A026150 -id:A026150 - cp's OEIS frontend

A083098 a(n) = 2a(n-1) + 6a(n-2).

1, 1, 8, 22, 92, 316, 1184, 4264, 15632, 56848, 207488, 756064, 2757056, 10050496, 36643328, 133589632, 487039232, 1775616256, 6473467904, 23600633344, 86042074112, 313687948288, 1143628341248, 4169384372224, 15200538791936

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

a(n+1) = a(n) + 7*A083099(n-1); a(n+1)/A083099(n) converges to sqrt(7).
Binomial transform of expansion of cosh(sqrt(7)x) (A000420 with interpolated zeros: 1, 0, 7, 0, 49, 0, 343, 0, ...).
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard, Sep 25 2005
a(n) is the number of compositions of n when there are 1 type of 1 and 7 types of other natural numbers. - Milan Janjic, Aug 13 2010

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Project Euler, Problem 752, sequence alpha(n).
Index entries for linear recurrences with constant coefficients, signature (2,6).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

CoefficientList[Series[(1+6x)/(1-2x-6x^2), {x, 0, 25}], x]
LinearRecurrence[{2, 6}, {1, 1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)
a[n_] := Simplify[((1 + Sqrt[7])^n + (1 - Sqrt[7])^n)/2]; Array[a, 25, 0] (* Robert G. Wilson v, Sep 18 2013 *)

x='x+O('x^30); Vec((1-x)/(1-2*x-6*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number2(n,2,-6)/2 for n in range(0, 25)] # Zerinvary Lajos, Apr 30 2009

G.f.: (1-x)/(1-2*x-6*x^2).
a(n) = (1+sqrt(7))^n/2 + (1-sqrt(7))^n/2.
E.g.f.: exp(x)*cosh(sqrt(7)x).
a(n) = Sum_{k=0..n} A098158(n,k)*7^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=7, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n) = det A. - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A125145 a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 15, 57, 216, 819, 3105, 11772, 44631, 169209, 641520, 2432187, 9221121, 34959924, 132543135, 502509177, 1905156936, 7222998339, 27384465825, 103822392492, 393620574951, 1492328902329, 5657848431840, 21450532002507

Offset: 0

Tanya Khovanova, Jan 11 2007

Number of aa-avoiding words of length n on the alphabet {a,b,c,d}.
Equals row 3 of the array shown in A180165, the INVERT transform of A028859 and the INVERTi transform of A086347. - Gary W. Adamson, Aug 14 2010
From Tom Copeland, Nov 08 2014: (Start)
This array is one of a family related by compositions of C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for A000108; its inverse Cinv(x) = x(1-x); and the special Mobius transformation P(x,t) = x / (1+t*x) with inverse P(x,-t) in x. Cf. A091867.
O.g.f.: G(x) = P[P[P[-Cinv(-x),-1],-1],-1] = P[-Cinv(-x),-3] = x*(1+x)/[1-3x(1-x)]= x*A125145(x).
Ginv(x) = -C[-P(x,3)] = [-1 + sqrt(1+4x/(1+3x))]/2 = x*A104455(-x).
G(-x) = -x(1-x) * [ 1 - 3*[x*(1+x)] + 3^2*[x*(1+x)]^2 - ...] , and so this array is related to finite differences in the row sums of A030528 * Diag((-3)^1,3^2,(-3)^3,..). (Cf. A146559.)
The inverse of -G(-x) is C[-P(-x,3)]= [1 - sqrt(1-4x/(1-3x))]/2 = x*A104455(x). (End)
Number of 3-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook)
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A028859 = a(n+2) = 2 a(n+1) + 2 a(n); A086347 = On a 3 X 3 board, number of n-move routes of chess king ending at a given side cell. a(n) = 4a(n-1) + 4a(n-2).
Cf. A128235.
Cf. A180165, A028859, A086347. - Gary W. Adamson, Aug 14 2010
Cf. A002605, A026150, A030195, A080040, A083337, A106435, A108898.
Cf. A000108, A091867, A125145, A104455, A030528, A146559.

a125145 n = a125145_list !! n
a125145_list =
   1 : 4 : map (* 3) (zipWith (+) a125145_list (tail a125145_list))
-- Reinhard Zumkeller, Oct 15 2011

I:=[1,4]; [n le 2 select I[n] else 3*Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

a[0]:=1: a[1]:=4: for n from 2 to 27 do a[n]:=3*a[n-1]+3*a[n-2] od: seq(a[n],n=0..27); # Emeric Deutsch, Feb 27 2007
A125145 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,4]) ;
    else
        3*(procname(n-1)+procname(n-2)) ;
    end if;
end proc: # R. J. Mathar, Feb 13 2022

nn=23;CoefficientList[Series[(1+x)/(1-3x-3x^2),{x,0,nn}],x] (* Geoffrey Critzer, Feb 09 2014 *)
LinearRecurrence[{3,3},{1,4},30] (* Harvey P. Dale, May 01 2022 *)

