A047849 -id:A047849 - cp's OEIS frontend

A155701 a(n) = (4^n + 8)/3.

3, 4, 8, 24, 88, 344, 1368, 5464, 21848, 87384, 349528, 1398104, 5592408, 22369624, 89478488, 357913944, 1431655768, 5726623064, 22906492248, 91625968984, 366503875928, 1466015503704, 5864062014808, 23456248059224, 93824992236888, 375299968947544

Offset: 0

Paul Curtz, Jan 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

[(4^n+8)/3: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

A155701 := proc(n) (4^n+8)/3 ; end: seq(A155701(n),n=0..80) ; # R. J. Mathar, Jul 23 2009

A155701[n_]:=(4^n+8)/3;Array[A155701,50,0] (* Paolo Xausa, Dec 05 2023 *)

a(n) = 3 + A002450(n).
a(n) = 5*a(n-1) - 4*a(n-2) = 4*a(n-1) - 8.
a(n) = A154879(2n) = A154890(2n).
a(n+1) - a(n) = A000302(n).
a(n+1) = 4*A047849(n) = 4*A078008(2n).
G.f.: (3-11*x)/((4*x-1)*(x-1)). - R. J. Mathar, Jul 23 2009

Edited and extended by R. J. Mathar, Jul 23 2009

A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 8, 9, 10, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 2, 5, 8, 9, 10, 17, 20, 21, 26, 29, 32, 33, 34, 33, 36, 37, 34, 37, 40, 41, 42, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 34, 37, 40, 41, 42, 49, 52, 53, 58, 61, 64, 65, 66, 65, 68

Offset: 0

Alex Ratushnyak, Jun 13 2012

Indices of 1's: A007583(n),
indices of 2's: A047849(n+1),
indices of 4's: A039301(n+2),
indices of 5's: A153643(n+3),
indices of 8's: A155701(n+2),
indices of 9's: A155701(n+2)+1 = A163868(n+2),
indices of 10's: A153643(n+4)+3^((n+1) mod 2),
indices of 13's: A039301(n+3)+3,
indices of 16's: A039301(n+3)+4,
indices of 17's: 17, 19, 27, 49, 51, 91, 177, 179, 347, 689, 691, 1371, 2737, 2739, 5467, 10929, 10931, 21851, 43697, 43699, 87387, 174769, 174771, 349531, 699057, 699059, 1398107, 2796209, 2796211, 5592411, 11184817, 11184819, 22369627, 44739249, 44739251, 89478491, ...
indices of 18's: A039301(n+3)+6,
n's such that a(n)<3: A005578, except the first term.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a:= proc(n) local i, k, m, r;
      k, m, r:= n, 3*n, 0;
      for i from 0 while (m>0 or k>0) do
        r:= r +2^i* irem(m, 2, 'm') *irem(k, 2, 'k')
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 22 2012

Table[BitAnd[n, 3*n], {n, 0, 68}] (* Arkadiusz Wesolowski, Jun 23 2012 *)

a(n)=bitand(n,3*n) \\ Charles R Greathouse IV, Feb 05 2013

for n in range(99):
    print(3*n & n, end=',')

A087432 Expansion of 1+x(1-x-4x^2)/((1+x)(1-2x)(1-3x)).

Original entry on oeis.org

1, 1, 3, 7, 19, 51, 143, 407, 1179, 3451, 10183, 30207, 89939, 268451, 802623, 2402407, 7196299, 21567051, 64657463, 193885007, 581480259, 1744091251, 5231574703, 15693326007, 47077181819, 141225953051, 423666674343

Offset: 0

Paul Barry, Sep 02 2003

Binomial transform of A047849 (with interpolated zeros, 1,0,2,0,6,0,...). Binomial transform is A087433.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

First differences of A093379.

