A026150 -id:A026150 - cp's OEIS frontend

A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.

1, 0, 4, 0, 2, 10, 0, 2, 6, 28, 0, 2, 6, 24, 76, 0, 2, 6, 28, 80, 208, 0, 2, 6, 32, 100, 264, 568, 0, 2, 6, 36, 120, 360, 840, 1552, 0, 2, 6, 40, 140, 464, 1232, 2624, 4240, 0, 2, 6, 44, 160, 576, 1680, 4128, 8064, 11584, 0, 2, 6, 48, 180, 696, 2184, 5952

Offset: 1

Clark Kimberling, Feb 26 2012

As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 28 2012

			First five rows:
  1;
  0,  4;
  0,  2, 10;
  0,  2,  6, 28;
  0,  2,  6, 24, 76;
First five polynomials u(n,x):
  1
      4x
      2x + 10x^2
      2x +  6x^2 + 28x^3
      2x +  6x^2 + 24x^3 + 76x^4.

Cf. A003946, A026150, A208332.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];
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[%]  (* A208332 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208333 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 28 2012: (Start)
As triangle with 0 <= k <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 4 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-x+2*y*x)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
T(n,n) = A026150(n+1).
Sum_{k=0..n} T(n,k) = A003946(n). (End)

A084157 a(n) = 8a(n-1) - 16a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

Original entry on oeis.org

0, 1, 4, 22, 112, 556, 2704, 13000, 62080, 295312, 1401664, 6644320, 31472896, 149017792, 705395968, 3338614912, 15800258560, 74772443392, 353840161792, 1674425579008, 7923565146112, 37494981225472, 177428889407488

Offset: 0

Paul Barry, May 16 2003

Binomial transform of A084156.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16,0,12).

Cf. A026150, A083881, A084155, A084156.

Cf. A002605, A003462, A080040.

I:=[0,1,4,22]; [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +12*Self(n-4): n in [1..41]]; // G. C. Greubel, Oct 11 2022

LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* Harvey P. Dale, Feb 19 2017 *)

A083881 = BinaryRecurrenceSequence(6,-6,1,3)
A026150 = BinaryRecurrenceSequence(2,2,1,1)
def A084157(n): return (A083881(n) - A026150(n))/2
[A084157(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

a(n) = (A083881(n) - A026150(n))/2.
a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).
a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.
G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).
E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).
From G. C. Greubel, Oct 11 2022: (Start)
a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.
a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)

A147518 Expansion of (1-x)/(1-4x-6x^2).

Original entry on oeis.org

1, 3, 18, 90, 468, 2412, 12456, 64296, 331920, 1713456, 8845344, 45662112, 235720512, 1216854720, 6281741952, 32428096128, 167402836224, 864179921664, 4461136704000, 23029626345984, 118885325607936, 613719060507648

Offset: 0

Philippe Deléham, Nov 06 2008

nonn

Binomial transform of [1,2,13,44,205,...] = A002534(n+1).

a(n) is the number of compositions of n when there are 3 types of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,6).

Cf. A026150, A122117.

a:=[1,3];; for n in [3..30] do a[n]:=4*a[n-1]+6*a[n-2]; od; a; # G. C. Greubel, Jan 09 2020

I:=[1,3]; [n le 2 select I[n] else 4*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 09 2020

seq(coeff(series((1-x)/(1-4*x-6*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 09 2020

CoefficientList[Series[(1-x)/(1-4x-6x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,6},{1,3},30] (* Harvey P. Dale, Aug 22 2016 *)

my(x='x+O('x^30)); Vec((1-x)/(1-4*x-6*x^2)) \\ G. C. Greubel, Jan 09 2020

def A147518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-4*x-6*x^2) ).list()
A147518_list(30) # G. C. Greubel, Jan 09 2020

a(n) = 4*a(n-1) + 6*a(n-2) with a(0)=1, a(1)=3.
a(n) = Sum_{k=0..n} A122016(n,k)*3^k.
a(n) = ((10+sqrt(10))/20)*(2+sqrt(10))^n + ((10-sqrt(10))/20)*(2-sqrt(10))^n. - Richard Choulet, Nov 20 2008

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A084594 a(n) = Sum_{r=0..2^(n-1)} Binomial(2^n,2r)*3^r.

