A105476 -id:A105476 - cp's OEIS frontend

A144021 Eigentriangle by rows, T(n,k) = A000034(n-k+1)*A105476(k-1).

1, 2, 1, 1, 2, 3, 2, 1, 6, 6, 1, 2, 3, 12, 15, 2, 1, 6, 6, 30, 33, 1, 2, 3, 12, 15, 66, 78, 2, 1, 6, 6, 30, 33, 156, 177, 1, 2, 3, 12, 15, 66, 78, 354, 411, 2, 1, 6, 6, 30, 33, 156, 177, 822, 942

Offset: 1

Gary W. Adamson, Sep 07 2008

nonn

Row sums = A105476: (1, 3, 6, 15, 33, 78,...).
Left column = A000034: (1, 2, 1, 2, 1, 2,...).
Right border = A105476 shifted: (1, 1, 3, 6, 15, 33, 78,...).

			First few rows of the triangle =
1;
2, 1;
1, 2, 3
2, 1, 6, 6
1, 2, 3, 12, 15
2, 1, 6, 6, 30, 33;
1, 2, 3, 12, 15, 66, 78
2, 1, 6, 6, 30, 33, 156, 177;
1, 2, 3, 12, 15, 66, 78, 354, 411;
...
Row 4 = (2, 1, 6, 6) = termwise product of (2, 1, 2, 1) and (1, 1, 3, 6) = (2*1, 1*1, 2*3, 1*6).

Cf. A000034, A105476.

Eigentriangle by rows, A000034(n-k+1)*A105476(k-1); where A105476(k-1) = A105476 shifted = (1, 1, 3, 6, 15, 33, 78, 177,...).

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

Original entry on oeis.org

1, 4, 8, 22, 50, 124, 290, 694, 1628, 3838, 8978, 21004, 48962, 114022, 265004, 615262, 1426658, 3305212, 7650722, 17697430, 40911740, 94528318, 218312114, 503994220, 1163124866, 2683496134, 6189647948, 14273690782

Offset: 0

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the corner squares to A175654.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences central square [decimal value A[5]]: A000007 [0], A000012 [16], A000045 [1], A011782 [2], A000079 [3], A003945 [42], A099036 [11], A175656 [7], A105476 [69], A168604 [26], A045891 [19], A078057 [21], A151821 [170], A175657 [43], 4*A172481 [15; n>=-1], A175655 [71, this sequence], 4*A026597 [325; n>=-1], A033484 [58], A087447 [27], A175658 [23], A026150 [85], A175661 [171], A036563 [186], A098156 [59], A046717 [341], 2*A001792 [187; n>=1 with a(0)=1], A175659 [343].

Cf. A000244, A006130, A075118, A175654.

I:=[1, 4, 8]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013

with(LinearAlgebra): nmax:=27; m:=5; A[5]:= [0,0,1,0,0,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1 + x - 5 x^2) / (1 - 3 x - x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{3,1,-6},{1,4,8},40] (* Harvey P. Dale, Dec 25 2024 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;4;8])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.
a(n) = ((10+8*A)*A^(-n-1) + (10+8*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n)-A006130(n-1)*sqrt(13)).
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 5*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Jan 31 2023

A006138 a(n) = a(n-1) + 3*a(n-2).

Original entry on oeis.org

1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, 8843, 20369, 46898, 108005, 248699, 572714, 1318811, 3036953, 6993386, 16104245, 37084403, 85397138, 196650347, 452841761, 1042792802, 2401318085, 5529696491, 12733650746, 29322740219, 67523692457, 155491913114

Offset: 0

N. J. A. Sloane

The binomial transform of a(n) is b(n) = A006190(n+1), which satisfies b(n) = 3*b(n-1) + b(n-2). - Paul Barry, May 21 2006
Partial sums of A105476. - Paul Barry, Feb 02 2007
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence. For the central square these vectors lead to the companion sequence A105476 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Equals the INVERTi transform of A063782: (1, 3, 10, 32, 104, ...). - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
The sequence is the INVERT transform of A016116: (1, 1, 2, 2, 4, 4, 8, 8, ...). - Gary W. Adamson, Jul 17 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
N. T. Gridgeman, A new look at Fibonacci generalization, Fib,. Quart., 11 (1973), 40-55.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A063782. - Gary W. Adamson, Aug 14 2010

