A081250 -id:A081250 - cp's OEIS frontend

A084213 Binomial transform of A081250.

1, 4, 18, 76, 312, 1264, 5088, 20416, 81792, 327424, 1310208, 5241856, 20969472, 83881984, 335536128, 1342160896, 5368676352, 21474770944, 85899214848, 343597121536, 1374389010432, 5497557090304, 21990230458368, 87960926027776

Offset: 0

Paul Barry, May 19 2003

When 5*2^n - 1 is prime, that is, n is in A001770, then a(n+1) is in A136539. - Farideh Firoozbakht and M. F. Hasler, Nov 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (6,-8)

[5*4^n/4-2^n/2+0^n/4: n in [0..30]]; // Vincenzo Librandi, Jun 15 2011

seq(coeff(series((1-2*x+2*x^2)/((1-2*x)*(1-4*x)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 09 2018

Table[If[n==0, 1, 2^(n-2)*(5*2^n - 2)], {n,0,30}] (* G. C. Greubel, Oct 08 2018 *)
CoefficientList[Series[(1 - 2*x + 2*x^2)/((1-2*x)*(1-4*x)), {x, 0, 50}], x] (* or *)
CoefficientList[Series[(5*Exp[4*x] - 2*Exp[2*x] + 1)/4, {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* Stefano Spezia, Oct 11 2018 *)

vector(30, n, n--; (5*4^n - 2^(n+1) + 0^n)/4) \\ G. C. Greubel, Oct 08 2018

a(n) = (5*4^n - 2^(n+1) + 0^n)/4.
G.f.: (1 - 2*x + 2*x^2)/((1-2*x)*(1-4*x)).
E.g.f.: (5*exp(4*x) - 2*exp(2*x) + 1)/4.
a(n+1) = 2^n*(5*2^n - 1) for all n >= 0. - M. F. Hasler, Nov 03 2012

A081134 Distance to nearest power of 3.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Klaus Brockhaus, Mar 08 2003

			a(7) = 2 since 9 is closest power of 3 to 7 and 9 - 7 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Index entries for sequences related to distance to nearest element of some set

Cf. A002452, A003605, A006166, A053646, A062153, A081249, A081250, A081251.

a:= n-> (h-> min(n-h, 3*h-n))(3^ilog[3](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2021

Flatten[Table[Join[Range[0,3^n],Range[3^n-1,1,-1]],{n,0,4}]] (* Harvey P. Dale, Dec 31 2013 *)

a(n) = my (p=#digits(n,3)); return (min(n-3^(p-1), 3^p-n)) \\ Rémy Sigrist, Mar 24 2018

def A081134(n):
    kmin, kmax = 0,1
    while 3**kmax <= n:
        kmax *= 2
    while True:
        kmid = (kmax+kmin)//2
        if 3**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return min(n-3**kmin, 3*3**kmin-n) # Chai Wah Wu, Mar 31 2021

a(n) = min(n-3^floor(log(n)/log(3)), 3*3^floor(log(n)/log(3))-n).
From Peter Bala, Sep 30 2022: (Start)
a(n) = n - A006166(n); a(n) = 2*n - A003605(n).
a(1) = 0, a(2) = 1, a(3) = 0; thereafter, a(3*n) = 3*a(n), a(3*n+1) = 2*a(n) + a(n+1) and a(3*n+2) = a(n) + 2*a(n+1). (End)

A081251 Numbers n such that A081249(m)/m^2 has a local maximum for m = n.

Original entry on oeis.org

2, 6, 20, 60, 182, 546, 1640, 4920, 14762, 44286, 132860, 398580, 1195742, 3587226, 10761680, 32285040, 96855122, 290565366, 871696100, 2615088300, 7845264902, 23535794706, 70607384120, 211822152360, 635466457082, 1906399371246

Offset: 1

Klaus Brockhaus, Mar 17 2003

nonn

The limit of the local maxima, lim A081249(n)/n^2 = 1/6. For local minima cf. A081250.
Also the number of different 4- and 3-colorings for the vertices of all triangulated planar polygons on a base with n+2 vertices, if the colors of the two base vertices are fixed. - Patrick Labarque, Mar 23 2010
From Toby Gottfried, Apr 18 2010: (Start)
a(n) = the number of ternary sequences of length n+1 where the numbers of (0's, 1's) are both odd.
A015518 covers the (odd, even) and (even, odd) cases, and A122983 covers (even, even). (End)

			6 is a term since A081249(5)/5^2 = 4/25 = 0.160, A081249(6)/6^2 = 7/36 = 0.194, A081249(7)/7^2 = 9/49 = 0.184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A008776, A033113, A054880, A081134, A081249, A081250.

