A180662 -id:A180662 - cp's OEIS frontend

A180847 a(n) = (27^n-4^n)/23.

0, 1, 31, 853, 23095, 623821, 16844191, 454797253, 12279542215, 331547705341, 8951788306351, 241698285320053, 6525853707835735, 176198050128342061, 4757347353532344511, 128448378545641737253, 3468106220733400647655

Offset: 0

Johannes W. Meijer, Sep 21 2010

For n>0, a(n) appears in several triangle sums of Nicomachus' table A036561, i.e. Ze1(2*n), Ze1(2*n+1)/2; Ze4(3*n), Ze4(3*n+1)/3 and Ze4(3*n+2)/9. See A180662 for information about these zebra and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (31,-108).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

Table[(27^n-4^n)/23,{n,0,20}] (* or *) LinearRecurrence[{31,-108},{0,1},20]  (* Harvey P. Dale, Sep 01 2011 *)

a(n)=(27^n-4^n)/23 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27^n-4^n)/23.
G.f.: x/((27*x-1)*(4*x-1)).
a(0)=0, a(1)=1, a(n) = 31*a(n-1)-108*a(n-2). - Harvey P. Dale, Sep 01 2011

A192928 The Gi1 and Gi2 sums of Losanitsch's triangle A034851.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 9, 11, 16, 20, 29, 37, 53, 69, 98, 130, 183, 245, 343, 463, 646, 877, 1220, 1664, 2310, 3161, 4381, 6009, 8319, 11430, 15811, 21751, 30070, 41405, 57216, 78836, 108906, 150130, 207346

Offset: 0

Johannes W. Meijer, Jul 14 2011

The Gi1 and Gi2 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal this sequence.
From Johannes W. Meijer, Aug 26 2013: (Start)
The a(n) are also the Ca1 and Ca2 sums of McGarvey’s triangle A102541.
Furthermore they are the Kn11 and Kn12 sums of triangle A228570.
And finally the terms of this sequence are the row sums of triangle A228572. (End)

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 25.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,0,-1,0,1,-1,0,0,-1).

Cf. A003269, A034851, A102541, A180662, A228570, A228572.

A192928 := proc(n): (A003269(n+1)+x(n)+x(n-1)+x(n-4))/2 end: A003269 := proc(n): sum(binomial(n-1-3*j, j), j=0..(n-1)/3) end: x:=proc(n): if type(n,even) then A003269(n/2+1) else 0 fi: end: seq(A192928(n),n=0..42);

LinearRecurrence[{1, 1, -1, 1, 0, -1, 0, 1, -1, 0, 0, -1}, {1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 9, 11}, 43] (* Jean-François Alcover, Nov 16 2017 *)

G.f.: (-1/2)*(1/(x^4+x-1) + (1+x+x^4)/(x^8+x^2-1))= -(1+x)*(x^7-x^6+x^5+x-1) / ( (x^4+x-1)*(x^8+x^2-1) ).
a(n) = (A003269(n+1)+x(n)+x(n-1)+x(n-4))/2 with x(2*n) = A003269(n+1) and x(2*n+1) = 0.
From Johannes W. Meijer, Aug 26 2013: (Start)
a(n) = sum(A228572(n, k), k=0..n)
a(n) = sum(A228570(n-k, k), k=0..floor(n/2))
a(n) = sum(A102541(n-2*k, k), k=0..floor(n/3))
a(n) = sum(A034851(n-3*k, k), k=0..floor(n/4)) (End)

A005683 Numbers of Twopins positions.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 22, 37, 63, 108, 186, 322, 559, 973, 1697, 2964, 5183, 9071, 15886, 27835, 48790, 85545, 150021, 263136, 461596, 809812, 1420813, 2492945, 4374273, 7675598, 13468787, 23634817, 41474548, 72780553, 127718046, 224125677, 393308019, 690200668