Cf. A006130, A006190, A016116, A105476, A122950, A175654, A274977.

a:=[1,2];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Nov 19 2019

[n le 2 select n else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 15 2016

A006138:=-(1+z)/(-1+z+3*z**2); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(1+z)/(1-z-3*z^2), {z,0,40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
Table[(I*Sqrt[3])^(n-1)*(I*Sqrt[3]*ChebyshevU[n, 1/(2*I*Sqrt[3])] + ChebyshevU[n-1, 1/(2*I*Sqrt[3])]), {n, 0, 40}]//Simplify (* G. C. Greubel, Nov 19 2019 *)
LinearRecurrence[{1,3},{1,2},40] (* Harvey P. Dale, May 29 2025 *)

main(size)={my(v=vector(size),i);v[1]=1;v[2]=2;for(i=3,size,v[i]=v[i-1]+3*v[i-2]);return(v);} /* Anders Hellström, Jul 17 2015 */

def A006138_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x)/(1-x-3*x^2)).list()
A006138_list(40) # G. C. Greubel, Nov 19 2019

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*2^(n+1-k). - Philippe Deléham, Jan 04 2008
G.f.: (1+x)/(1-x-3*x^2). - Paul Barry, May 21 2006
a(n) = Sum_{k=0..n} C(floor((2n-k)/2),n-k)*3^floor(k/2). - Paul Barry, Feb 02 2007
a(n) = A006130(n) + A006130(n-1). - R. J. Mathar, Jun 22 2011
a(n) = (i*sqrt(3))^(n-1)*(i*sqrt(3)*ChebyshevU(n, 1/(2*i*sqrt(3))) + ChebyshevU(n-1, 1/(2*i*sqrt(3)))), where i=sqrt(-1). - G. C. Greubel, Nov 19 2019

Typo in formula corrected by Johannes W. Meijer, Aug 15 2010

A189732 a(1)=1, a(2)=5, a(n) = a(n-1) + 5*a(n-2).

Original entry on oeis.org

1, 5, 10, 35, 85, 260, 685, 1985, 5410, 15335, 42385, 119060, 330985, 926285, 2581210, 7212635, 20118685, 56181860, 156775285, 437684585, 1221561010, 3409983935, 9517788985, 26567708660, 74156653585, 206995196885, 577778464810, 1612754449235, 4501646773285

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 33.
Index entries for linear recurrences with constant coefficients, signature (1, 5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

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

a[1]:1$ a[2]:5$ a[n]:=a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 29);  /* Bruno Berselli, May 24 2011 */

a(n)=([0,1; 5,1]^(n-1)*[1;5])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

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

a(n+1) = Sum_{k=0..n} A119473(n,k)*4^k. - Philippe Deléham, Oct 05 2012

A189741 a(1)=4, a(2)=2, a(n) = 4a(n-1) + 2a(n-2).

Original entry on oeis.org

4, 2, 16, 68, 304, 1352, 6016, 26768, 119104, 529952, 2358016, 10491968, 46683904, 207719552, 924246016, 4112423168, 18298184704, 81417585152, 362266710016, 1611902010368, 7172141461504, 31912369866752, 141993762390016, 631799789293568, 2811186681954304

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 (4,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{4,2},{4,2},40]
```

a[1]:4$ a[2]:2$ a[n]:=4*a[n-1]+2*a[n-2]$ makelist(a[n], n, 1, 25);  /* Bruno Berselli, May 24 2011 */

G.f.: 2*x*(2-7*x)/(1-4*x-2*x^2). - Bruno Berselli, May 24 2011

A189734 a(1)=2, a(2)=5, a(n)=2a(n-1) + 5a(n-2).

Original entry on oeis.org

2, 5, 20, 65, 230, 785, 2720, 9365, 32330, 111485, 384620, 1326665, 4576430, 15786185, 54454520, 187839965, 647952530, 2235104885, 7709972420, 26595469265, 91740800630, 316458947585, 1091621898320, 3765538534565, 12989186560730, 44806065794285

Offset: 1

Harvey P. Dale, Apr 26 2011

nonn

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

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

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

a[1]:2$ a[2]:5$ a[n]:=2*a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 26); /* Bruno Berselli, May 24 2011 */

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

A189736 a(1)=3, a(2)=2, a(n)=3a(n-1) + 2a(n-2).

Original entry on oeis.org

3, 2, 12, 40, 144, 512, 1824, 6496, 23136, 82400, 293472, 1045216, 3722592, 13258208, 47219808, 168175840, 598967136, 2133253088, 7597693536, 27059586784, 96374147424, 343241615840, 1222473142368, 4353902658784, 15506654261088, 55227768100832

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 (3,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{3,2},{3,2},40]
```