G.f.: (1+z)/(1-3z-3z^2). - Emeric Deutsch, Feb 27 2007
a(n) = (5*sqrt(21)/42 + 1/2)*(3/2 + sqrt(21)/2)^n + (-5*sqrt(21)/42 + 1/2)*(3/2 - sqrt(21)/2)^n. - Antonio Alberto Olivares, Mar 20 2008
a(n) = A030195(n)+A030195(n+1). - R. J. Mathar, Feb 13 2022
E.g.f.: exp(3*x/2)*(21*cosh(sqrt(21)*x/2) + 5*sqrt(21)*sinh(sqrt(21)*x/2))/21. - Stefano Spezia, Aug 04 2022
a(n) = (((3 + sqrt(21)) / 2)^(n+2) - ((3 - sqrt(21)) / 2)^(n+2)) / (3 * sqrt(21)). - Werner Schulte, Dec 17 2024

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

A083100 a(n) = 2a(n-1) + 7a(n-2).

Original entry on oeis.org

1, 9, 25, 113, 401, 1593, 5993, 23137, 88225, 338409, 1294393, 4957649, 18976049, 72655641, 278143625, 1064876737, 4076758849, 15607654857, 59752621657, 228758827313, 875786006225, 3352883803641, 12836269650857, 49142725927201

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003

a(n) = a(n-1) + 8*A015519(n). a(n)/A015519(n+1) converges to sqrt(8).

a(n-1) is the number of compositions of n when there is 1 type of 1 and 8 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 7).

Essentially a duplicate of A084058.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) + 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

CoefficientList[Series[(1 + 7 x)/(1 - 2 x - 7 x^2), {x, 0, 25}], x] (* Or *) a[n_] := Simplify[((1 + Sqrt[8])^n + (1 - Sqrt[8])^n)/2]; Array[a, 25, 0] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 28] (* Or *) Table[ MatrixPower[{{1, 2}, {4, 1}}, n][[1, 1]], {n, 0, 25}] (* Robert G. Wilson v, Sep 18 2013 *)

a(n)=([0,1; 7,2]^n*[1;9])[1,1] \\ Charles R Greathouse IV, Apr 06 2016

x='x+O('x^30); Vec((1+7*x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Jan 08 2018

G.f.: (1+7*x)/(1-2*x-7*x^2).
a(n) = binomial transform of 1,8,8,64,64,512. - Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
If p[1]=1, and p[i]=8,(i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det A. - Milan Janjic, Apr 29 2010
G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

A028860 a(n+2) = 2a(n+1) + 2a(n); a(0) = -1, a(1) = 1.

Original entry on oeis.org

-1, 1, 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, 36544, 99840, 272768, 745216, 2035968, 5562368, 15196672, 41518080, 113429504, 309895168, 846649344, 2313089024, 6319476736, 17265131520, 47169216512, 128868696064, 352075825152, 961889042432, 2627929735168

Offset: 0

N. J. A. Sloane

a(n+1) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 04 2014

(A002605, a(.+1)) is the canonical basis of the space of linear recurrent sequences with signature (2, 2), i.e., any sequence s(n) = 2(s(n-1) + s(n-2)) is given by s = s(0)*A002605 + s(1)*a(.+1). - M. F. Hasler, Aug 06 2018

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 924
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A002605, A026150, A030195, A080040, A083337, A106435, A108898, A125145.

a:=[-1,1];; for n in [3..30] do a[n]:=2*a[n-1]+2*a[n-2]; od; a; # Muniru A Asiru, Aug 07 2018

a028860 n = a028860_list !! n
a028860_list =
   -1 : 1 : map (* 2) (zipWith (+) a028860_list (tail a028860_list))
-- Reinhard Zumkeller, Oct 15 2011

I:=[-1,1]; [n le 2 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2018

seq(coeff(series((3*x-1)/(1-2*x-2*x^2), x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 07 2018

(With a different offset) M = {{0, 2}, {1, 2}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}] (* Roger L. Bagula, May 29 2005 *)
LinearRecurrence[{2,2},{-1,1},40] (* Harvey P. Dale, Dec 13 2012 *)
CoefficientList[Series[(-3 x + 1)/(2 x^2 + 2 x - 1), {x, 0, 27}], x] (* Robert G. Wilson v, Aug 07 2018 *)

apply( A028860(n)=([2,2;1,0]^n)[2,]*[1,-1]~, [0..30]) \\ 15% faster than (A^n*[1,-1]~)[2]. - M. F. Hasler, Aug 06 2018

