A005248 -id:A005248 - cp's OEIS frontend

A153175 a(n) = L(7*n)/L(n) where L(n) = Lucas number A000204(n).

29, 281, 6119, 101521, 1875749, 33281921, 599786069, 10745088481, 192933544679, 3461223997001, 62114818827629, 1114566304366081, 20000347407134669, 358889844987430121, 6440029487834912999, 115561554399692896321

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 9 mod 10 (iff n is odd),
congruent to 1 mod 10 (iff n is even).

G. C. Greubel, Table of n, a(n) for n = 1..795
Index entries for linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).

Cf. A000032, A000204, A110391, A153173.

Cf. A153177, A153179, A153180. [From R. J. Mathar, Oct 22 2010]

[Lucas(7*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[7*n]/LucasL[n], {n, 1, 50}]

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(7*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = +13*a(n-1) +104*a(n-2) -260*a(n-3) -260*a(n-4) +104*a(n-5) +13*a(n-6) -a(n-7).
G.f.: -x*(-29+96*x+550*x^2-290*x^3-200*x^4+16*x^5+x^6) / ( (1+x)*(x^2-3*x+1)*(x^2-18*x+1)*(x^2+7*x+1) ).
a(n) = A005248(n) +A087215(n) -(-1)^n*A056854(n) - (-1)^n. (End)

A153177 a(n) = L(9*n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

76, 1926, 109801, 4769326, 230701876, 10716675201, 505618944676, 23714405408926, 1114769987764201, 52357935173823126, 2459933168462154076, 115560463558534156801, 5428954301161174383676, 255043991670277234750326

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 1 mod 100 (iff n is congruent to 0 mod 3),
congruent to 26 mod 100 (iff n is congruent to 2 or 4 mod 6),
congruent to 76 mod 100 (iff n is congruent to 1 or 5 mod 6).

G. C. Greubel, Table of n, a(n) for n = 1..595
Index entries for linear recurrences with constant coefficients, signature (34,714,-4641,-12376,12376,4641,-714,-34,1).

Cf. A000032, A000204, A110391, A153173, A153175.

Cf. A153179, A153180. - R. J. Mathar, Oct 22 2010

[Lucas(9*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[9*n]/LucasL[n], {n, 1, 50}]
LinearRecurrence[{34,714,-4641,-12376,12376,4641,-714,-34,1},{76,1926,109801,4769326,230701876,10716675201,505618944676,23714405408926,1114769987764201},20] (* Harvey P. Dale, Aug 12 2012 *)

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(9*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = 34*a(n-1) +714*a(n-2) -4641*a(n-3) -12376*a(n-4) +12376*a(n-5) +4641*a(n-6) -714*a(n-7) -34*a(n-8) +a(n-9).
G.f.: -x*(76-658*x-9947*x^2+13644*x^3+26020*x^4-5306*x^5-1372*x^6+42*x^7 +x^8) / ((x-1)*(x^2+18*x+1)*(x^2-47*x+1)*(x^2+3*x+1)*(x^2-7*x+1)).
a(n) = 1-(-1)^n*A087215(n) -(-1)^n*A005248(n) +A056854(n) +A087265(n). (End)

A153179 a(n) = L(11*n)/L(n) where L(n) = A000204(n).

Original entry on oeis.org

199, 13201, 1970299, 224056801, 28374454999, 3450736132801, 426236170575799, 52337681992411201, 6441140796368008699, 792018481913198430001, 97420733208491869044199, 11981539981561372141075201

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 99 mod 100 (iff n is odd),
congruent to 1 mod 100 (iff n is even).

G. C. Greubel, Table of n, a(n) for n = 1..475
Index entries for linear recurrences with constant coefficients, signature (89,4895,-83215,-582505,1514513,1514513,-582505,-83215,4895,89,-1).

Cf. A000032, A000204, A110391, A153173, A153175, A153177.

