A105476 -id:A105476 - cp's OEIS frontend

A189745 a(n) = 5*a(n-1) + a(n-2); with a(1)=5, a(2)=1.

5, 1, 10, 51, 265, 1376, 7145, 37101, 192650, 1000351, 5194405, 26972376, 140056285, 727253801, 3776325290, 19608880251, 101820726545, 528712512976, 2745383291425, 14255628970101, 74023528141930, 384373269679751, 1995889876540685, 10363822652383176

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 (5,1).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,1},{5,1},40]
```

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

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

A189749 a(1)=5, a(2)=5, a(n)=5a(n-1) + 5a(n-2).

Original entry on oeis.org

5, 5, 50, 275, 1625, 9500, 55625, 325625, 1906250, 11159375, 65328125, 382437500, 2238828125, 13106328125, 76725781250, 449160546875, 2629431640625, 15392960937500, 90111962890625, 527524619140625, 3088182910156250, 18078537646484375, 105833602783203125

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 (5,5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748.

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

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

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

a(n) = 5*A188168(n). - R. J. Mathar, Feb 13 2020

A189737 a(1)=3, a(2)=3, a(n)=3a(n-1) + 3a(n-2).

Original entry on oeis.org

3, 3, 18, 63, 243, 918, 3483, 13203, 50058, 189783, 719523, 2727918, 10342323, 39210723, 148659138, 563609583, 2136806163, 8101247238, 30714160203, 116446222323, 441481147578, 1673782109703, 6345789771843, 24058715644638, 91213516249443, 345816695682243

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 (3,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

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

a[1]:3$ a[2]:3$ a[n]:=3*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 26); /* Bruno Berselli, May 24 2011 */

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

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

Original entry on oeis.org

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).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189748, A189749.

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).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189749.

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

A274977 a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, 19575, 45009, 103734, 238761, 549963, 1266246, 2916135, 6714873, 15463278, 35607897, 81997731, 188821422, 434814615, 1001278881, 2305722726, 5309559369, 12226727547, 28155405654, 64835588295, 149301805257, 343808570142

Offset: 0

Bruno Berselli, Sep 13 2016

a(n)/a(n+1) converges to 1/A209927 as n approaches infinity.

			Table of similar sequences (not extendable on the left side) where this recurrence can be applied to the first two terms:
----------------------------------------------------------------------
(*)      -  -  1, -1,  2, -1,  5,   2,  17,  23,   74,  143,  365, ...
A052533: -  -  1,  0,  3,  3, 12,  21,  57, 120,  291,  651, 1524, ...
(^)      -  0, 1,  1,  4,  7, 19,  40,  97, 217,  508, 1159, 2683, ...
A006138: -  -  1,  2,  5, 11, 26,  59, 137, 314,  725, 1667, 3842, ...
A105476: -  -  1,  3,  6, 15, 33,  78, 177, 411,  942, 2175, 5001, ...
(^)      0, 1, 1,  4,  7, 19, 40,  97, 217, 508, 1159, 2683, 6160, ...
A105963: -  -  1,  5,  8, 23, 47, 116, 257, 605, 1376, 3191, 7319, ...
A274977: -  -  1,  6,  9, 27, 54, 135, 297, 702, 1593, 3699, 8478, ...
A075118: -  2, 1,  7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, ...
----------------------------------------------------------------------
(*) see version A140165.
(^) see A006130 and the signed versions A140167, A182228.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A006130, A006138, A052533, A075118, A105476, A105963.

a:=[1,6];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020

[n le 2 select 5*n-4 else Self(n-1)+3*Self(n-2): n in [1..40]];

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!((1 + 5*x)/(1- x-3*x^2))); // Marius A. Burtea, Jan 15 2020

seq(coeff(series((1+5*x)/(1-x-3*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 15 2020

RecurrenceTable[{a[n]==a[n-1] +3a[n-2], a[0]==1, a[1]==6}, a, {n,0,40}]