Offset: 3

N. J. A. Sloane

nonn

Appears to be a bisection of A068930. - Ralf Stephan, Apr 20 2004
The Ze3 and Ze4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence with a(1) = 1 and a(2) = 1; the recurrence relation below confirms these values and gives a(0) = 0. - Johannes W. Meijer, Jul 14 2011
The complete sequence by R. K. Guy in "Anyone for Twopins?" starts with a(0)=0, a(1)=1 and a(2)=1 and has g.f. x*(1-x-x^2)/(1-2*x+x^4+x^6). - Johannes W. Meijer, Aug 14 2011
a(n) is the number of equivalence classes of subsets of {1..n-2} without isolated elements up to reflection. The reflection of a subset is the set obtained by mapping each element i to n + 1 - i. For example, the a(6)=5 equivalence classes of subsets of {1..4} are {}, {1,2}/{3,4}, {2,3}, {1,2,3}/{2,3,4}, {1,2,3,4}. If reflections are not considered equivalent then A005251(n) gives the number of subsets of {1..n-2} without isolated elements. - Andrew Howroyd, Dec 24 2019

R. K. Guy, "Anyone for Twopins?", in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
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 (2, 0, 0, -1, 0, -1).

Cf. A000931, A005251.

A005683:=-(-1+z**2+z**3+z**4+z**5)/(z**3-z**2+2*z-1)/(z**3+z**2-1); [Conjectured by Simon Plouffe in his 1992 dissertation.]

CoefficientList[Series[(1-x^2-x^3-x^4-x^5)/(1-2x+x^4+x^6),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,0,-1,0,-1},{1,2,3,5,8,13},40] (* Harvey P. Dale, Jun 20 2011 *)

G.f.: x^3*(1-x^2-x^3-x^4-x^5)/(1-2*x+x^4+x^6). - Ralf Stephan, Apr 20 2004
a(3)=1, a(4)=2, a(5)=3, a(6)=5, a(7)=8, a(8)=13, a(n)=2*a(n-1)- a(n-4)- a(n-6). - Harvey P. Dale, Jun 20 2011
a(n) = (A005251(n) + A000931(n+4))/2. - Andrew Howroyd, Dec 24 2019

More terms from Harvey P. Dale, Jun 20 2011

A102543 Antidiagonal sums of the antidiagonals of Losanitsch's triangle.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 24, 33, 49, 69, 102, 145, 214, 307, 452, 653, 960, 1393, 2046, 2978, 4371, 6376, 9354, 13665, 20041, 29307, 42972, 62884, 92191, 134974, 197858, 289772, 424746, 622198, 911970, 1336121, 1958319, 2869417, 4205538, 6162579, 9031996, 13235661, 19398240

Offset: 0

Gerald McGarvey, Feb 24 2005

nonn

The Ca1 and Ca2 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal this sequence. - Johannes W. Meijer, Jul 14 2011

For n >= 5, a(n+1)-1 is the number of non-isomorphic snake polyominoes with n cells that can be inscribed in a rectangle of height 2. - Christian Barrientos and Sarah Minion, Jul 29 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 25.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,1,-1,0,-1).

Cf. A034851 A068927, A102451.

A102543 := proc(n): (A000930(n)+x(n)+x(n-1)+x(n-3))/2 end: A000930:=proc(n): sum(binomial(n-2*i, i), i=0..n/3) end: x:=proc(n): if type(n, even) then A000930(n/2) else 0 fi: end: seq(A102543(n), n=0..38); # Johannes W. Meijer, Jul 14 2011

CoefficientList[Series[(1 - x^2 - x^4 - x^6)/((x^3 + x - 1)*(x^6 + x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)
LinearRecurrence[{1,1,0,0,-1,1,-1,0,-1},{1,1,1,2,2,3,4,6,8},50] (* Harvey P. Dale, Dec 14 2023 *)

x='x+O('x^50); Vec((1 - x^2 - x^4 - x^6)/((x^3 + x - 1)*(x^6 + x^2 - 1))) \\ G. C. Greubel, Apr 27 2017

a(n) = A068927(n-1), n>3.
From Johannes W. Meijer, Jul 14 2011: (Start)
G.f.: (-1/2)*(1/(x^3+x-1)+(1+x+x^3)/(x^6+x^2-1))= ( 1-x^2-x^4-x^6 ) / ( (x^3+x-1)*(x^6+x^2-1) ).
a(n) = (A000930(n)+x(n)+x(n-1)+x(n-3))/2 with x(2*n) = A000930(n) and x(2*n+1) = 0. (End)