[Lucas(11*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[11*n]/LucasL[n], {n, 1, 50}]

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(11*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = +89*a(n-1) +4895*a(n-2) -83215*a(n-3) -582505*a(n-4) +1514513*a(n-5) +1514513*a(n-6) -582505*a(n-7) -83215*a(n-8) +4895*a(n-9) +89*a(n-10) -a(n-11).
G.f.: -1 -1/(1+x) +(-2-47*x)/(x^2+47*x+1) +(2-3*x)/(x^2-3*x+1) +(-2-7*x)/(x^2+7*x+1) +(2-123*x)/(x^2-123*x+1) +(2-18*x)/(x^2-18*x+1).
a(n) = -(-1)^n -(-1)^n*A087265(n) +A005248(n) -(-1)^n*A056854(n) +A065705(n) +A087215(n). (End)

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 14, 54, 204, 774, 2934, 11124, 42174, 159894, 606204, 2298294, 8713494, 33035364, 125246574, 474845814, 1800277164, 6825368934, 25876938294, 98106921684, 371951579934, 1410175504854, 5346381254364, 20269670277654, 76848154596054, 291353474621124

Offset: 0

Johannes W. Meijer, Aug 13 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to 16 A[5] vectors with decimal values between 3 and 384. These vectors lead for the corner squares to A123620 and for the central square to A155116.
This sequence appears among the members of a family of sequences with g.f. (1 + x - k*x^2)/(1 - 3*x + (k-4)*x^2). Berserker sequences that are members of this family are 4*A007482 (k=2; with leading 1 added), A180142 (k=1; this sequence), A000302 (k=0), A180140 (k=-1) and 4*A154964 (k=-2; n>=1 and a(0)=1). Some other members of this family are 2*A180148 (k=3; with leading 1 added), 4*A025192 (k=4; with leading 1 added), 2*A005248 (k=5; with leading 1 added) and A123932 (k=6).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,0,0,0,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);
# second Maple program:
a:= n-> ceil((<<0|1>, <3|3>>^n. <<2/3, 4>>)[1,1]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 14 2021

LinearRecurrence[{3, 3}, {1, 4, 14}, 26] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).
a(n) = 3*a(n-1) + 3*a(n-2) for n >= 2 with a(0)=1, a(1)=4 and a(2)=14.
a(n) = (6-2*A)*A^(-n-1)/21 + (6-2*B)*B^(-n-1)/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6.
Lim_{k->infinity} a(2*n+k)/a(k) = 2*A000244(n)/(A003501(n) - A004254(n)*sqrt(21)) for n >= 1.
Lim_{k->infinity} a(2*n-1+k)/a(k) = 2*A000244(n)/(A004253(n)*sqrt(21) - 3*A030221(n-1)) for n >= 1.

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 13, 18, 1, 6, 9, 19, 38, 47, 1, 7, 11, 25, 58, 117, 123, 1, 8, 13, 31, 78, 187, 370, 322, 1, 9, 15, 37, 98, 257, 622, 1186, 843, 1, 10, 17, 43, 118, 327, 874, 2110, 3827, 2207, 1, 11, 19, 49, 138, 397, 1126, 3034, 7252, 12389, 5778, 1

Offset: 0

L. Edson Jeffery, Jan 23 2012

Row indices q begin with 1, column indices n begin with 0.

			Array begins as
1,  1,  1,  1,   1,    1, ...
2,  3,  7, 18,  47,  123, ...
3,  5, 13, 38, 117,  370, ...
4,  7, 19, 58, 187,  622, ...
5,  9, 25, 78, 257,  874, ...
6, 11, 31, 98, 327, 1126, ...
...

S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Conjecture. Transpose of array A186740.
Conjecture. Rows 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
Conjecture. Columns 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
Cf. A209235.

