A007070 -id:A007070 - cp's OEIS frontend

A087161 Records in A087159, i.e., A087159(a(n)) = n, and satisfies the recurrence a(n+3) = 5a(n+2) - 6 a(n+1) + 2*a(n) with a(1) = 1, a(2) = 2, and a(3) = 4.

1, 2, 4, 10, 30, 98, 330, 1122, 3826, 13058, 44578, 152194, 519618, 1774082, 6057090, 20680194, 70606594, 241065986, 823050754, 2810071042, 9594182658, 32756588546, 111837988866, 381838778370, 1303679135746, 4451038986242

Offset: 1

Paul D. Hanna, Aug 22 2003

nonn

Binomial transform of A001333 (which, with an extra leading 1, is the expansion of (1 - x - 2*x^2)/(1 - 2*x - x^2)). - Paul Barry, Aug 26 2003

Partial sums of the binomial transform of Pell(n-1). - Paul Barry, Apr 24 2004

Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Cf. A000129, A007070, A087159, A087160, A100631.

CoefficientList[Series[(1-3x)/(1-5x+6x^2-2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{5,-6,2},{1,2,4},30] (* Harvey P. Dale, Oct 12 2015 *)

G.f.: x*(1 - 3*x)/(1 - 5*x + 6*x^2 - 2*x^3).
a(n) = 2 + 2*A007070(n-3) for n > 2.
a(n) = ((2 - sqrt(2))^(n)/(1 - sqrt(2)) + (2 + sqrt(2))^(n)/(1 + sqrt(2)))/2 + 2 (offset 0) - Paul Barry, Aug 26 2003
a(n+1) - a(n) = A006012(n-1) for n >= 2. - Philippe Deléham, Feb 01 2012
a(1) = 1, a(2) = 2, a(3) = 4, a(n) = 5*a(n-1) - 6*a(n-2) + 2*a(n-3) for n >= 4. - Harvey P. Dale, Oct 12 2015
a(n+1) = Sum_{k=0..n} A100631(n,k) for n >= 0. - Petros Hadjicostas, Feb 09 2021

More terms from Paul Barry, Apr 24 2004

A094806 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 5.

Original entry on oeis.org

1, 5, 20, 74, 264, 924, 3200, 11016, 37792, 129392, 442496, 1512224, 5165952, 17643456, 60250112, 205729920, 702452224, 2398414592, 8188884992, 27958972928, 95458646016, 325917686784, 1112755552256, 3799191029760, 12971261403136, 44286680330240

Offset: 2

Herbert Kociemba, Jun 11 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

Harvey P. Dale, Table of n, a(n) for n = 2..1000
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

f[n_] := FullSimplify[ TrigToExp[(1/4)Sum[ Sin[Pi*k/8]Sin[5Pi*k/8](2Cos[Pi*k/8])^(2n), {k, 1, 7}]]]; Table[ f[n], {n, 2, 25}] (* Robert G. Wilson v, Jun 18 2004 *)
LinearRecurrence[{6,-10,4},{1,5,20},30] (* Harvey P. Dale, Mar 04 2015 *)

a(n) = (1/4)*Sum_{k=1..7} sin(Pi*k/8)*sin(5*Pi*k/8)*(2*cos(Pi*k/8))^(2n).
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
G.f.: x^2*(x-1) / ( (2*x-1)*(2*x^2-4*x+1) ).
a(n) = (-2^n+(-(2-sqrt(2))^n+(2+sqrt(2))^n)/sqrt(2))/4. - Colin Barker, Apr 27 2016
4*a(n) = 2*A007070(n-1) - 2^n.- R. J. Mathar, Nov 14 2019

A140071 Triangle read by rows: iterates of X * [1,0,0,0,...]; where X = an infinite lower bidiagonal matrix with [3,1,3,1,3,1...] in the main diagonal and [1,1,1,...] in the subdiagonal.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 13, 7, 1, 81, 40, 34, 8, 1, 243, 121, 142, 42, 11, 1, 729, 364, 547, 184, 75, 12, 1, 2187, 1093, 2005, 731, 409, 87, 15, 1, 6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1, 19683, 9841, 24604, 9844, 8610, 2454, 892, 148, 19, 1

Offset: 1

Gary W. Adamson and Roger L. Bagula, May 04 2008

Companion triangle A140070 uses an analogous operation with the main diagonal [1,3,1,3,1,3,...].

			First few rows of the triangle are:
1;
3, 1;
9, 4, 1;
27, 13, 7, 1;
81, 40, 34, 8, 1;
243, 121, 142, 42, 11, 1;
729, 364, 547, 184, 75, 12, 1;
2187, 1093, 2005, 731, 409, 87, 15, 1;
6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1;
...

Cf. A140070, A007070 (row sums), A157751.

From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 + (x - 1)*z)/(1 - 4*z - (x^2 - 3)*z^2) = 1 + (x + 3)*z + (x^2 + 4*x + 9)*z^2 + ....
Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).
Recurrence equation for row polynomials: R(n,x) = 4*R(n-1,x) + (x^2 - 3)*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.
Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) + R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)

A209691 Triangle of coefficients of polynomials u(n,x) jointly generated with A209692; see the Formula section.