List([1..30], n-> (9*3^(n-1) -(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019

[Floor(3^(n+1)/4) : n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

seq(floor(3^(n+1)/4), n=1..30). # Mircea Merca, Dec 27 2010

a[n_]:= Floor[3^(n+1)/4]; Array[a, 30]
Table[(9*3^(n-1) -(-1)^n -2)/4, {n, 1, 30}] (* G. C. Greubel, Jul 14 2019 *)

vector(30, n, (9*3^(n-1) -(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019

[(9*3^(n-1) -(-1)^n -2)/4 for n in (1..30)] # G. C. Greubel, Jul 14 2019

G.f.: 2/((1-x)*(1+x)*(1-3*x)).
a(n) = a(n-2) + 2*3^(n) for n > 1.
a(n+2) - a(n) = A008776(n).
a(n) = 2*A033113(n+1).
a(2*n+1) = A054880(n+1).
a(n) = floor(3^(n+1)/4). - Mircea Merca, Dec 26 2010
From G. C. Greubel, Jul 14 2019: (Start)
a(n) = (9*3^(n-1) -(-1)^n -2)/4.
E.g.f.: (3*exp(3*x) - 2*exp(x) - exp(-x))/4. (End)

A081249 Partial sums of A081134.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 9, 10, 10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 63, 70, 76, 81, 85, 88, 90, 91, 91, 92, 94, 97, 101, 106, 112, 119, 127, 136, 146, 157, 169, 182, 196, 211, 227, 244, 262, 281, 301, 322, 344, 367, 391, 416, 442, 469, 495, 520, 544, 567, 589, 610, 630

Offset: 1

Klaus Brockhaus, Mar 17 2003

			First seven terms of A081134 are 0,1,0,1,2,3,2, so a(7) = 9.

Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251

Accumulate[Flatten[Table[Join[Range[0,3^n],Range[3^n-1,1,-1]],{n,0,4}]]] (* Harvey P. Dale, Sep 21 2016 *)

{s=0; for(n=1,62,p=3^floor(0.1^25+log(n)/log(3)); print1(s=s+min(n-p,3*p-n),","))}

a(n) = sum{j=1..n, A081134(j)}.

A209763 Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 3, 9, 13, 8, 4, 15, 31, 35, 16, 5, 23, 61, 97, 85, 32, 6, 33, 107, 219, 279, 203, 64, 7, 45, 173, 433, 717, 761, 469, 128, 8, 59, 263, 779, 1583, 2195, 1991, 1067, 256, 9, 75, 381, 1305, 3141, 5361, 6381, 5049, 2389, 512, 10, 93, 531

Offset: 1

Clark Kimberling, Mar 14 2012

Row n begins with n and ends with 2^(n-1).
Row sums:  1,3,11,33,101,303,911,... A081250
Alternating row sums: 1,-1,1,-1,1,.. A033999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
2...5....4
3...9....13...8
4...15...31...35...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.

Cf. A209764, 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_] := 2 x*u[n - 1, 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[%]    (* A209763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209764 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442 *)

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

A209764 Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 8, 14, 8, 5, 14, 32, 34, 16, 6, 22, 62, 96, 86, 32, 7, 32, 108, 218, 280, 202, 64, 8, 44, 174, 432, 718, 760, 470, 128, 9, 58, 264, 778, 1584, 2194, 1992, 1066, 256, 10, 74, 382, 1304, 3142, 5360, 6382, 5048, 2390, 512, 11, 92, 532

Offset: 1

Clark Kimberling, Mar 14 2012

Row n begins with n and ends with 2^(n-1).
Row sums:  1,4,11,34,101,304,911,2734,... A060925.
Alternating row sums: 1,0,3,2,5,4,7,6,... A060925.
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...2
3...4....4
4...8....14...8
5...14...32...34...16
First three polynomials v(n,x): 1, 2 + 2x , 3 + 4x + 4x^2.

Cf. A209663, 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_] := 2 x*u[n - 1, 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[%]    (* A209763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209764 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442*)

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

A209765 Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 5, 15, 12, 1, 5, 21, 45, 29, 1, 5, 21, 77, 129, 70, 1, 5, 21, 89, 265, 361, 169, 1, 5, 21, 89, 353, 865, 991, 408, 1, 5, 21, 89, 377, 1325, 2717, 2681, 985, 1, 5, 21, 89, 377, 1549, 4733, 8281, 7169, 2378, 1, 5, 21, 89, 377, 1597, 6125

Offset: 1

Clark Kimberling, Mar 14 2012

Limiting row: F(2+3k), where F=A000045 (Fibonacci numbers)
Coefficient of x^n in u(n,x): 1,2,5,12,.... A000129(n)
Row sums:  1,3,11,33,101,303,911,2733,..... A081250
Alternating row sums: 1,-1,1,-1,1,-1,,..... A033999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...5...5
1...5...15...12
1...5...21...45...29
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.

