A001835 -id:A001835 - cp's OEIS frontend

A125077 #4 in an infinite set of generalized Pascal's triangles with trigonometric properties.

1, 1, 3, 1, 4, 11, 1, 7, 15, 41, 1, 8, 38, 56, 153, 1, 11, 46, 186, 209, 571, 1, 12, 81, 232, 859, 780, 2131, 1, 15, 93, 499, 1091, 3821, 7953, 1, 16, 140, 592, 2774, 4912, 16556, 10864, 29681, 1, 19, 156, 1044, 3366

Offset: 1

Gary W. Adamson, Nov 18 2006

Row sums are powers of 4. The triangle is #4 in an infinite of generalized Pascal's triangles constrained by two rules: row sums are powers of N and upward sloping diagonals (as coefficients to polynomials with alternating signs) have roots N + 2*cos(2*Pi/Q).

Right border, A001835, and next to right border, A001353 = bisections of denominator of continued fraction [1, 2, 1, 2, 1, 2, 1, 2]; i.e., bisection of A002530. - Gary W. Adamson, Jun 21 2009

			First few rows of the triangle are:
  1;
  1,  3;
  1,  4, 11;
  1,  7, 15,  41;
  1,  8, 38,  56, 153;
  1, 11, 46, 186, 209, 571;
  1, 12, 81, 232, 859, 780, 2131;
  ...
The upward-sloping diagonal (1, 11, 38, 41) relates to the heptagon and in the form x^3 - 11x^2 + 38x - 41 has a root 5.24697960... = 4 + 2*cos(2*Pi/7). The corresponding matrix is [3, 1, 0; 1, 4, 1; 0, 1, 4]. The next upward-sloping diagonal relates to the octagon, with a characteristic polynomial x^3 - 12x^2 + 46x - 56 and a root 5.414213562... = 4 + 2*cos(2*Pi/8). The corresponding matrix is [4, 1, 0; 1, 4, 1; 0, 1, 4].

Cf. A125076, A125078.

Cf. A001835, A001353. - Gary W. Adamson, Jun 21 2009

Upward-sloping diagonals of the triangle are derived from (alternating) characteristic polynomials of two types of matrices: those of the form: (all 1's in the super and subdiagonals and 3,4,4,4,... in the main diagonal) and (all 1's in the super and subdiagonals and 4,4,4,... in the main diagonal.

A126124 Triangle, matrix inverse of A124733, companion to A123965.

Original entry on oeis.org

1, -2, 1, 5, -5, 1, -13, 19, -8, 1, 34, -65, 42, -11, 1, -89, 210, -183, 74, -14, 1, 233, -654, 717, -394, 115, -17, 1, -610, 1985, -2622, 1825, -725, 165, -20, 1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1

Offset: 1

Gary W. Adamson, Dec 17 2006

Left border (unsigned) = odd-indexed Fibonacci numbers. Left border (unsigned) of A123965 = even-indexed Fibonacci numbers.
Subtriangle of the triangle T(n,k) given by [0,-2,-1/2,-1/2,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2007
Equals A129818*A130595 as lower triangular matrices. - Philippe Deléham, Oct 26 2007
Reversals = bisection of triangle A152063: (1; 1,2; 1,5,5; ...) having the following property: Product_{k=1..floor((n-1)/2)} (1 + 4*cos^2 k*2Pi/n) = the odd-indexed Fibonacci numbers. Example: x^3 - 8x^2 - 19x + 13 relates to the heptagon, and with k=1,2,3,..., the product = 13. - Gary W. Adamson, Aug 15 2010
Apart from signs, equals A123971.
Matrix inverse of A124733.

			First few rows of the triangle are:
    1;
   -2,    1;
    5,   -5,    1;
  -13,   19,   -8,    1;
   34,  -65,   42,  -11,    1;
  -89,  210, -183,   74,  -14,    1;
  ...
Triangle (n >= 0 and 0 <= k <= n) [0,-2,-1/2,-1/2,0,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] begins:
  1;
  0,    1;
  0,   -2,    1;
  0,    5,   -5,    1;
  0,  -13,   19,   -8,    1;
  0,   34,  -65,   42,  -11,    1;
  0,  -89,  210, -183,   74,  -14,    1;
  0,  233, -654,  717, -394,  115,  -17,    1;

Cf. A123965, A124733, A123971, A152063.

