A159612 -id:A159612 - cp's OEIS frontend

A189746 a(1)=5, a(2)=2, a(n) = 5a(n-1) + 2a(n-2).

5, 2, 20, 104, 560, 3008, 16160, 86816, 466400, 2505632, 13460960, 72316064, 388502240, 2087143328, 11212721120, 60237892256, 323614903520, 1738550302112, 9339981317600, 50177007192224, 269564998596320, 1448179007366048, 7780025034022880, 41796483184846496

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,2},{5,2},40]
```

a[1]:5$ a[2]:2$ a[n]:=5*a[n-1]+2*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-23*x)/(1-5*x-2*x^2). - Bruno Berselli, May 24 2011

A189747 a(1)=5, a(2)=3, a(n)=5a(n-1) + 3a(n-2).

Original entry on oeis.org

5, 3, 30, 159, 885, 4902, 27165, 150531, 834150, 4622343, 25614165, 141937854, 786531765, 4358472387, 24151957230, 133835203311, 741631888245, 4109665051158, 22773220920525, 126195099756099, 699295161542070, 3875061106978647, 21473191019519445

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,3).

Mathematica
```
LinearRecurrence[{5,3},{5,3},40]
```

a[1]:5$ a[2]:3$ a[n]:=5*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-22*x)/(1-5*x-3*x^2). - Bruno Berselli, May 24 2011

A189748 a(n) = 5a(n-1) + 4a(n-2) with a(1)=5, a(2)=4.

Original entry on oeis.org

5, 4, 40, 216, 1240, 7064, 40280, 229656, 1309400, 7465624, 42565720, 242691096, 1383718360, 7889356184, 44981654360, 256465696536, 1462255100120, 8337138286744, 47534711834200, 271022112317976, 1545249408926680, 8810335493905304, 50232675105233240

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,4).

Mathematica
```
LinearRecurrence[{5,4},{5,4},40]
```

a[1]:5$ a[2]:4$ a[n]:=5*a[n-1]+4*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-21*x)/(1-5*x-4*x^2). - Bruno Berselli, May 24 2011

A119473 Triangle read by rows: T(n,k) is number of binary words of length n and having k runs of 0's of odd length, 0 <= k <= ceiling(n/2). (A run of 0's is a subsequence of consecutive 0's of maximal length.)

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 1, 5, 8, 3, 8, 15, 8, 1, 13, 28, 19, 4, 21, 51, 42, 13, 1, 34, 92, 89, 36, 5, 55, 164, 182, 91, 19, 1, 89, 290, 363, 216, 60, 6, 144, 509, 709, 489, 170, 26, 1, 233, 888, 1362, 1068, 446, 92, 7, 377, 1541, 2580, 2266, 1105, 288, 34, 1, 610, 2662, 4830

Offset: 0

Emeric Deutsch, May 22 2006

Row n has 1+ceiling(n/2) terms.
T(n,0) = Fibonacci(n+1) = A000045(n+1).
T(n,1) = A029907(n).
Sum_{k>=0} k*T(n,k) = A059570(n).
Triangle, with zeros included, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 07 2011
T(n,k) is the number of compositions of n+1 that have exactly k even parts. - Geoffrey Critzer, Mar 03 2012

			T(5,2)=8 because we have 00010, 01000, 01011, 01101, 01110, 10101, 10110 and 11010.
T(5,2)=8 because there are 8 compositions of 6 that have 2 even parts: 4+2, 2+4, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2. - _Geoffrey Critzer_, Mar 03 2012
Triangle starts:
  1;
  1,  1;
  2,  2;
  3,  4,  1;
  5,  8,  3;
  8, 15,  8,  1;
From _Philippe Deléham_, Dec 07 2011: (Start)
Triangle (1,1,-1,0,0,0...) DELTA (1,-1,0,0,0,...) begins:
   1;
   1,  1;
   2,  2,  0;
   3,  4,  1,  0;
   5,  8,  3,  0,  0;
   8, 15,  8,  1,  0,  0;
  13, 28, 19,  4,  0,  0,  0;
  21, 51, 42, 13,  1,  0,  0,  0;
  34, 92, 89, 36,  5,  0,  0,  0,  0; ... (End)

I. Goulden and D. Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 54.