Conjecture. The n-th entry in row q is given by R_q(n) = 2^(2*n)*(sum_{j=1,...,n+1} (cos(j*Pi/(2*q+1)))^(2*n)), q >= 1, n >= 0.
Conjecture. G.f. for column n is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x, n >= 0.
Conjecture. 2*A185095(q,n) = A198632(2*q,n), q >= 1, n >= 0. - L. Edson Jeffery, Nov 23 2013

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 7, 5, 4, 1, 18, 13, 7, 5, 1, 47, 38, 19, 9, 6, 1, 123, 117, 58, 25, 11, 7, 1, 322, 370, 187, 78, 31, 13, 8, 1, 843, 1186, 622, 257, 98, 37, 15, 9, 1, 2207, 3827, 2110, 874, 327, 118, 43, 17, 10, 1, 5778, 12389, 7252, 3034, 1126, 397, 138, 49, 19, 11

Offset: 0

L. Edson Jeffery, Jan 21 2012

Row indices n begin with 0, column indices q begin with 1.

			Array begins:
1    2     3     4     5     6     7     8     9 ...
1    3     5     7     9    11    13    15    17 ...
1    7    13    19    25    31    37    43    49 ...
1   18    38    58    78    98   118   138   158 ...
1   47   117   187   257   327   397   467   537 ...
1  123   370   622   874  1126  1378  1630  1882 ...
1  322  1186  2110  3034  3958  4882  5806  6730 ...
1  843  3827  7252 10684 14116 17548 20980 24412 ...
1 2207 12389 25147 38017 50887 63757 76627 89497 ...
...
As a triangle:
1,
1,  2,
1,  3,  3,
1,  7,  5,  4,
1, 18, 13,  7, 5,
1, 47, 38, 19, 9, 6,
...

Andrew Howroyd, Table of n, a(n) for n = 0..1275
S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Conjecture: Transpose of array A185095.
Conjecture: Columns 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
Conjecture: Rows 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
Cf. A209235.

Conjecture: G.f. for column q is F_q(x) = (Sum_{r=0..q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r)) / (Sum_{s=0..q} ((-1)^s*binomial(2*q-s,s)*x^s)), q >= 1.

Conjecture: G.f. for n-th row is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x.

A198635 Total number of round trips, each of length 2*n on the graph P_5 (o-o-o-o-o).

Original entry on oeis.org

5, 8, 20, 56, 164, 488, 1460, 4376, 13124, 39368, 118100, 354296, 1062884, 3188648, 9565940, 28697816, 86093444, 258280328, 774840980, 2324522936, 6973568804, 20920706408, 62762119220, 188286357656, 564859072964, 1694577218888, 5083731656660, 15251194969976

Offset: 0

Wolfdieter Lang, Nov 02 2011

See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l = 2*k).

			With the graph P_5 as 1-2-3-4-5:
n=0: 5, from the length 0 walks starting at 1,2,3,4 and 5.
n=1: 8, from the walks of length 2, namely 121, 212, 232, 323, 343, 434, 454 and 545.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A005248, A198632, A198633.

Essentially the same as A115099.

a[0] = 5; a[n_] := 2*3^n + 2; Array[a, 28, 0] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
CoefficientList[Series[(5 - 12 x + 3 x^2)/(1 - 4 x + 3 x^2), {x, 0, 27}], x] (* Michael De Vlieger, Dec 18 2017 *)
LinearRecurrence[{4,-3},{5,8,20},30] (* Harvey P. Dale, Nov 27 2024 *)

a(n) = w(5,2*n), n >= 0, with w(5,l) the total number of closed walks on the graph P_5 (the simple path with 5 points (vertices) and 4 lines (or edges)).
O.g.f. for w(5,l) (with zeros for odd l): y*(d/dy)S(5,y)/S(5,y) with y = 1/x and Chebyshev S-polynomials (coefficients A049310). See also A198632 for a rewritten form.
G.f.: (5-12*x+3*x^2)/(1-4*x+3*x^2). - Colin Barker, Jan 02 2012
a(n) = 3*a(n-1) - 4, n > 1. - Vincenzo Librandi, Jan 02 2012
a(n) = 2*3^n + 2 for n > 0. - Andrew Howroyd, Mar 18 2017
a(n) = 2*A034472(n) for n > 0. - Andrew Howroyd, Mar 18 2017

A219233 Alternating row sums of Riordan triangle A110162.