A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 5, 14, 32, 67, 133, 256, 484, 905, 1681, 3110, 5740, 10579, 19481, 35856, 65976, 121377, 223277, 410702, 755432, 1389491, 2555709, 4700720, 8646012, 15902537, 29249369, 53798022, 98950036, 181997539, 334745713

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn13 and Kn23 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=31: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*n-8 od: seq(a(n),n=0..nmax);

LinearRecurrence[{3,-2,0,-1,1},{0,0,1,5,14},40] (* Harvey P. Dale, Dec 15 2023 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+3)-2 with a(0)=0.
a(n) = sum(A008574(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+2,k+2),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^2)/((1-x)^2*(1-x-x^2-x^3)).
Contribution from Bruno Berselli, Sep 23 2010: (Start)
a(n) = 2*a(n-1)-a(n-4)+4 for n>4.
a(n)-3*a(n-1)+2a(n-2)+a(n-4)-a(n-5) = 0 for n>4. (End)

A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4n^2-16n+18 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 7, 26, 72, 171, 371, 760, 1500, 2889, 5475, 10266, 19116, 35435, 65495, 120832, 222664, 410017, 754671, 1388650, 2554784, 4699707, 8644907, 15901336, 29248068, 53796617, 98948523, 181995914, 334743972, 615691547

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn14 and Kn24 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-1,2,-1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

Cf. A000073, A005899, A008288.

nmax:=30: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 od: seq(a(n),n=0..nmax);

nxt[{n_,a_,b_,c_}]:={n+1,b,c,a+b+c+4n(n-2)+6}; NestList[nxt,{2,0,0,1},30][[;;,2]] (* or *) LinearRecurrence[{4,-5,2,-1,2,-1},{0,0,1,7,26,72},40] (* Harvey P. Dale, Jul 13 2024 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+5)-2-4*n with a(0)=0.
a(n) = Sum_{m=0..n} A005899(m)*A000073(n-m).
a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+3,k+3).
GF(x) = (x^2*(1+x)^3)/((1-x)^3*(1-x-x^2-x^3)).
From Bruno Berselli, Sep 23 2010: (Start)
a(n) = 3*a(n-1)-2a(n-2)-a(n-4)+a(n-5)+8 for n>4.
a(n)-4*a(n-1)+5a(n-2)-2*a(n-3)+a(n-4)-2*a(n-5)+a(n-6) = 0 for n>5. (End)

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 9, 42, 140, 383, 925, 2056, 4316, 8705, 17069, 32810, 62192, 116743, 217673, 404000, 747496, 1380177, 2544865, 4688186, 8631620, 15886111, 29230725, 53776968, 98926372, 181971057, 334716197, 615660634, 1132400520

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn15 and Kn25 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-3,3,-3,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=29: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 od: seq(a(n),n=0..nmax);

RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+(8n^3-48n^2+112n-96)/3},a,{n,30}] (* or *) LinearRecurrence[{5,-9,7,-3,3,-3,1},{0,0,1,9,42,140,383},30] (* Harvey P. Dale, Dec 04 2019 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+7)-(12+4*n+4*n^2) with a(0)=0.
a(n) = sum(A008412(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+4,k+4),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^4)/((1-x)^4*(1-x-x^2-x^3)).

A180671 a(n) = Fibonacci(n+6) - Fibonacci(6).

Original entry on oeis.org

0, 5, 13, 26, 47, 81, 136, 225, 369, 602, 979, 1589, 2576, 4173, 6757, 10938, 17703, 28649, 46360, 75017, 121385, 196410, 317803, 514221, 832032, 1346261, 2178301, 3524570, 5702879, 9227457, 14930344, 24157809, 39088161, 63245978, 102334147, 165580133