CoefficientList[Series[1+x (1-x-4x^2)/((1+x)(1-2x)(1-3x)),{x,0,30}],x] (* or *) LinearRecurrence[{4,-1,-6},{1,1,3,7},30] (* Harvey P. Dale, Aug 23 2017 *)

Vec((x-1)*(2*x^2+2*x-1)/((1+x)*(1-2*x)*(1-3*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012, corrected Nov 27 2014

a(n) = (-1)^n/6+2^n/3+3^n/6, n>0.

For n>4, a(n) = 6*a(n-1) - 9*a(n-2) - 4*a(n-3) + 12*a(n-4). - Gary W. Adamson, Jun 14 2006

A094233 Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 20, 0, 70, 2, 252, 22, 924, 156, 3432, 910, 12870, 4760, 48622, 23256, 184796, 108528, 705894, 490314, 2708204, 2163150, 10430500, 9373652, 40313160, 40060078, 156305070, 169345560, 607812102, 709645552, 2369918628, 2952780320

Offset: 0

Herbert Kociemba, May 29 2004

In general, a(n,m) = (2^n/m)*Sum_{k=0..m-1} cos(2*Pi*k/m)^n gives the number of closed walks of length n at a vertex of the cyclic graph on m nodes C_m.

Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Cf. A078008, A054877, A047849.

f[n_] := FullSimplify[ TrigToExp[ 2^n/9 Sum[ Cos[2Pi*k/9]^n, {k, 0, 8}]]]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Jun 01 2004 *)

a(n) = (2^n/9)*Sum_{k=0..8} cos(2*Pi*k/9)^n.
G.f.: -(x-1)*(x^3+3*x^2-1)/((2*x-1)*(x+1)*(x^3-3*x^2+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
9*a(n) = 2*(-1)^n +2^n +6*(-1)^n*A188048(n). - R. J. Mathar, Nov 03 2020

More terms from Robert G. Wilson v, Jun 01 2004

A097165 Expansion of (1-3x)/((1-x)(1-4x)(1-5x)).

Original entry on oeis.org

1, 7, 41, 227, 1221, 6447, 33601, 173467, 889181, 4533287, 23015961, 116477907, 587981941, 2962279327, 14900875121, 74862289547, 375743103501, 1884442140567, 9445117195081, 47317211944387, 236952563597861

Offset: 0

Paul Barry, Jul 30 2004

Partial sums of A085351. Convolution of A034478 and 4^n. Convolution of A047849 and 5^n. a(n)=A097162(2n+1)/3. Third binomial transform of A097164.

Index entries for linear recurrences with constant coefficients, signature (10,-29,20).

CoefficientList[Series[(1-3x)/((1-x)(1-4x)(1-5x)),{x,0,30}],x] (* or *) LinearRecurrence[{10,-29,20},{1,7,41},30] (* Harvey P. Dale, Jan 24 2012 *)

a(n)=5*5^n/2-4*4^n/3-1/6; a(n)=sum{k=0..n, (5^k+1)4^(n-k)/2}; a(n)=sum{k=0..n, (4^k+2)5^(n-k)/3}; a(n)=10a(n-1)-29a(n-2)+20a(n-3).

A123490 Triangle whose k-th column satisfies a(n) = (k+3)a(n-1)-(k+2)a(n-2).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 16, 14, 6, 2, 1, 32, 41, 22, 7, 2, 1, 64, 122, 86, 32, 8, 2, 1, 128, 365, 342, 157, 44, 9, 2, 1, 256, 1094, 1366, 782, 260, 58, 10, 2, 1, 512, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1

Offset: 0

Paul Barry, Oct 01 2006

			Triangle begins
     1;
     2,    1;
     4,    2,     1;
     8,    5,     2,     1;
    16,   14,     6,     2,    1;
    32,   41,    22,     7,    2,    1;
    64,  122,    86,    32,    8,    2,   1;
   128,  365,   342,   157,   44,    9,   2,  1;
   256, 1094,  1366,   782,  260,   58,  10,  2,  1;
   512, 3281,  5462,  3907, 1556,  401,  74, 11,  2, 1;
  1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Columns include A000079, A007051, A047849, A047850, A047851.

Cf. A047848, A103439 (row sums), A123491 (diagonal sums).