Original entry on oeis.org

1, 4, 28, 1552, 4817152, 46409906716672, 4307758882900393634270543872, 37113573186414494550922197215584520229965687291643953152

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), May 31 2003

a(n)/A084595(n) converges to sqrt(3). Related to Newton's iteration.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Eric Weisstein's World of Mathematics, Newton's Iteration.

Cf. A001146, A026150, A084595,

Table[Sum[Binomial[2^n, 2 r]3^r, {r, 0, 2^(n - 1)}], {n, 0, 8}]
Table[Simplify[Expand[(1/2) ((1 + Sqrt[3])^(2^n) + (1 - Sqrt[3])^(2^n))]], {n, 0, 7}] (* Artur Jasinski, Oct 11 2008 *)

a(n) = ( (1+sqrt(3))^(2^n) + (1-sqrt(3))^(2^n) )/2.
a(n) = A026150(2^n).
a(n) = 2*a(n-1)^2 - A001146(n-1), n>1.
a(n) = a(n-1)^2 + 3*A084595(n-1)^2.

A094286 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2187, 5787, 15435, 41419, 111659, 302059, 819243, 2226219, 6058155, 16503211, 44991659, 122727595, 334914219, 914235051, 2496201387, 6816678571, 18617371307, 50851322539, 138903833259, 379443202731, 1036559854251

Offset: 0

Herbert Kociemba, Jun 02 2004

a(n) is the number of Motzkin n-paths of height <= 4. - Alois P. Heinz, Nov 24 2023

S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-2,4).

Cf. A001006.

LinearRecurrence[{5,-6,-2,4},{1,2,4,9},30] (* Harvey P. Dale, Feb 01 2012 *)

a(n) = (1/12)*(4 + 3*2^n + (1-sqrt(3))^n + (1+sqrt(3))^n).
a(n) = 1/3 + 2^(n-2) + A026150(n)/6.
G.f.: -x*(1-3*x+3*x^3) / ( (x-1)*(2*x-1)*(2*x^2+2*x-1) ). - R. J. Mathar, Dec 20 2011

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023

A094297 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 2.

Original entry on oeis.org

1, 3, 7, 18, 46, 120, 316, 840, 2248, 6048, 16336, 44256, 120160, 326784, 889792, 2424960, 6613120, 18043392, 49247488, 134450688, 367134208, 1002645504, 2738510848, 7480215552, 20433258496, 55818559488, 152486858752

Offset: 1

Herbert Kociemba, Jun 02 2004

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

Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4).

First differences of A038508.

a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/3)^2*(1+2*cos(Pi*k/6))^n or a(n) = (2^n + (1-sqrt(3))^n + (1 + sqrt(3))^n)/4.
(a(n)) seems to be given by tesseq(- 2'i + 2'j + 2'k - 2i' + 2j' + 2k' - 2'ii' + 2'jj' - 'kk' - 2.5'ik' - 1.5'jk' - 2.5'ki' - 1.5'kj' - e) (disregarding signs) - Creighton Dement, Nov 17 2004
G.f.: ( 1-x-3*x^2 )*x / ( (2*x-1)*(2*x^2+2*x-1) ). - R. J. Mathar, Sep 11 2019
4*a(n) = 2^n + 2*A026150(n). - R. J. Mathar, Oct 25 2022

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027, 40897, 91257, 198697, 423777, 894081, 1886011, 4007301, 8594411, 18560081, 40181493, 86872293, 187197193, 402060793, 861827743, 1846685729, 3960390059, 8504658049, 18283290609, 39325827729

Offset: 0

Alexander Samokrutov, Sep 20 2014

a(n)/a(n-1) tends to 2.1486... = 1 + 2^(1/5), the real root of the polynomial x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 3.
If x^5 = 2 and n >= 0, then there are unique integers a, b, c, d, g  such that (1 + x)^n = a + b*x + c*x^2 + d*x^3 + g*x^4. The coefficient a is a(n) (from A052102). - Alexander Samokrutov, Jul 11 2015
If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then x, y, z, s, t satisfies Diophantine equation (see link). - Alexander Samokrutov, Jul 11 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..48 from Alexander Samokrutov)
Alexander Samokrutov, Diophantine equation
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,3).

The following sequences belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A052101, A052102, A052103, A063727, A083098, A083099, A083100, A084057, A093406, A247344.

Cf. A005531.