Sum_{k=1..n} (-1)^(n-k)*T(n,k) = A001835(n). - Philippe Deléham, Jul 14 2007
T(n,k) = T(n-1,k-1) - 3*T(n-1,k) - T(n-2,k). - Philippe Deléham, Dec 13 2011
T(n,k) = (-1)^(n+k)*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012
G.f.: (1+x)*x*y/(1+3*x+x^2-x*y). - R. J. Mathar, Aug 11 2015

Corrected by Philippe Deléham, Jul 14 2007

More terms from Philippe Deléham, Dec 13 2011

A140827 Interleave denominators and numerators of convergents to sqrt(3).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 15, 26, 41, 56, 97, 153, 209, 362, 571, 780, 1351, 2131, 2911, 5042, 7953, 10864, 18817, 29681, 40545, 70226, 110771, 151316, 262087, 413403, 564719, 978122, 1542841, 2107560, 3650401, 5757961, 7865521, 13623482, 21489003, 29354524, 50843527, 80198051, 109552575

Offset: 0

Peter H van der Kamp, Jul 18 2008, Jul 22 2008

Coefficients of (1+r)^m modulo r^4-r^2+1.
The first few principal and intermediate convergents to 3^(1/2) are 1/1, 2/1, 3/2, 5/3, 7/4, 12/7; essentially, numerators=A143642 and denominators=A140827. - Clark Kimberling, Aug 27 2008
From Michel Dekking, Mar 11 2020: (Start)
This sequence can be seen as a generalization of the Fibonacci numbers A000045. The Zeckendorf expansion of a natural number uses the Fibonacci numbers as constituents. The Zeckendorf expansion is called a 2-bin decomposition in the paper by Demontigny et al.
The numbers a(n) are the constituents of the 3-bin decomposition of a natural number. See Example 4.2 and Proposition 4.3 in the Demontigny et al. paper.
Any natural number N can be uniquely expanded as
      N = Sum_{i=0..k} d(i)*a(i)
under the requirement d(i)d(i+1) = 0, and d(3i)d(3i+2) = 0 for all i.
Here k is the largest integer such that a(k) < N+1.
(End)

			(1+r)^(2+12*q)=(-1)^q*(a(1+18*q)*(1+r^2)+a(2+18*q)*r).
Here we write N = [d(k)d(k-1)...d(0)] for the 3-bin expansion of N.
0=[0], 1 =[1], 2=[10], 3=[100], 4=[1000], 5=[1001], 6=[1010], 7=[10000], 8=[10001], 9=[10010], 10=[10100], 11=[100000]. - _Michel Dekking_, Mar 11 2020

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Demontigny et al., Generalizing Zeckendorf's Theorem to f-decompositions, Journal of Number Theory 141, 135-158 (2014).
Peter H. van der Kamp, Global classification of two-component approximately integrable evolution equations, arXiv:0710.2233 [nlin.SI], 2007-2008.
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997), 122-126.
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-1).

Cf. A001075, A001835, A001353, A002965, A002530.

N:=100: a[0]:=1: a[1]:=1: for i from 2 to N do if i mod 3 = 1 then a[i]:=a[i-1]+a[i-3] else a[i]:=a[i-1]+a[i-2] fi od:

idnc[n_]:=Module[{cvrgts=Convergents[Sqrt[3],n],num,den},num=Take[ Numerator[ cvrgts],{2,-1,2}];den=Denominator[cvrgts]; Riffle[den, num,3]]; idnc[30] (* Harvey P. Dale, Mar 17 2012 *)

a(n) = 4*a(n-3) - a(n-6).
G.f.: ( 1+x+2*x^2-x^3-x^5 ) / ( 1-4*x^3+x^6 ).
a(n) = a(n-1)+a(n-3) if 3 |(n-1), else a(n)=a(n-1)+a(n-2), with n>1.
a(3*n-1) = A001075(n); a(3*n) = A001835(n-1); a(3*n+1) = A001353(n+1).
a(n)^2 = 1+3*a(n-1)^2 if n==2 (mod 3).

A209760 Triangle of coefficients of polynomials v(n,x) jointly generated with A209759; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 11, 21, 1, 3, 11, 38, 55, 1, 3, 11, 41, 124, 144, 1, 3, 11, 41, 150, 389, 377, 1, 3, 11, 41, 153, 533, 1187, 987, 1, 3, 11, 41, 153, 568, 1838, 3549, 2584, 1, 3, 11, 41, 153, 571, 2084, 6168, 10447, 6765, 1, 3, 11, 41, 153, 571, 2128

Offset: 1

Clark Kimberling, Mar 14 2012

Limiting row: A001835
Coefficient of x^n in v(n,x):  even-indexed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...3...8
1...3...11...21
1...3...11...38...55
First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 8x^2.