Alois P. Heinz, Rows n = 0..200, flattened
Rigoberto Flórez, Javier González, Mateo Matijasevick, Cristhian Pardo, José Luis Ramírez, Lina Simbaqueba, and Fabio Velandia, Lattice paths in corridors and cyclic corridors, Contrib. Disc. Math. (2024) Vol. 19. No. 2, 36-55. See p. 17.
Ralph Grimaldi and Silvia Heubach, Binary strings without odd runs of zeros, Ars Combinatoria 75 (2005), 241-255.

Cf. A000045, A029907, A059570.

G:=(1+t*z)/(1-z-z^2-t*z^2): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)->x+y, %,
      [`if`(irem(j, 2)=0, 0, NULL), b(n-j)], 0) od; %[] fi
    end:
T:= n-> b(n+1):
seq(T(n), n=0..14);  # Alois P. Heinz, May 23 2013

f[list_] := Select[list, # > 0 &]; nn = 15; a = (x + y x^2)/(1 - x^2); Map[f, Drop[CoefficientList[Series[1/(1 - a), {x, 0, nn}], {x, y}], 1]] // Flatten  (* Geoffrey Critzer, Mar 03 2012 *)

G.f.: (1+t*z)/(1-z-z^2-t*z^2).
G.f. of column k (k>=1): z^(2*k-1)*(1-z^2)/(1-z-z^2)^(k+1).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Dec 07 2011
Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A000079(n), A105476(n+1), A159612(n+1), A189732(n+1) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Dec 07 2011
G.f.: (1+x*y)*T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013

A210637 Triangle T(n,k), read by rows, given by (2, 1/2, -1/2, 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.

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 12, 27, 20, 5, 29, 84, 91, 44, 8, 70, 248, 352, 251, 90, 13, 169, 708, 1240, 1164, 618, 176, 21, 408, 1973, 4106, 4771, 3344, 1414, 334, 34, 985, 5400, 13010, 18000, 15645, 8748, 3073, 620, 55

Offset: 0

Philippe Deléham, Mar 26 2012

Row sums are powers of 4 (A000302).

			Triangle begins :
1
2, 2
5, 8, 3
12, 27, 20, 5
29, 84, 91, 44, 8
70, 248, 352, 251, 90, 13
169, 708, 1240, 1164, 618, 176, 21
408, 1973, 4106, 4771, 3344, 1414, 334, 34
985, 5400, 13010, 18000, 15645, 8748, 3073, 620, 55
2378, 14574, 39880, 63966, 66282, 46014, 21400, 6429, 1132, 89
5741, 38896, 119129, 217232, 261185, 216348, 125028, 49772, 13061, 2040, 144

Cf. A000045, A000129, A000302, A261056 (2nd column).

G.f.: (1+y*x)/(1-(y+2)*x-(y+1)^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A159612(n+1), (-1)^n*A000034(n), A000007(n), A000129(n+1), A000302(n) for x = -3, -2, -1, 0, 1 respectively.
T(n,0) = A000129(n+1), T(n,n) = A000045(n+2), T(n+1,n) = 2*A004798(n+1).

cp's OEIS Frontend

A189746 a(1)=5, a(2)=2, a(n) = 5*a(n-1) + 2*a(n-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A189747 a(1)=5, a(2)=3, a(n)=5*a(n-1) + 3*a(n-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A189748 a(n) = 5*a(n-1) + 4*a(n-2) with a(1)=5, a(2)=4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A119473 Triangle read by rows: T(n,k) is number of binary words of length n and having k runs of 0's of odd length, 0 <= k <= ceiling(n/2). (A run of 0's is a subsequence of consecutive 0's of maximal length.)

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A210637 Triangle T(n,k), read by rows, given by (2, 1/2, -1/2, 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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A189746 a(1)=5, a(2)=2, a(n) = 5a(n-1) + 2a(n-2).

A189747 a(1)=5, a(2)=3, a(n)=5a(n-1) + 3a(n-2).

A189748 a(n) = 5a(n-1) + 4a(n-2) with a(1)=5, a(2)=4.