

Original entry on oeis.org

4, 14, 53, 197, 736, 2746, 10249, 38249, 142748, 532742, 1988221, 7420141, 27692344, 103349234, 385704593, 1439469137, 5372171956, 20049218686, 74824702789, 279249592469, 1042173667088, 3889445075882, 14515606636441, 54172981469881, 202176319243084

Offset: 0

Clark Kimberling, Nov 10 2012

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2 + sqrt(3), and the limit p3(r) = (23 + 13*sqrt(3))/12.

			a(0) = ceiling(r) =  4, where r = 2+sqrt(3);
a(1) = floor(4*r) = 14; a(2) = ceiling(14*r) = 53.

Clark Kimberling, Table of n, a(n) for n = 0..250
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A214992, A001835, A109437, A001353, A001654.

x = 2 + Sqrt[3]; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A001835 *)
Table[p2[n], {n, 0, z}]  (* A109437 *)
Table[p3[n], {n, 0, z}]  (* A214998 *)
Table[p4[n], {n, 0, z}]  (* A001353 *)

Vec((4 + 2*x - x^2) / ((1 + x)*(1 - 4*x + x^2)) + O(x^30)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(3) and a(0) = ceiling(x).
a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3).
G.f.: (4 + 2*x - x^2)/(1 - 3*x - 3*x^2 + x^3).
a(n) = (-1)^n + 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 14. - Peter Bala, Nov 12 2017
a(n) = (1/12)*(2*(-1)^n + (23-13*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(23+13*sqrt(3))). - Colin Barker, Nov 13 2017

A387489 Number of packing 1X1X2 bricks into 2X2Xn boxes considering packings obtained by rigid motions equivalent.

Original entry on oeis.org

1, 1, 2, 7, 26, 71, 258, 857, 3148, 11300, 41841, 154140, 573201, 2129726, 7935779, 29569762, 110281431, 411333271, 1534676318, 5726191937, 21367848168, 79738762725, 297573920356, 1110521036955, 4144432037026, 15467004104026, 57723125759179, 215424338586742, 803971544759711, 3000455162798396, 11197833423648453, 41790839930063492, 155965434740272813, 582070675232252525

Offset: 0

R. J. Mathar, Aug 31 2025

There seem to be several typos in Jepsen's equations. The enumeration here is derived from the expression of p(n) as 1/8ths of Psi(e)+2*Psi(rho)+Psi(rho^2)+2*Psi(sigma)+2*Psi(rho*sigma) if n>=3.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Charles H. Jepsen, Packing a box with bricks, Math. Mag. 64 (2) (1991) 92-97, Table 1.
Index entries for linear recurrences with constant coefficients, signature (6,-4,-26,33,8,-8,24,-31,-14,12,2,-1).

Cf. A109437 (is Jepsen's b(n)/4), A006253 (rigid motion symmetry ignored, Jepsen's a(n)).

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 +x +2*x^2 -x^3*(-7 +16*x +57*x^2 -118*x^3 -38*x^4 +30*x^5 -53*x^6 +127*x^7 +42*x^8 -49*x^9 -7*x^10 +4*x^11) / ( (x-1)*(1+x) *(x^2+2*x-1) *(x^2+1) *(x^2-4*x+1) *(x^4-4*x^2+1)) )); // Vincenzo Librandi, Sep 02 2025

CoefficientList[Series[1+x+2*x^2-x^3*(-7+16*x+57*x^2-118*x^3-38*x^4+30*x^5-53*x^6+127*x^7+42*x^8-49*x^9-7*x^10+4*x^11)/((x-1)*(1+x)*(x^2+2*x-1)*(x^2+1)*(x^2-4*x+1)*(x^4-4*x^2+1)),{x,0,33}],x] (* Vincenzo Librandi, Sep 02 2025 *)

G.f.: 1 +x +2*x^2 -x^3*(-7 +16*x +57*x^2 -118*x^3 -38*x^4 +30*x^5 -53*x^6 +127*x^7 +42*x^8 -49*x^9 -7*x^10 +4*x^11) / ( (x-1)*(1+x) *(x^2+2*x-1) *(x^2+1) *(x^2-4*x+1) *(x^4-4*x^2+1) ).

cp's OEIS Frontend

A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.

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A214992 Power ceiling-floor sequence of (golden ratio)^4.

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A011916 a(n) = ((b(n)-1)+sqrt(3*b(n)^2-4*b(n)+1))/2, where b(n) is A011922(n).

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A217233 Expansion of (1-2*x+x^2)/(1-3*x-3*x^2+x^3).

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A109438 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).

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A214998 Power ceiling-floor sequence of 2 + sqrt(3).

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A011916 a(n) = ((b(n)-1)+sqrt(3b(n)^2-4b(n)+1))/2, where b(n) is A011922(n).

A217233 Expansion of (1-2x+x^2)/(1-3x-3*x^2+x^3).