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

G.f.: x*(3-7*x)/(1-3*x-2*x^2). - Bruno Berselli, May 24 2011

A189742 a(1)=4, a(2)=3, a(n) = 4a(n-1) + 3a(n-2).

Original entry on oeis.org

4, 3, 24, 105, 492, 2283, 10608, 49281, 228948, 1063635, 4941384, 22956441, 106649916, 495468987, 2301825696, 10693709745, 49680316068, 230802393507, 1072250522232, 4981409269449, 23142388644492, 107513782386315, 499482295478736, 2320470529073889

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 (4,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{4,3},{4,3},40]
```

a[1]:4$ a[2]:3$ a[n]:=4*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

G.f.: x*(4-13*x)/(1-4*x-3*x^2). - Bruno Berselli, May 24 2011

A189743 a(1)=4, a(2)=4, a(n) = 4a(n-1) + 4a(n-2).

Original entry on oeis.org

4, 4, 32, 144, 704, 3392, 16384, 79104, 381952, 1844224, 8904704, 42995712, 207601664, 1002389504, 4839964672, 23369416704, 112837525504, 544827768832, 2630661177344, 12701955784704, 61330467848192, 296129694531584, 1429840649519104, 6903881376202752

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 (4,4).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189744, A189745, A189746, A189747, A189748, A189749.

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

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

G.f.: 4*x*(1-3*x)/(1-4*x-4*x^2). - Bruno Berselli, May 24 2011

A189744 a(1)=4, a(2)=5, a(n) = 4a(n-1) + 5a(n-2).

Original entry on oeis.org

4, 5, 40, 185, 940, 4685, 23440, 117185, 585940, 2929685, 14648440, 73242185, 366210940, 1831054685, 9155273440, 45776367185, 228881835940, 1144409179685, 5722045898440, 28610229492185, 143051147460940, 715255737304685, 3576278686523440, 17881393432617185

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734-A189739, A189741, A189742, A189743, A189745-A189749.

[5/2*(-1)^(n-1)+3/2*5^(n-1): n in [1..30]]; // Vincenzo Librandi, Jul 15 2011

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

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

a(n)=5/2*(-1)^(n-1)+3/2*5^(n-1) \\ Charles R Greathouse IV, Jul 02 2013

G.f.: x*(4-11*x)/(1 - 4*x - 5*x^2). - Bruno Berselli, May 24 2011

cp's OEIS Frontend

A144021 Eigentriangle by rows, T(n,k) = A000034(n-k+1)*A105476(k-1).

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A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).

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A006138 a(n) = a(n-1) + 3*a(n-2).

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A189732 a(1)=1, a(2)=5, a(n) = a(n-1) + 5*a(n-2).

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A189741 a(1)=4, a(2)=2, a(n) = 4*a(n-1) + 2*a(n-2).

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A189734 a(1)=2, a(2)=5, a(n)=2*a(n-1) + 5*a(n-2).

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A189736 a(1)=3, a(2)=2, a(n)=3*a(n-1) + 2*a(n-2).

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Formula

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

A189741 a(1)=4, a(2)=2, a(n) = 4a(n-1) + 2a(n-2).

A189734 a(1)=2, a(2)=5, a(n)=2a(n-1) + 5a(n-2).

A189736 a(1)=3, a(2)=2, a(n)=3a(n-1) + 2a(n-2).

A189742 a(1)=4, a(2)=3, a(n) = 4a(n-1) + 3a(n-2).

A189743 a(1)=4, a(2)=4, a(n) = 4a(n-1) + 4a(n-2).

A189744 a(1)=4, a(2)=5, a(n) = 4a(n-1) + 5a(n-2).