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 1, 8, 8, 0, 1, 5, 24, 16, 0, 1, 4, 21, 64, 32, 0, 1, 4, 15, 77, 160, 64, 0, 1, 4, 14, 58, 253, 384, 128, 0, 1, 4, 14, 49, 221, 765, 896, 256, 0, 1, 4, 14, 48, 177, 800, 2173, 2048, 512, 0, 1, 4, 14, 48, 165, 654, 2723, 5885, 4608, 1024, 0, 1, 4

Offset: 1

Clark Kimberling, Mar 12 2012

Combinatorial limit of rows: A007070.

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
0...2
0...2...4
0...1...8...8
0...1...5...24...16
First three polynomials v(n,x): 1, 2x, 2x + 4x^2.

Cf. A209692, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, 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[%]   (* A209691 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209692 *)

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

A210218 Triangle of coefficients of polynomials v(n,x) jointly generated with A210217; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 4, 7, 1, 4, 13, 15, 1, 4, 14, 38, 31, 1, 4, 14, 47, 103, 63, 1, 4, 14, 48, 151, 264, 127, 1, 4, 14, 48, 163, 462, 649, 255, 1, 4, 14, 48, 164, 544, 1348, 1546, 511, 1, 4, 14, 48, 164, 559, 1768, 3769, 3595, 1023, 1, 4, 14, 48, 164, 560, 1893, 5564

Offset: 1

Clark Kimberling, Mar 19 2012

Limiting row: A007070

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...4...7
1...4...13...15
1...4...14...38...31
First three polynomials v(n,x): 1, 1 + 3x , 1 + 4x + 7x^2.

Cf. A210217, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
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[%]    (* A210217 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210218 *)

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

A210741 Triangle of coefficients of polynomials u(n,x) jointly generated with A210742; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 7, 19, 21, 1, 9, 34, 65, 55, 1, 11, 53, 141, 210, 144, 1, 13, 76, 257, 534, 654, 377, 1, 15, 103, 421, 1111, 1905, 1985, 987, 1, 17, 134, 641, 2041, 4447, 6512, 5911, 2584, 1, 19, 169, 925, 3440, 9038, 16837, 21557, 17345, 6765, 1

Offset: 1

Clark Kimberling, Mar 24 2012

Rows end with even-indexed Fibonacci numbers
Row sums: A007070
Alternating row sums:  signed powers of 2
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...5...8
1...7...19...21
1...9...34...65...55
First three polynomials u(n,x): 1, 1+ 3x, 1 + 5x + 8x^2.

Cf. A210742, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]     (* A210741 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210742 *)

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

A210753 Triangle of coefficients of polynomials u(n,x) jointly generated with A210754; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 7, 4, 4, 16, 20, 8, 5, 30, 61, 52, 16, 6, 50, 146, 198, 128, 32, 7, 77, 301, 575, 584, 304, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 9, 156, 966, 3060, 5641, 6328, 4272, 1600, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584

Offset: 1

Clark Kimberling, Mar 25 2012

Row n starts with n and ends with 2^(n-1).
Row sums: A007070
Alternating row sums: 1,0,0,0,0,0,0,0,0,....
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...2
3...7....4
4...16...20...8
5...30...61...52...16
First three polynomials u(n,x): 1, 2 + 2x, 3 + 7x + 4x^2.

Cf. A210754, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

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

A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 1, 5, 9, 5, 0, 0, 0, 0, 6, 14, 14, 0, 0, 0, 0, 0, 6, 20, 28, 14, 0, 0, 0, 0, 0, 0, 26, 48, 42, 0, 0, 0, 0, 0, 0, 0, 26, 74, 90, 42, 0, 0, 0, 0, 0, 0, 0, 0, 100, 164, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 264, 296, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364, 560, 428, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 17 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
0, 1, 2, 3, 4, 5, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
0, 0, 2, 5, 9, 14, 20, 26, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=2
0, 0, 0, 5, 14, 28, 48, 74, 100, 100, 0, 0, 0, 0, 0, 0, 0, ... row n=3
0, 0, 0, 0, 14, 42, 90, 162, 264, 364, 364, 0, 0, 0, 0, 0, ... row n=4
0, 0, 0, 0, 0, 42, 132, 296, 560, 924, 1288, 1288, 0, 0, 0, ... row n=5
...

E. Lucas, Théorie des nombres, A. Blanchard, Paris, 1958, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, p. 89

Cf. similar sequences: A216230, A216228, A216226, A216238, A216054.

T(n,n) = A024175(n).
T(n,n+1) = A024175(n+1).
T(n,n+2) = A094803(n+1).
T(n,n+3) = A007070(n).
T(n,n+4) = A094806(n+2).
T(n,n+5) = T(n,n+6) = A094811(n+2).
Sum_{k, 0<=k<=n} T(n-k,k) = A030436(n).

a(69) = 0 deleted by Georg Fischer, Oct 16 2021

A106731 Expansion of -2x/(1 - 4x + 2*x^2).