Cf. A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209760 *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210563 Triangle of coefficients of polynomials u(n,x) jointly generated with A210564; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 3, 10, 13, 1, 3, 11, 32, 34, 1, 3, 11, 40, 99, 89, 1, 3, 11, 41, 141, 299, 233, 1, 3, 11, 41, 152, 482, 887, 610, 1, 3, 11, 41, 153, 556, 1604, 2595, 1597, 1, 3, 11, 41, 153, 570, 1998, 5217, 7508, 4181, 1, 3, 11, 41, 153, 571, 2113, 7042

Offset: 1

Clark Kimberling, Mar 23 2012

Last terms in rows: odd-indexed Fibonacci numbers
Limiting row:  A001835
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...3...5
1...3...10...13
1...3...11...32...34
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.

Cf. A210564, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A210563 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210564 *)

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A237250 Values of x in the solutions to x^2 - 4xy + y^2 + 11 = 0, where 0 < x < y.

Original entry on oeis.org

2, 3, 5, 10, 18, 37, 67, 138, 250, 515, 933, 1922, 3482, 7173, 12995, 26770, 48498, 99907, 180997, 372858, 675490, 1391525, 2520963, 5193242, 9408362, 19381443, 35112485, 72332530, 131041578, 269948677, 489053827, 1007462178, 1825173730, 3759900035

Offset: 1

Colin Barker, Feb 05 2014

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 14xy + y^2 + 176 = 0.

			10 is in the sequence because (x, y) = (10, 37) is a solution to x^2 - 4xy + y^2 + 11 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A001075, A001835, A003500, A082841.

Vec(-x*(x-1)*(x+2)*(2*x+1)/(x^4-4*x^2+1) + O(x^100))

a(n) = 4*a(n-2)-a(n-4).

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

A357115 T(n,m) is the numerator of the resistance between two nodes located at the end of a side of length n of a rectangular electric network of n*m quadratic meshes in which all edges are replaced by one-ohm resistors, where T(n,m) is a square array read by descending antidiagonals.

Original entry on oeis.org

3, 11, 4, 41, 5, 13, 153, 26, 267, 26, 571, 68, 181, 192, 149, 2131, 89, 10609, 1506, 1171, 138, 7953, 466, 25059, 251, 155927, 246, 375, 29681, 1220, 3869723, 13852, 759435, 77948, 75255, 668, 110771, 1597, 1334085, 781778, 109897, 1949020, 982871, 24995, 3523

Offset: 1

Hugo Pfoertner, Sep 15 2022

			The array of resistances starts:
    3/4,      11/15,        41/56,         153/209, ... A001835(n-1)/A001353(n-1)
    4/3,       5/4,         26/21,          68/55,   ...
   13/7,     267/161,      181/112,      10609/6603,  ...
   26/11,    192/95,      1506/781,        251/132,   ...
  149/52,   1171/495,   155927/70616,   759435/352583,  ...
  138/41,    246/91,     77948/31529,  1949020/817991,  ...
.
T(1,3)/A357116(1,3) = 41/56:
.      _____       _____       _____
   O--|__1__|--O--|__1__|--O--|__1__|--O-----O
   |           |           |           |     |
  | |         | |         | |         | |    |
  |1|         |1|         |1|         |1|  41/56 ohms
  |_|         |_|         |_|         |_|    |
   |   _____   |   _____   |   _____   |     |
   O--|__1__|--O--|__1__|--O--|__1__|--O-----O
.
T(3,1)/A357116(3,1) = 13/7    T(2,2)/A357116(2,2) = 5/4
.      _____                    _____       _____
   O--|__1__|--O-----O      O--|__1__|--O--|__1__|--O-----O
   |           |     |      |           |           |     |
  | |         | |    |     | |         | |         | |    |
  |1|         |1|    |     |1|         |1|         |1|    |
  |_|         |_|    |     |_|         |_|         |_|    |
   |   _____   |     |      |   _____   |   _____   |     |
   O--|__1__|--O     |      O--|__1__|--O--|__1__|--O    5/4 ohms
   |           |     |      |           |           |     |
  | |         | |  13/7    | |         | |         | |    |
  |1|         |1|  ohms    |1|         |1|         |1|    |
  |_|         |_|    |     |_|         |_|         |_|    |
   |   _____   |     |      |   _____   |   _____   |     |
   O--|__1__|--O     |      O--|__1__|--O--|__1__|--O-----O
   |           |     |
  | |         | |    |
  |1|         |1|    |
  |_|         |_|    |
   |   _____   |     |
   O--|__1__|--O-----O

A357116 are the corresponding denominators.