A028860 = BinaryRecurrenceSequence(2,2,-1,1)
[A028860(n) for n in range(51)] # G. C. Greubel, Dec 08 2022

a(n) = 4*A028859(n-4), for n > 3.
From R. J. Mathar, Nov 27 2008: (Start)
G.f.: -(1 - 3*x)/(1 - 2*x - 2*x^2).
a(n) = 3*A002605(n-1) - A002605(n). (End)
a(n) = det A, where A is the Hessenberg matrix of order n+1 defined by: A[i,j] = p(j - i + 1) (i <= j), A[i,j] = -1 (i = j + 1), A[i,j] = 0 otherwise, with p(i) = fibonacci(2i - 4). - Milan Janjic, May 08 2010, edited by M. F. Hasler, Aug 06 2018
a(n) = (2*sqrt(3) - 3)/6*(1 + sqrt(3))^n - (2*sqrt(3) + 3)/6*(1 - sqrt(3))^n. - Sergei N. Gladkovskii, Jul 18 2012
a(n) = 2*A002605(n-2) for n >= 2. - M. F. Hasler, Aug 06 2018
E.g.f.: exp(x)*(2*sqrt(3)*sinh(sqrt(3)*x) - 3*cosh(sqrt(3)*x))/3. - Franck Maminirina Ramaharo, Nov 11 2018

Edited by N. J. A. Sloane, Apr 11 2009

Edited and initial values added in definition by M. F. Hasler, Aug 06 2018

A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 1

Klaus Brockhaus, Jul 03 2009, Jul 05 2009

Interleaving of A000244 and 3*A000244.
Unsigned version of A128019.
Partial sums are in A164123.
Apparently a(n) = A056449(n-1) for n > 1. a(n) = A108411(n) for n >= 1.
Binomial transform is A026150 without initial 1, second binomial transform is A001834, third binomial transform is A030192, fourth binomial transform is A161728, fifth binomial transform is A162272.

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

Cf. A000244 (powers of 3), A128019 (expansion of (1-3x)/(1+3x^2)), A164123, A026150, A001834, A030192, A161728, A162272.

Essentially the same as A056449 (3^floor((n+1)/2)) and A108411 (powers of 3 repeated).

[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..35] ];

CoefficientList[Series[(-3*x - 1)/(3*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]] (* or *) With[{c= 3^Range[20]},Join[{1},Riffle[c,c]]](* Harvey P. Dale, Feb 17 2012 *)

a(n)=3^(n>>1) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = 3^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 3*x)/(1 - 3*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
E.g.f.: cosh(sqrt(3)*x) - 1 + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

G.f. corrected, formula simplified, comments added by Klaus Brockhaus, Sep 18 2009

A106435 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 9, 36, 135, 513, 1944, 7371, 27945, 105948, 401679, 1522881, 5773680, 21889683, 82990089, 314639316, 1192888215, 4522582593, 17146412424, 65006985051, 246460192425, 934401532428, 3542585174559, 13430960120961

Offset: 0

Roger L. Bagula, May 29 2005

The first entry of the vector v[n] = M*v[n-1], where M is the 2 x 2 matrix [[0,3],[1,3]] and v[1] is the column vector [0,1]. The characteristic polynomial of the matrix M is x^2-3x-3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,3).

Equals 3*A030195(n).
Cf. A028860.
Cf. A002605, A026150, A028859, A080040, A083337, A108898, A125145.

a106435 n = a106435_list !! n
a106435_list = 0 : 3 : map (* 3) (zipWith (+) a106435_list (tail
a106435_list))
-- Reinhard Zumkeller, Oct 15 2011

a:=[0,3]; [n le 2 select a[n] else    3*Self(n-1) + 3*Self(n-2) : n in [1..24]]; // Marius A. Burtea, Jan 21 2020

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!(3*x/(1-3*x-3*x^2))); // Marius A. Burtea, Jan 21 2020

seq(coeff(series(3*x/(1-3*x-3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Mar 12 2020

LinearRecurrence[{3,3}, {0,3}, 30] (* G. C. Greubel, Mar 12 2020 *)

PARI
```
a(n)=([0,3;1,3]^n)[1,2]
```

[3^((n+1)/2)*i^(1-n)*chebyshev_U(n-1, i*sqrt(3)/2) for n in (0..30)] # G. C. Greubel, Mar 12 2020

G.f.: 3*x/(1-3*x-3*x^2). - Philippe Deléham, Nov 19 2008
From G. C. Greubel, Mar 12 2020: (Start)
a(n) = 3^((n+1)/2) * Fibonacci(n, sqrt(3)), where F(n, x) is the Fibonacci polynomial.
a(n) = 3^((n+1)/2)*i^(1-n)*ChebyshevU(n-1, i*sqrt(3)/2). (End)

Edited by N. J. A. Sloane, May 20 2006 and May 29 2006

Offset corrected by Reinhard Zumkeller, Oct 15 2011

A083337 a(n) = 2a(n-1) + 2a(n-2); a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 20064, 54816, 149760, 409152, 1117824, 3053952, 8343552, 22795008, 62277120, 170144256, 464842752, 1269974016, 3469633536, 9479215104, 25897697280, 70753824768, 193303044096, 528113737728, 1442833563648, 3941894602752, 10769456332800

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,2).