Cf. A209766, 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_] := 2 x*u[n - 1, x] + 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[%]    (* A209765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209766 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

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

A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 13, 17, 1, 3, 13, 43, 41, 1, 3, 13, 55, 133, 99, 1, 3, 13, 55, 209, 391, 239, 1, 3, 13, 55, 233, 739, 1113, 577, 1, 3, 13, 55, 233, 939, 2469, 3095, 1393, 1, 3, 13, 55, 233, 987, 3589, 7903, 8457, 3363, 1, 3, 13, 55, 233, 987, 4085

Offset: 1

Clark Kimberling, Mar 14 2012

Limiting row: F(1+3k), where F=A000045 (Fibonacci numbers)
Coefficient of x^n in u(n,x): A001333(n)
Row sums:  1,4,11,34,101,304,... A060925.
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...3...7
1...3...13...17
1...3...13...43...41
First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 7x^2.

Cf. A209665, 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_] := 2 x*u[n - 1, x] + 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[%]    (* A209765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209766 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

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

A209769 Triangle of coefficients of polynomials u(n,x) jointly generated with A209770; see the Formula section.

Original entry on oeis.org

1, 1, 2, 3, 5, 3, 5, 12, 11, 5, 9, 26, 34, 24, 8, 15, 53, 88, 86, 48, 13, 25, 104, 210, 258, 200, 93, 21, 41, 198, 470, 695, 680, 440, 175, 34, 67, 369, 1007, 1737, 2043, 1671, 929, 323, 55, 109, 676, 2085, 4107, 5625, 5529, 3895, 1901, 587, 89, 177

Offset: 1

Clark Kimberling, Mar 15 2012

Column 1: A001595
Row n ends with F(n+1), where F=A000045 (Fibonacci numbers).
Row sums: 1,3,11,33,101,303,911,2733,... A081250
Alternating row sums: 1,-1,1,-1,1,-1,... A033999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
3...5....3
5...12...11...5
9...26...34...24...8
First three polynomials u(n,x): 1, 1 + 2x, 3 + 5x + 3x^2.

Cf. A209770, 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 + 1)*u[n - 1, 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[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

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

A104522 Expansion of (-1+x+3x^2-x^3)/((x+1)(3x-1)(x-1)^2).

Original entry on oeis.org

1, 3, 7, 19, 53, 155, 459, 1371, 4105, 12307, 36911, 110723, 332157, 996459, 2989363, 8968075, 26904209, 80712611, 242137815, 726413427, 2179240261, 6537720763, 19613162267, 58839486779, 176518460313, 529555380915, 1588666142719, 4765998428131

Offset: 0

Creighton Dement, Apr 20 2005

A floretion-generated sequence relating to A081250 (Numbers n such that A081249(m)/m^2 has a local minimum for m = n).
Binomial transform starts: 1, 4, 14, 50, 184, 696, 2688, 10528, 41600, 165248, ... - Wesley Ivan Hurt, Sep 12 2014
Floretion Algebra Multiplication Program, FAMP Code: 1famforrokseq[ - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki']

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Cf. A081250, A105792.

[(5*3^n+4*(n+1)-(-1)^n)/8 : n in [0..30]]; // Wesley Ivan Hurt, Sep 12 2014

A104522:=n->(5*3^n+4*(n+1)-(-1)^n)/8: seq(A104522(n), n=0..30); # Wesley Ivan Hurt, Sep 12 2014

CoefficientList[Series[(-1 + x + 3 x^2 - x^3)/((x + 1) (3*x - 1) (x - 1)^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 12 2014 *)

a(n) = (1/8) * (5*3^n + 4*(n+1) - (-1)^n). - Ralf Stephan, Nov 13 2010.
a(n+2) - 2a(n+1) + a(n) = A081250(n+1) - A081250(n).
a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3)+3*a(n-4). - Wesley Ivan Hurt, Sep 12 2014

cp's OEIS Frontend

A084213 Binomial transform of A081250.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A081134 Distance to nearest power of 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A081251 Numbers n such that A081249(m)/m^2 has a local maximum for m = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A081249 Partial sums of A081134.

Views

Author

Keywords

Examples

Links

Programs

Mathematica

PARI

Formula

A209763 Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A209764 Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A209765 Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

A104522 Expansion of (-1+x+3x^2-x^3)/((x+1)(3x-1)(x-1)^2).