[n le 5 select 1 else 5*Self(n-1) -10*Self(n-2) +10*Self(n-3) -5*Self(n-4) +3*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jul 11 2015

m:=50; S:=series( (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)

makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */

Vec((1-x)^4/(1-5*x+10*x^2-10*x^3+5*x^4-3*x^5) + O(x^100)) \\ Colin Barker, Sep 22 2014

[sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5).
a(n) = Sum_{k=0...floor(n/5)} (2^k*binomial(n,5*k)). - Alexander Samokrutov, Jul 11 2015
G.f.: (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5). - Colin Barker, Sep 22 2014

A292848 a(n) is the smallest prime of form (1/2)((1 + sqrt(2n))^k + (1 - sqrt(2*n))^k).

Original entry on oeis.org

3, 5, 7, 113, 11, 13, 43, 17, 19, 61, 23, 73, 79, 29, 31, 97, 103, 37, 1241463763, 41, 43, 664973, 47, 2593, 151, 53, 163, 14972833, 59, 61, 4217, 193, 67, 23801, 71, 73, 223, 229, 79, 241, 83, 7561, 61068909859, 89, 271, 277, 283, 97, 10193, 101, 103, 313

Offset: 1

XU Pingya, Sep 24 2017

nonn

When 2n + 1 = p is prime, a(n) = p.
From Robert Israel, Sep 26 2017: (Start)
a(n) is also the first prime in the sequence defined by the recursion x(k+2)=2*x(k+1)+(2*n-1)*x(k) with x(0)=x(1)=1.
a(307), if it exists, has more than 10000 digits.
It appears that x(n*k) is divisible by x(k) if n is odd.  Thus a(n) (if it exists) must be x(k) where k is either a power of 2 or a prime. (End)

			For k = {1, 2, 3, 4}, (1/2)((1 + sqrt(8))^k + (1 - sqrt(8))^k) = {1, 9, 25, 113}. 113 is prime, so a(4) = 113.

Robert Israel, Table of n, a(n) for n = 1..306

Cf. A001333, A026150, A046717, A084057, A002533, A083098, A083100, A003665, A002535, A133294, A090042, A125816, A133343, A133345, A120612, A133356, A125818.

f:= proc(n) local a,b,t;
  a:= 1; b:= 1;
  do
    t:= a; a:= 2*a + (2*n-1)*b;
    if isprime(a) then return a fi;
    b:= t;
  od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 26 2017

f[n_, k_] := ((1 + Sqrt[n])^k + (1 - Sqrt[n])^k)/2;
Table[k = 1; While[! PrimeQ[Expand@f[2n, k]], k++]; Expand@f[2n, k], {n, 52}]

A108476 Expansion of (1-4x)/(1-6x-12x^2+8x^3).

Original entry on oeis.org

1, 2, 24, 160, 1232, 9120, 68224, 508928, 3799296, 28357120, 211662848, 1579868160, 11792306176, 88018952192, 656982441984, 4903783628800, 36602339459072, 273203580764160, 2039219289063424, 15220939987877888

Offset: 0

Paul Barry, Jun 04 2005

In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3).

Index entries for linear recurrences with constant coefficients, signature (6,12,-8).

CoefficientList[Series[(1-4x)/(1-6x-12x^2+8x^3),{x,0,30}],x] (* or *) LinearRecurrence[{6,12,-8},{1,2,24},30] (* Harvey P. Dale, Feb 21 2012 *)

G.f.: (1-4*x)/((1+2*x)*(1-8*x+4*x^2)).
a(n) = 6*a(n-1)+12*a(n-2)-8*a(n-3).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*3^j.
Conjecture: a(n) = A002605(n+1)*A026150(n). - R. J. Mathar, Jul 08 2009
a(0)=1, a(1)=2, a(2)=24, a(n)=6*a(n-1)+12*a(n-2)-8*a(n-3). - Harvey P. Dale, Feb 21 2012
a(n) = (-2)^n/2 +A102591(n)/2. - R. J. Mathar, Sep 20 2012

cp's OEIS Frontend

A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A147518 Expansion of (1-x)/(1-4*x-6*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A084594 a(n) = Sum_{r=0..2^(n-1)} Binomial(2^n,2r)*3^r.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A094286 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A094297 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A084157 a(n) = 8a(n-1) - 16a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

A147518 Expansion of (1-x)/(1-4x-6x^2).

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

A292848 a(n) is the smallest prime of form (1/2)((1 + sqrt(2n))^k + (1 - sqrt(2*n))^k).

A108476 Expansion of (1-4x)/(1-6x-12x^2+8x^3).