Original entry on oeis.org

0, -2, -8, -28, -96, -328, -1120, -3824, -13056, -44576, -152192, -519616, -1774080, -6057088, -20680192, -70606592, -241065984, -823050752, -2810071040, -9594182656, -32756588544, -111837988864, -381838778368, -1303679135744, -4451038986240, -15196797673472

Offset: 0

Roger L. Bagula, May 30 2005

See a Oct 01 2013 comment on A007070 where it is pointed out that this sequence, interspersed with zeros, appears, together with A007070, also interspersed with zeros, in the representation of nonnegative powers of the algebraic number rho(8) = 2*cos(Pi/8) in the power basis of the number field Q(rho(8)) of degree 4, known from the octagon. - Wolfdieter Lang, Oct 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A000129, A002203, A007070, A060995.

[n le 2 select -(1+(-1)^n) else 4*Self(n-1) - 2*Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 10 2021

a[0]:=0: a[1]:=-2: for n from 2 to 27 do a[n]:=4*a[n-1]-2*a[n-2] od: seq(a[n], n=0..30);

M= {{0,-2}, {1,4}}; v[1]= {0,1}; v[n_]:= v[n]= M.v[n-1]; Table[Abs[v[n][[1]]], {n, 30}]
CoefficientList[Series[-2x/(1 -4x +2x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2013 *)

def a(n): return -2^((n+2)/2)*lucas_number1(n,2,-1) if (n%2==0) else -2^((n-1)/2)*lucas_number2(n,2,-1)
[a(n) for n in (0..30)] # G. C. Greubel, Sep 10 2021

G.f.: -2*x/(1-4*x+2*x^2).
a(n) = -2*A007070(n-1) for n>=1.
a(n) = 4*a(n-1) - 2*a(n-2); a(0)=0, a(1)=-2.
From G. C. Greubel, Sep 10 2021: (Start)
a(2*n) = -2^(n+1)*Pell(2*n) = -2^(n+1)*A000129(2*n).
a(2*n+1) = -2^n*Q(2n+1) = -2^n*A002203(2*n+1). (End)
E.g.f.: -sqrt(2)*exp(2*x)*sinh(sqrt(2)*x). - Stefano Spezia, May 20 2024

Edited by N. J. A. Sloane, Apr 30 2006

Further editing and simpler name, Joerg Arndt, Oct 02 2013

A111955 a(n) = A078343(n) + (-1)^n.

Original entry on oeis.org

0, 1, 4, 7, 20, 45, 112, 267, 648, 1561, 3772, 9103, 21980, 53061, 128104, 309267, 746640, 1802545, 4351732, 10506007, 25363748, 61233501, 147830752, 356895003, 861620760, 2080136521, 5021893804, 12123924127, 29269742060, 70663408245

Offset: 0

Creighton Dement, Aug 25 2005

This sequence is a companion sequence to A111954 (compare formula / program code). Three other companion sequences (i.e., they are generated by the same floretion given in the program code) are A105635, A097076 and A100828.

Floretion Algebra Multiplication Program, FAMP Code: 4kbasejseq[J*D] with J = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and D = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e. (an initial term 0 was added to the sequence)

Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A078343, A000129, A001333, A111954, A111956, A007070, A077995, A100828, A097076, A105635, A048655.

LinearRecurrence[{1,3,1},{0,1,4},40] (* Harvey P. Dale, Mar 12 2015 *)

a(n) + a(n+1) = A048655(n).
a(n) = a(n-1) + 3*a(n-2) + a(n-3), n >= 3; a(n) = (-1/4*sqrt(2)+1)*(1-sqrt(2))^n + (1/4*sqrt(2)+1)*(1+sqrt(2))^n - (-1)^n;
G.f.: -x*(1+3*x) / ( (1+x)*(x^2+2*x-1) ). - R. J. Mathar, Oct 02 2012
E.g.f.: cosh(x) - exp(x)*cosh(sqrt(2)*x) - sinh(x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, May 26 2024

cp's OEIS Frontend

A087161 Records in A087159, i.e., A087159(a(n)) = n, and satisfies the recurrence a(n+3) = 5*a(n+2) - 6* a(n+1) + 2*a(n) with a(1) = 1, a(2) = 2, and a(3) = 4.

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A094806 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 5.

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A140071 Triangle read by rows: iterates of X * [1,0,0,0,...]; where X = an infinite lower bidiagonal matrix with [3,1,3,1,3,1...] in the main diagonal and [1,1,1,...] in the subdiagonal.

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A209691 Triangle of coefficients of polynomials u(n,x) jointly generated with A209692; see the Formula section.

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A210218 Triangle of coefficients of polynomials v(n,x) jointly generated with A210217; see the Formula section.

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A210741 Triangle of coefficients of polynomials u(n,x) jointly generated with A210742; see the Formula section.

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A210753 Triangle of coefficients of polynomials u(n,x) jointly generated with A210754; see the Formula section.

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A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A087161 Records in A087159, i.e., A087159(a(n)) = n, and satisfies the recurrence a(n+3) = 5a(n+2) - 6 a(n+1) + 2*a(n) with a(1) = 1, a(2) = 2, and a(3) = 4.

A106731 Expansion of -2x/(1 - 4x + 2*x^2).