[((k+2)^(n-k) +k)/(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 15 2021

Table[((k+2)^(n-k) +k)/(k+1), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 14 2017 *)

for(n=0, 10, for(k=0,n, print1(((k+2)^(n-k)+k)/(k+1), ", "))) \\ G. C. Greubel, Oct 14 2017

flatten([[((k+2)^(n-k) +k)/(k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Column k has g.f.: x^k*(1-x(1+k))/((1-x)*(1-x(2+k))).
T(n,k) = ((k+2)^(n-k) + k)/(k+1), for 0 <= k <= n.
Sum_{k=0..n} T(n, k) = A103439(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A123491(n).

A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 8, 10, 3, 1, 22, 37, 21, 5, 1, 63, 125, 100, 45, 8, 1, 185, 409, 410, 260, 88, 13, 1, 550, 1321, 1562, 1240, 598, 169, 21, 1, 1644, 4238, 5706, 5331, 3258, 1319, 315, 34, 1, 4925, 13534, 20284, 21507, 15651, 8071, 2776, 578, 55, 1

Offset: 1

Clark Kimberling, Mar 27 2012

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 1:  1,1,1,1,1,1,1,1,1,1,1,...
Column 2:  A047849
Row sums:  A003462
Alternating row sums:  1,0,0,0,0,0,0,0,0,...
For a discussion and guide to related arrays, see A208510.
Essentially the same triangle as (1, 0, 3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 11 2012

			First five rows:
1
1...1
1...3....2
1...8....10...3
1...22...37...21...5
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.

Cf. A210804, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 3; f = 0;
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[%]    (* A210803 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210804 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

u(n,x)=u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x-1)*u(n-1,x)+(x+3)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = T(3,0) = 1, T(3,1) = 3, T(3,2) = 2, T(n,k) = 0 if k<0 or if k >= n. - Philippe Deléham, Jul 11 2012
G.f.: (-1+3*x)*x*y/(-1+4*x-3*x^2-2*x^2*y+x*y+x^2*y^2). - R. J. Mathar, Aug 12 2015

A210804 Triangle of coefficients of polynomials v(n,x) jointly generated with A210803; see the Formula section.

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 14, 27, 18, 5, 41, 88, 79, 40, 8, 122, 284, 310, 215, 80, 13, 365, 912, 1152, 980, 510, 156, 21, 1094, 2917, 4144, 4091, 2660, 1150, 294, 34, 3281, 9296, 14578, 16176, 12393, 6752, 2461, 544, 55, 9842, 29526, 50436, 61638, 53730

Offset: 1

Clark Kimberling, Mar 27 2012

Row n ends with F(n), where F=A000045 (Fibonacci numbers).
Column 1:  A007051.
Row sums:  A000302 (powers of 4).
Alternating row sums:  1,0,0,0,0,0,0,0,0,...
For a discussion and guide to related arrays, see A208510.
Essentially the same triangle as given by (2, 1/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham Jul 11 2012

			First five rows:
   1;
   2,  2;
   5,  8,  3;
  14, 27, 18,  5;
  41, 88, 79, 40,  8;
First three polynomials v(n,x):
  1
  2 + 2x
  5 + 8x + 3x^2

Cf. A210803, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 3; f = 0;
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[%]    (* A210803 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210804 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

u(n,x) = u(n-1,x) + x*v(n-1,x) + 1, v(n,x) = (x-1)*u(n-1,x) + (x+3)*v(n-1,x), where u(1,x)=1, v(1,x)=1.
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(3,0) = 5, T(3,1) = 8, T(3,2) = 3, T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Jul 11 2012
G.f.: (-1+2*x-x*y)*x*y/(-1+4*x+x*y-3*x^2-2*x^2*y+x^2*y^2). - R. J. Mathar, Aug 12 2015

A088556 Numbers of the form (4^n + 4^(n-1) + ... + 1) + (n mod 2).