Cf. A357113, A357114.

Limit_{n->oo} T(1,n)/A357116(1,n) = sqrt(3) - 1.

A097947 Expansion of g.f. (2+7x+2x^2)/((x^2-1)(1+4x+x^2)).

Original entry on oeis.org

-2, 1, -6, 16, -62, 225, -842, 3136, -11706, 43681, -163022, 608400, -2270582, 8473921, -31625106, 118026496, -440480882, 1643897025, -6135107222, 22896531856, -85451020206, 318907548961, -1190179175642, 4441809153600, -16577057438762, 61866420601441, -230888624967006

Offset: 0

N. J. A. Sloane, following a suggestion of Creighton Dement, Sep 06 2004

One of 4 related sequences. This is the sequence "les(n)". "jes(n)" = [1, -4, 15, -56, ...] is (-1)^(n+1)*A001353(n+1), "tes(n)" is A097948 and "ves(n)" is A099949.

Index entries for linear recurrences with constant coefficients, signature (-4,0,4,1).

Cf. A001353, A097948, A097949.

LinearRecurrence[{-4, 0, 4, 1}, {-2, 1, -6, 16}, 27] (* Robert P. P. McKone, Aug 25 2023 *)

Properties (from Creighton Dement, Sep 06 2004):
I: jes(n) + les(n) + tes(n) = ves(n)
II: All of the following are perfect squares: {les(2n+1); tes(2n+1); ves(2n+1); ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1; 3*les(2n+1) + 1 = 3*jes(n)^2 + 1}.
III: les(2n+1) divides ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1
IV: (jes(n))^2 = les(2n+1)
V: tes(2n) = A001570(n), sqrt( tes(2n+1) ) = A001075(n)
VI: sqrt( ves(2n+1) ) = A001835(n)
VII: sqrt( les(2n+1) ) = A001353(n)
VIII: les(n) + tes(n) = ves(2+n) + jes(n)
IX: lim n |jes(n+1)/jes(n)| = lim n |les(n+1)/les(n)| = lim n |tes(n+1)/tes(n)| = lim n |ves(n+1)/ves(n)| = 2 + sqrt(3)
Comment from Roland Bacher, Sep 07 2004: These 4 sequences satisfy jes(n+1)=-4*jes(n)-jes(n-1), les(n+1)=les(n-1)+jes(n), ves(n+1)=les(n-1)-jes(n-1)+tes(n-1), tes(n+1)=les(n-1)+3*jes(n), plus initial conditions for n=0, 1.
12*a(n) = -11 -9*(-1)^n -2*(-1)^n*A001075(n+1). - R. J. Mathar, May 21 2019
From Eric Simon Jacob, Aug 26 2023: (Start)
a(n) = ( ( sqrt(3) - 2 )^(n+1) + ( -sqrt(3) - 2 )^(n+1) + 9*(-1)^(n+1) - 11 )/12.
a(n) = ( 2*cosh( (n+1)*log(sqrt(3) - 2) ) + 9*(-1)^(n+1) - 11 )/12. (End)

A100245 Triangle read by rows: T(n,k) is the number of k-matchings in the P_3 X P_n lattice graph.

Original entry on oeis.org

1, 1, 2, 1, 7, 11, 3, 1, 12, 44, 56, 18, 1, 17, 102, 267, 302, 123, 11, 1, 22, 185, 758, 1597, 1670, 757, 106, 1, 27, 293, 1654, 5256, 9503, 9401, 4603, 908, 41, 1, 32, 426, 3080, 13254, 35004, 56456, 53588, 27688, 6716, 540, 1, 37, 584, 5161, 28191, 99183

Offset: 0

Emeric Deutsch, Dec 28 2004

Row n contains 1+floor(3n/2) terms. Row sums yield A033506.

			T(2,2)=11 because in the P_3 X P_ 2 lattice graph with vertex set {O(0,0),A(1,0),B(1,1),C(1,2),D(0,2),E(0,1)} and edge set {OA,EB,DC,OE,ED,AB,BC} we have the following eleven 2-matchings: {OA,EB},{OA,DC},{EB,DC},{OA,ED},{OA,BC},{DC,OE},{DC,AB},{OE,AB},{OE,BC},{ED,AB} and {ED,BC}.
Triangle starts:
1;
1,2;
1,7,11,3;
1,12,44,56,18;
1,17,102,267,302,123,11;

H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (26) and Table V).

Cf. A033506, A001835 (bisection diagonal).