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn15 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+6)-8); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+6)-Fibonacci(6): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+6)-fibonacci(6) od: seq(a(n),n=0..nmax);

f[n_]:= Fibonacci[n+6] - Fibonacci[6]; Array[f, 40, 0] (* or *)
LinearRecurrence[{2,0,-1}, {0,5,13}, 41] (* or *)
CoefficientList[Series[x(3x+5)/(x^3-2x+1), {x,0,40}], x] (* Robert G. Wilson v, Apr 11 2017 *)

for(n=1,40,print(fibonacci(n+6)-fibonacci(6))); \\ Anton Mosunov, Mar 02 2017

concat(0, Vec(x*(5+3*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Apr 20 2017

[fibonacci(n+6)-8 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+6) - F(6) with F = A000045.
a(n) = a(n-1) + a(n-2) + 8 for n>1, a(0)=0, a(1)=5, and where 8 = F(6).
From Colin Barker, Apr 13 2012: (Start)
G.f.: x*(5 + 3*x)/((1 - x)*(1 - x - x^2)).
a(n) = 2*a(n-1) - a(n-3). (End)
a(n) = (-8 + (2^(-n)*((1-sqrt(5))^n*(-9+4*sqrt(5)) + (1+sqrt(5))^n*(9+4*sqrt(5)))) / sqrt(5)). - Colin Barker, Apr 20 2017

A180672 a(n) = Fibonacci(n+7) - Fibonacci(7).

Original entry on oeis.org

0, 8, 21, 42, 76, 131, 220, 364, 597, 974, 1584, 2571, 4168, 6752, 10933, 17698, 28644, 46355, 75012, 121380, 196405, 317798, 514216, 832027, 1346256, 2178296, 3524565, 5702874, 9227452, 14930339, 24157804, 39088156, 63245973

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn16 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+7)-13 ); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+7) - Fibonacci(7): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+7)-fibonacci(7) od: seq(a(n),n=0..nmax);

Fibonacci[7 +Range[0, 40]] -13 (* G. C. Greubel, Jul 13 2019 *)

concat(0, Vec(x*(8+5*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n)=fibonacci(n+7)-fibonacci(7) \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+7)-13 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+7) - F(7) with F = A000045.
a(n) = a(n-1) + a(n-2) + 13 for n>1, a(0)=0, a(1)=8, and where 13 = F(7).
G.f.: x*(8 + 5*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-13 + (2^(-1-n)*((1-sqrt(5))^n*(-29+13*sqrt(5)) + (1+sqrt(5))^n*(29+13*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 8*A000071(n+2) + 5*A000071(n+1). - Bruno Berselli, Feb 24 2017

A180673 a(n) = Fibonacci(n+8) - Fibonacci(8).

Original entry on oeis.org

0, 13, 34, 68, 123, 212, 356, 589, 966, 1576, 2563, 4160, 6744, 10925, 17690, 28636, 46347, 75004, 121372, 196397, 317790, 514208, 832019, 1346248, 2178288, 3524557, 5702866, 9227444, 14930331, 24157796, 39088148, 63245965, 102334134

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn17 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..275
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+8)-21); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+8) - Fibonacci(8): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+8)-fibonacci(8) od: seq(a(n),n=0..nmax);

Fibonacci[8 +Range[0, 40]] -21 (* G. C. Greubel, Jul 13 2019 *)

concat(0, Vec(x*(13+8*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n)=fibonacci(n+8)-21 \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+8)-21 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+8) - F(8) with F(n) the Fibonacci numbers A000045.
a(n) = a(n-1) + a(n-2) + 21 for n>1, a(0)=0, a(1)=13, and where 21 = F(8).
G.f.: x*(13 + 8*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 13*A000071(n+2) + 8*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-21 + (2^(-1-n)*((1-sqrt(5))^n*(-47+21*sqrt(5)) + (1+sqrt(5))^n*(47+21*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

cp's OEIS Frontend

A180847 a(n) = (27^n-4^n)/23.

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A192928 The Gi1 and Gi2 sums of Losanitsch's triangle A034851.

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A005683 Numbers of Twopins positions.

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A102543 Antidiagonal sums of the antidiagonals of Losanitsch's triangle.

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A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

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A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n^2-16*n+18 with a(0)=0, a(1)=0 and a(2)=1.

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A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

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A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4n^2-16n+18 with a(0)=0, a(1)=0 and a(2)=1.

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.