Original entry on oeis.org

1, -3, 7, -18, 47, -123, 322, -843, 2207, -5778, 15127, -39603, 103682, -271443, 710647, -1860498, 4870847, -12752043, 33385282, -87403803, 228826127, -599074578, 1568397607, -4106118243, 10749957122, -28143753123, 73681302247, -192900153618, 505019158607

Offset: 0

Wolfdieter Lang, Nov 16 2012

If a(0) is put to 2 instead of 1 this becomes a(n) = (-1)^n*A005248(n), n >= 0. These are then the alternating row sums of triangle A127677.

Also abs(a(n)) is the number of rounded area of pentagon or pentagram in series arrangement. - Kival Ngaokrajang, Mar 27 2013

Colin Barker, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 8.
Kival Ngaokrajang, Pentagram for n = 1..6
Eric Weisstein's World of Mathematics, Pentagram
Index entries for linear recurrences with constant coefficients, signature (-3,-1).

Cf. A099837 (row sums of A110162).

Cf. A000032, A000045, A001254, A005248, A075150, A127677.

A219233:= func< n | n eq 0 select 1 else (-1)^n*Lucas(2*n) >; // G. C. Greubel, Jun 13 2025

A219233[n_]:= (-1)^n*LucasL[2*n] - Boole[n==0]; (* G. C. Greubel, Jun 13 2025 *)

Vec((1-x^2)/(1+3*x+x^2) + O(x^40)) \\ Colin Barker, Oct 14 2015

def A219233(n): return (-1)**n*lucas_number2(2*n,1,-1) - int(n==0) # G. C. Greubel, Jun 13 2025

a(0) = 1 and a(n) = (-1)^n*(F(2*(n+1)) - F(2*(n-1))) = (-1)^n*L(2*n), n>=1, with F=A000045 (Fibonacci) and L=A000032 (Lucas).
O.g.f.: (1-x^2)/(1+3*x+x^2).
G.f.: (W(0) -6)/(5*x) -1 , where W(k) = 5*x*k + x + 6 - 6*x*(5*k-9)/W(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
From Colin Barker, Oct 14 2015: (Start)
a(n) = -3*a(n-1) - a(n-2) for n>2.
a(n) = (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n for n>0. (End)
E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) - 1. - Stefano Spezia, Dec 26 2021
From G. C. Greubel, Jun 13 2025: (Start)
a(-n) = a(n).
a(n) = (-1)^n*A001254(n) - 2 - [n=0] = A075150(n) - 2 - [n=0]. (End)

A263575 Stirling transform of Lucas numbers (A000032).

Original entry on oeis.org

2, 1, 4, 14, 53, 227, 1092, 5791, 33350, 206511, 1365563, 9590847, 71216713, 556861216, 4569168866, 39222394456, 351304769679, 3275433717440, 31723522878974, 318571978752719, 3311400814816987, 35573458376435132, 394404160256111139, 4507130777468928696

Offset: 0

Vladimir Reshetnikov, Oct 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..563
Eric Weisstein's MathWorld, Lucas Number.
Eric Weisstein's MathWorld, Stirling Transform.
Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A000032, A213593, A005248, A061084, A263576.