G:=(1+t*z-t^3*z^2)*(1-2*t*z-t^3*z^2)/(1-(1+3*t)*z-t*(1+t)*(2+5*t)*z^2-t^2*(1+2*t)*(1-t)*z^3+t^4*(2+3*t+5*t^2)*z^4-t^6*(1-t)*z^5-t^9*z^6): Gser:=simplify(series(G,z=0,11)): P[0]:=1: for n from 1 to 8 do P[n]:=coeff(Gser,z^n) od:for n from 0 to 8 do seq(coeff(t*P[n],t^k),k=1..floor(3*n/2)+1) od; # yields sequence in triangular form

G.f.=(1+tz-t^3*z^2)(1-2tz-t^3*z^2)/[1-(1+3t)z-t(1+t)(2+5t)z^2-t^2*(1+2t)(1-t)z^3+t^4*(2+3t+5t^2)z^4-t^6*(1-t)z^5-t^9*z^6]. The row generating polynomials A[n] satisfy A[n]=(1+3t)A[n-1]+t(2+7t+5t^2)A[n-2]+t^2*(1+t-2t^2)A[n-3]-t^4*(2+3t+5t^2)A[n-4]+t^6*(1-t)A[n-5]+t^9*A[n-6].

Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013

A123520 Number of vertical dominoes in all possible tilings of a 2n X 3 grid by dominoes.

Original entry on oeis.org

4, 28, 152, 744, 3436, 15284, 66224, 281424, 1178196, 4874444, 19973192, 81189688, 327817404, 1316035940, 5257118560, 20909651104, 82849544868, 327163551612, 1288036695544, 5057236343176, 19807689093644, 77408388584724

Offset: 1

Emeric Deutsch, Oct 16 2006

			a(1) = 4 because a 2 X 3 grid can be tiled in 3 ways with dominoes: 3 horizontal dominoes, 1 horizontal domino above two adjacent vertical dominoes and 1 horizontal domino below two adjacent vertical dominoes; these have altogether 4 vertical dominoes.

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

Cf. A001835, A123519.

a:=n->sum(k*2^(k+1)*binomial(n+k,2*k),k=0..n): seq(a(n),n=1..24);

FullSimplify[Table[(2+Sqrt[3])^n*((1+Sqrt[3])*n+1/Sqrt[3])/3 + (2-Sqrt[3])^n*((1-Sqrt[3])*n-1/Sqrt[3])/3,{n,1,20}]] (* Vaclav Kotesovec, Nov 29 2012 *)
Table[Sum[2^(k + 1)*k*Binomial[n + k, 2 k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 14 2017 *)

z='z+O('z^50); Vec(4*z*(1-z)/(1-4*z+z^2)^2) \\ G. C. Greubel, Oct 14 2017

a(n) = Sum_{k=0..n} 2^(k+1) * k * C(n+k,2*k).
a(n) = Sum_{k=0..n} k * A123519(n,k).
G.f.: 4*z*(1-z)/(1-4*z+z^2)^2.
a(n) = (2+sqrt(3))^n*((1+sqrt(3))*n+1/sqrt(3))/3 + (2-sqrt(3))^n*((1-sqrt(3))*n-1/sqrt(3))/3. - Vaclav Kotesovec, Nov 29 2012

cp's OEIS Frontend

A125077 #4 in an infinite set of generalized Pascal's triangles with trigonometric properties.

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A126124 Triangle, matrix inverse of A124733, companion to A123965.

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Extensions

A140827 Interleave denominators and numerators of convergents to sqrt(3).

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Maple

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A209760 Triangle of coefficients of polynomials v(n,x) jointly generated with A209759; see the Formula section.

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Mathematica

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A210563 Triangle of coefficients of polynomials u(n,x) jointly generated with A210564; see the Formula section.

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Mathematica

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A237250 Values of x in the solutions to x^2 - 4xy + y^2 + 11 = 0, where 0 < x < y.

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PARI

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A357115 T(n,m) is the numerator of the resistance between two nodes located at the end of a side of length n of a rectangular electric network of n*m quadratic meshes in which all edges are replaced by one-ohm resistors, where T(n,m) is a square array read by descending antidiagonals.

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A097947 Expansion of g.f. (2+7*x+2*x^2)/((x^2-1)*(1+4*x+x^2)).

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A100245 Triangle read by rows: T(n,k) is the number of k-matchings in the P_3 X P_n lattice graph.

A097947 Expansion of g.f. (2+7x+2x^2)/((x^2-1)(1+4x+x^2)).