Equals 3 * A002605.
Cf. A026150.
Cf. A028859, A028860, A030195, A080040, A106435, A108898, A125145.

a083337 n = a083337_list !! n
a083337_list =
   0 : 3 : map (* 2) (zipWith (+) a083337_list (tail a083337_list))
-- Reinhard Zumkeller, Oct 15 2011

CoefficientList[Series[3x/(1-2x-2x^2), {x, 0, 25}], x]
s = Sqrt[3]; a[n_] := Simplify[s*((1 + s)^n - (1 - s)^n)/2]; Array[a, 30, 0] (* or *)
LinearRecurrence[{2, 2}, {0, 3}, 31] (* Robert G. Wilson v, Aug 07 2018 *)

apply( a(n)=([1,1;3,1]^n)[2,1], [0..30]) \\ or: ([2,2;1,0]^n)[2,1]*3. - M. F. Hasler, Aug 06 2018

G.f.: 3x/(1 - 2x - 2x^2).
a(n) = a(n-1) + 3*A026150(n-1). a(n)/A026150(n) converges to sqrt(3).
a(n) = lower left term of [1,1; 3,1]^n. - Gary W. Adamson, Mar 12 2008

Edited and definition completed by M. F. Hasler, Aug 06 2018

A108898 a(n+3) = 3a(n+2) - 2a(n), a(0) = -1, a(1) = 1, a(2) = 3.

Original entry on oeis.org

-1, 1, 3, 11, 31, 87, 239, 655, 1791, 4895, 13375, 36543, 99839, 272767, 745215, 2035967, 5562367, 15196671, 41518079, 113429503, 309895167, 846649343, 2313089023, 6319476735, 17265131519, 47169216511, 128868696063, 352075825151, 961889042431, 2627929735167, 7179637555199

Offset: 0

Creighton Dement, Jul 16 2005

In reference to the program code, "ibasek" corresponds to the floretion 'ik'. Sequences in this same batch are "kbase" = A005665 (Tower of Hanoi with cyclic moves only.) and "ibase" = A077846.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Cf. A005665, A077846, A028860.

Cf. A002605, A026150, A028859, A030195, A080040, A083337, A106435, A125145.

a108898 n = a108898_list !! n
a108898_list = -1 : 1 : 3 :
   zipWith (-) (map (* 3) $ drop 2 a108898_list) (map (* 2) a108898_list)
-- Reinhard Zumkeller, Oct 15 2011

seriestolist(series((-1+4*x)/((x-1)*(2*x^2+2*x-1)), x=0,31)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2ibaseksumseq[A*B] with A = + 'i + 'ii' + 'ij' + 'ik' and B = + .5'i + .5'j - .5'k + .5i' - .5j' + .5k' + .5'ij' + .5'ik' - .5'ji' - .5'ki'; Sumtype is set to:sum[(Y[0], Y[1], Y[2]),mod(3)

LinearRecurrence[{3, 0, -2}, {-1, 1, 3}, 40] (* Paolo Xausa, Aug 21 2024 *)

Vec(-(1 - 4*x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Apr 29 2019

a(n) = A028860(n+2)-1.
G.f.: (-1+4*x)/((x-1)*(2*x^2+2*x-1)).
From Colin Barker, Apr 29 2019: (Start)
a(n) = (-1 + (-(1-sqrt(3))^n + (1+sqrt(3))^n)/sqrt(3)).
a(n) = 3*a(n-1) - 2*a(n-3) for n>2.
(End)

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

cp's OEIS Frontend

A083098 a(n) = 2*a(n-1) + 6*a(n-2).

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A125145 a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4.

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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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A083100 a(n) = 2*a(n-1) + 7*a(n-2).

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A028860 a(n+2) = 2*a(n+1) + 2*a(n); a(0) = -1, a(1) = 1.

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A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

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Magma

A083098 a(n) = 2a(n-1) + 6a(n-2).

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).

A083100 a(n) = 2a(n-1) + 7a(n-2).

A028860 a(n+2) = 2a(n+1) + 2a(n); a(0) = -1, a(1) = 1.

A106435 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=3.

A083337 a(n) = 2a(n-1) + 2a(n-2); a(0)=0, a(1)=3.

A108898 a(n+3) = 3a(n+2) - 2a(n), a(0) = -1, a(1) = 1, a(2) = 3.