Table[Sum[LucasL[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
Table[Simplify[BellB[n, GoldenRatio] + BellB[n, 1 - GoldenRatio]], {n, 0, 23}]

a(n) = Sum_{k=0..n} A000032(k)*Stirling2(n,k).
Let phi = (1+sqrt(5))/2.
a(n) = B_n(phi)+B_n(1-phi), where B_n(x) is n-th Bell polynomial.
2*B_n(phi) = a(n) + A263576*sqrt(5).
E.g.f.: exp((exp(x)-1)*phi)+exp((exp(x)-1)*(1-phi)).
Sum_{k=0..n} a(k)*Stirling1(n,k) = A000032(n).
G.f.: Sum_{j>=0} Lucas(j)*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019

A081069 a(n) = Lucas(4n)+2 = Lucas(2n)^2.

Original entry on oeis.org

4, 9, 49, 324, 2209, 15129, 103684, 710649, 4870849, 33385284, 228826129, 1568397609, 10749957124, 73681302249, 505019158609, 3461452808004, 23725150497409, 162614600673849, 1114577054219524, 7639424778862809

Offset: 0

R. K. Guy, Mar 04 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Pridon Davlianidze, Problem B-1270, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 2 (2020), p. 179; Four Telescopic Infinite Products, Solution to Problem B-1270 by Jason L. Smith, ibid., Vol. 59, No. 2 (2021), pp. 183-184.
Emrah Kılıç, Yücel Türker Ulutaş, and Neşe Ömür, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, Table 2, k=2.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000032 (Lucas numbers), A001622, A005248, A056854.

[ Lucas(2*n)^2: n in [0..70] ]; // Vincenzo Librandi, Apr 16 2011

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d,`,luc(4*n)+2) od: # James Sellers, Mar 05 2003
G:=(x,n)-> cos(x)^n +cos(3*x)^n: seq(simplify(2^(4*n)*G(Pi/5,2*n)^2), n=0..19) # Gary Detlefs, Dec 05 2010
t:= n-> sum(fibonacci(4*k+2),k=0..n):seq(5*t(n)+4,n=-1..18); # Gary Detlefs, Dec 06 2010

LucasL[4*Range[0,20]]+2 (* Harvey P. Dale, Sep 09 2012 *)

a(n) = A005248(n)^2 = A056854(n)+2.
a(n) = 8a(n-1) - 8a(n-2) + a(n-3).
a(n) = 2^(4*n)*(cos(Pi/5)^(2*n)+cos(3*Pi/5)^(2*n))^2. - Gary Detlefs, Dec 05 2010
From Gary Detlefs, Dec 06 2010: (Start)
a(n) = 7*a(n-1)-a(n-2)-10, n>1.
a(n) = 5*Sum_{k=0..n}(Fibonacci(4*k+2))+4, with offset -1. (End)
G.f.: -(9*x^2-23*x+4)/((x-1)*(x^2-7*x+1)). - Colin Barker, Jun 24 2012
Product_{n>=0} (1 + 5/a(n)) = 3*phi^2/2, where phi is the golden ratio (A001622) (Davlianidze, 2020). - Amiram Eldar, Dec 04 2024
a(n) = Sum_{k>=0} Lucas(2*n*k)/(Lucas(2*n)^k). - Diego Rattaggi, Jan 12 2025
E.g.f.: 2*(cosh(x) + exp(7*x/2)*cosh(3*sqrt(5)*x/2) + sinh(x)). - Stefano Spezia, Jan 20 2025

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A153175 a(n) = L(7*n)/L(n) where L(n) = Lucas number A000204(n).

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Magma

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A153177 a(n) = L(9*n)/L(n) where L(n) = Lucas number A000204(n).

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Magma

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A153179 a(n) = L(11*n)/L(n) where L(n) = A000204(n).

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Magma

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A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).

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Maple

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A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r) / sum_{s=0,...,q} ((-1)^s*binomial(2*q-s,s)*x^s), where q=1,2,....

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A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2*n)*(Sum_{j=1..n+1} (cos(j*Pi/(2*q+1)))^(2*n)), n >= 0, q >= 1.

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A198635 Total number of round trips, each of length 2*n on the graph P_5 (o-o-o-o-o).

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Mathematica

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A219233 Alternating row sums of Riordan triangle A110162.

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A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.