Table[Round[Sqrt[3]^(n-1)*(Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] + 5*Fibonacci[n, 1/Sqrt[3]])], {n,0,40}] (* G. C. Greubel, Jan 15 2020 *)
LinearRecurrence[{1,3},{1,6},40] (* Harvey P. Dale, Jul 11 2023 *)

v=vector(40); v[1]=1; v[2]=6; for(n=3, #v, v[n]=v[n-1]+3*v[n-2]); v

from sage.combinat.sloane_functions import recur_gen2
a = recur_gen2(1, 6, 1, 3)
[next(a) for n in range(40)]

G.f.: (1 + 5*x)/(1 - x - 3*x^2).
a(n) = ((13 + 11*sqrt(13))*(1 + sqrt(13))^n + (13 - 11*sqrt(13))*(1 - sqrt(13))^n)/(26*2^n).
3*a(n) + a(n+1) =  9*A105476(n+1).
3*a(n) - a(n+1) = 27*A006130(n-3) with n>1, A006130(-1) = 0.
a(n+1) - a(n)   = 27*A105476(n-3) with n>2.
a(n) = 3^((n-1)/2)*( sqrt(3)*Fibonacci(n+1, 1/sqrt(3)) + 5*Fibonacci(n, 1/sqrt(3)) ). - G. C. Greubel, Jan 15 2020
E.g.f.: (1/13)*exp(x/2)*(13*cosh((sqrt(13)*x)/2) + 11*sqrt(13)*sinh((sqrt(13)*x)/2)). - Stefano Spezia, Jan 15 2020

A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 2, 6, 6, 1, 1, 8, 15, 8, 1, 2, 11, 26, 28, 10, 1, 1, 12, 42, 64, 45, 12, 1, 2, 16, 60, 122, 130, 66, 14, 1, 1, 16, 82, 208, 295, 232, 91, 16, 1, 2, 21, 108, 324, 582, 621, 378, 120, 18, 1, 1, 20, 135, 480, 1035, 1404, 1176, 576, 153, 20, 1, 2, 26, 170, 675

Offset: 1

Emeric Deutsch, Apr 09 2005

Riordan array ((1+2x)/(1-x^2),x(1+2x)/(1-x^2)). Factorizes as ((1+2x)/(1-x^2),x)*(1,x(1+2x)/(1-x^2)). Row sums A105476 form an eigensequence for ((1+2x)/(1-x^2),x). - Paul Barry, Feb 10 2011
Triangle T(n,k), 1<=k<=n, given by (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012
Also the convolution triangle of A000034. - Peter Luschny, Oct 08 2022

			T(4,2) = 6 because we have (1,3), (3,1), (2,2), (2,2'), (2',2) and (2',2').
Triangle begins:
1;
2, 1;
1, 4,  1;
2, 6,  6, 1;
1, 8, 15, 8, 1;
Triangle (0, 2, -3/2, -1/2, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 1, 4, 1
0, 2, 6, 6, 1
0, 1, 8, 15, 8, 1
0, 2, 11, 26, 28, 10, 1
0, 1, 12, 42, 64, 45, 12, 1

Alois P. Heinz, Rows n = 1..141, flattened

Row sums yield A105476.

Cf. Diagonals: A000007, A000034, A000012, A005843, A000384, A100504.

G:=t*z*(1+2*z)/(1-t*z-z^2-2*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      expand(add((2-irem(i, 2))*b(n-i)*x, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, k), k=1..n))(b(n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 16 2013
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> [1, 2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022

max = 14; g = t*z*(1 + 2*z)/(1 - t*z - z^2 - 2*t*z^2); gser = Series[g, {z, 0, max}]; coes = CoefficientList[gser, {z, t}]; Table[ Table[ coes[[n, k]], {k, 2, n}], {n, 2, max}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Maple *)

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

A152167 a(n)=-a(n-1)+3*a(n-2), n>1 ; a(0)=1, a(1)=-3 .

Original entry on oeis.org

1, -3, 6, -15, 33, -78, 177, -411, 942, -2175, 5001, -11526, 26529, -61107, 140694, -324015, 746097, -1718142, 3956433, -9110859, 20980158, -48312735, 111253209, -256191414, 589951041, -1358525283, 3128378406, -7203954255, 16589089473

Offset: 0

Philippe Deléham, Nov 27 2008

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

Cf. A105476, A147703.

LinearRecurrence[{-1,3},{1,-3},30] (* Harvey P. Dale, Dec 24 2013 *)

Vec((1-2*x)/(1+x-3*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

G.f.: (1-2*x)/(1+x-3*x^2).
a(n) = (-1)^n*A105476(n+1).
a(n) = Sum{k=0..n} A147703(n,k)*(-4)^k.

a(19) corrected by Charles R Greathouse IV, Jan 11 2012

cp's OEIS Frontend

A189745 a(n) = 5*a(n-1) + a(n-2); with a(1)=5, a(2)=1.

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A189749 a(1)=5, a(2)=5, a(n)=5*a(n-1) + 5*a(n-2).

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A189737 a(1)=3, a(2)=3, a(n)=3*a(n-1) + 3*a(n-2).

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

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

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

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A274977 a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.

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A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.

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A189749 a(1)=5, a(2)=5, a(n)=5a(n-1) + 5a(n-2).

A189737 a(1)=3, a(2)=3, a(n)=3a(n-1) + 3a(n-2).

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.