Original entry on oeis.org

6, 21, 86, 341, 1366, 5461, 21846, 87381, 349526, 1398101, 5592406, 22369621, 89478486, 357913941, 1431655766, 5726623061, 22906492246, 91625968981, 366503875926, 1466015503701, 5864062014806, 23456248059221, 93824992236886, 375299968947541, 1501199875790166

Offset: 1

Cino Hilliard, Nov 17 2003

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

Cf. A002450, A047849.

I:=[6,21,86]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 14 2015

LinearRecurrence[{4, 1, -4}, {6, 21, 86}, 50] (* Vincenzo Librandi, Jun 14 2015 *)

trajpolypn(n1) = { for(x1=1,n1, y1 = polypn(4,x1); print1(y1",") ) }
polypn(n,p) = { x=n; if(p%2,y=2,y=1); for(m=1,p, y=y+x^m; ); return(y) }

Vec(x*(6-3*x-4*x^2)/((1-x)*(1+x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Jun 13 2015

If n is even, then 4^n + ... + 1 = (4^(n+1) - 1)/3 = (2^(n+1) - 1)*(2^(n+1) + 1)/3. - R. K. Guy, Nov 17 2003
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Colin Barker, Apr 02 2012
G.f.: x*(6-3*x-4*x^2) / ((1-x)*(1+x)*(1-4*x)). - Colin Barker, Apr 02 2012

A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 20, 2, 70, 18, 252, 110, 924, 572, 3434, 2730, 12902, 12376, 48926, 54264, 187036, 232562, 720062, 980674, 2789164, 4086550, 10861060, 16878420, 42484682, 69242082, 166823430, 282580872, 657178982, 1148548016, 2595874468

Offset: 0

Herbert Kociemba, Jun 06 2004

In general, a(n,m) = (2^n/m)*Sum_{k=0..m-1} cos(2*Pi*k/m)^n gives the number of closed walks of length n at a vertex of the cyclic graph on m nodes C_m.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-2).

Cf. A199572 (m=2), A078008 (m=3), A199573 (m=4), A054877 (m=5), A047849 (bisection of m=6), A063376 (bisection of m=8), A094233 (m=9), A095929 (bisection of m=10), A087433 (bisection of m=12).

f[n_] := FullSimplify[ TrigToExp[ 2^n/7 Sum[Cos[2Pi*k/7]^n, {k, 0, 6}]]]; Table[ f[n], {n, 0, 36}] (* Robert G. Wilson v, Jun 09 2004 *)
LinearRecurrence[{1,4,-3,-2},{1,0,2,0},40] (* Harvey P. Dale, Jun 12 2014 *)

a(n) = (2^n/7)*Sum_{k=0..6} cos(2*Pi*k/7)^n.
a(n) = 7(a(n-2) - 2a(n-4) + a(n-6)) + 2a(n-7).
G.f.: (1-x-2x^2+x^3)/((2x-1)(-1-x+2x^2+x^3)).
a(0)=1, a(1)=0, a(2)=2, a(3)=0, a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-2*a(n-4). - Harvey P. Dale, Jun 12 2014
7*a(n) = 2^n + 2*A094648(n). - R. J. Mathar, Nov 03 2020

More terms from Robert G. Wilson v, Jun 09 2004

cp's OEIS Frontend

A155701 a(n) = (4^n + 8)/3.

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A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

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A087432 Expansion of 1+x*(1-x-4*x^2)/((1+x)*(1-2*x)*(1-3*x)).

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A094233 Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.

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A097165 Expansion of (1-3x)/((1-x)(1-4x)(1-5x)).

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A123490 Triangle whose k-th column satisfies a(n) = (k+3)*a(n-1)-(k+2)*a(n-2).

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A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.

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A210804 Triangle of coefficients of polynomials v(n,x) jointly generated with A210803; see the Formula section.

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A087432 Expansion of 1+x(1-x-4x^2)/((1+x)(1-2x)(1-3x)).

A123490 Triangle whose k-th column satisfies a(n) = (k+3)a(n-1)-(k+2)a(n-2).