A003463 -id:A003463 - cp's OEIS frontend

A016218 Expansion of 1/((1-x)(1-4x)(1-5x)).

1, 10, 71, 440, 2541, 14070, 75811, 400900, 2091881, 10808930, 55442751, 282806160, 1436400421, 7271480590, 36715316891, 185008240220, 930767824161, 4676745613050, 23475354034231, 117743274047080, 590182385739101, 2956775990710310, 14807336201610771

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-29,20).

Cf. A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256.

Vec(1/((1-x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

From Vincenzo Librandi, Feb 10 2011: (Start)
a(n) = a(n-1) + 5^(n+1) - 4^(n+1), n >= 1.
a(n) = 9*a(n-1) - 20*a(n-2) + 1, n >= 2. (End)
a(n) = 1/12 - 4^(n+2)/3 + 5^(n+2)/4. - R. J. Mathar, Mar 15 2011

A016256 Expansion of 1/((1-x)(1-8x)(1-9x)).

Original entry on oeis.org

1, 18, 235, 2700, 28981, 298278, 2984095, 29253600, 282456361, 2695498938, 25486623955, 239196683700, 2231306698141, 20710052641998, 191416812647815, 1762962024789000, 16188343910770321, 148268580698287458, 1355005110295423675, 12359749064745505500

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (18,-89,72)

Cf. A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125.

a:=n->sum(9^(n-j)-8^(n-j),j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 04 2007

Table[(-8^(n + 2) + 7*9^(n + 1) + 1)/56, {n, 40}] (* and *) CoefficientList[Series[1/((1 - z) (1 - 8*z) (1 - 9*z)), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)

Vec(1/((1-x)*(1-8*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

G.f.: 1/((1-x)*(1-8*x)*(1-9*x)).
a(n) = 17*a(n-1) - 72*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
a(n) = 9^(n+2)/8 - 8^(n+2)/7 + 1/56. - R. J. Mathar, Mar 14 2011
a(n) = 18*a(n-1) - 89*a(n-2) + 72*a(n-3). - Wesley Ivan Hurt, Apr 20 2023

A125831 a(n) = (5^n - 1)/2.

Original entry on oeis.org

0, 2, 12, 62, 312, 1562, 7812, 39062, 195312, 976562, 4882812, 24414062, 122070312, 610351562, 3051757812, 15258789062, 76293945312, 381469726562, 1907348632812, 9536743164062, 47683715820312, 238418579101562, 1192092895507812, 5960464477539062, 29802322387695312

Offset: 0

Zerinvary Lajos, Feb 03 2007

Number of compositions of odd numbers into n parts < 5. - Adi Dani, Jun 11 2011

Numbers whose base 5 representation is 22222...2 (n times).

			a(2)=12: there are 12 compositions of odd numbers into 2 parts < 5:
1: (0,1),(1,0);
3: (0,3),(3,0),(1,2),(2,1);
5: (1,4),(4,1),(2,3),(3,2);
7: (3,4),(4,3). - _Adi Dani_, Jun 11 2011

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), pp. 55-70, eqs. (6) and (7) on p. 58.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Adi Dani, Restricted compositions of natural numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A003463, A024049, A121177 (same with different offset).

List([0..30], n-> (5^n-1)/2); # G. C. Greubel, Aug 03 2019

[(5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 11 2011

Maple
```
seq((5^n-1)/2, n=0..30);
```

Table[(5^n -1)/2, {n, 0, 30}] (* Harvey P. Dale, Dec 03 2010 *)

a(n)=5^n\2 \\ Charles R Greathouse IV, Jun 11 2011

[(5^n-1)/2 for n in (0..30)] # G. C. Greubel, Aug 03 2019

a(n) = 5*a(n-1) + 2 for n > 0, a(0)=0. - Vincenzo Librandi, Sep 30 2010
From Colin Barker, May 16 2013: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: 2*x/((1-x)*(1-5*x)). (End)
a(n) = 2*A003463(n). - Joerg Arndt, Aug 03 2019
From Elmo R. Oliveira, Dec 10 2023: (Start)
a(n) = A024049(n)/2.
E.g.f.: (1/2)*(exp(5*x) - exp(x)). (End)

Offset corrected by N. J. A. Sloane, Oct 02 2010

Major edit by Joerg Arndt, Jun 11 2011

A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-7x^2).

Original entry on oeis.org

1, 6, 37, 227, 1394, 8559, 52553, 322678, 1981261, 12165051, 74694082, 458625767, 2815987409, 17290317414, 106163498933, 651849716563, 4002393075346, 24574913392671, 150891318490777, 926480986202582, 5688644160448349

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner or side square (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white chess queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.
On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the central square (we assume here that a red queen might behave like a white queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner and side squares the 512 red queens lead to 17 red queen sequences, see the cross-references for the complete set.
The sequence above corresponds to 8 red queen vectors, i.e., A[5] vectors, with decimal values 239, 367, 431, 463, 487, 491, 493 and 494. The central square leads for these vectors to A152240.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 5*x - k*x^2). The members of this family that are red queen sequences are A180030 (k=8), A180032 (k=7; this sequence), A000400 (k=6), A180033 (k=5), A126501 (k=4), A180035 (k=3), A180037 (k=2) A015449 (k=1) and A003948 (k=0). Other members of this family are A030221 (k=-1), A109114 (k=-3), A020989 (k=-4), A166060 (k=-6).
Inverse binomial transform of A054413.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wikipedia, Alice in Wonderland (2010 film)
Index entries for linear recurrences with constant coefficients, signature (5, 7).

Cf. A180028 (Central square).

Cf. Red queen sequences corner and side squares [decimal value A[5]]: A090018 [511], A135030 [255], A180030 [495], A005668 [127], A180032 [239], A000400 [63], A180033 [47], A001109 [31], A126501 [15], A154244 [23], A180035 [7], A138395 [19], A180037 [3], A084326 [17], A015449 [1], A003463 [16], A003948 [0].

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [1,1,1,1,0,1,1,1,0]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): 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);

LinearRecurrence[{5,7},{1,6},40] (* Vincenzo Librandi, Nov 15 2011 *)
CoefficientList[Series[(1+x)/(1-5x-7x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 04 2024 *)

G.f.: (1+x)/(1 - 5*x - 7*x^2).
a(n) = 5*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+9*A)*A^(-n-1) + (7+9*B)*B^(-n-1))/53 with A = (-5+sqrt(53))/14 and B = (-5-sqrt(53))/14.

A086122 Primes of the form (5^k-1)/4.

Original entry on oeis.org

31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781

Offset: 1

Labos Elemer, Jul 23 2003

nonn

Corresponding exponents k are listed in A004061. - Alexander Adamchuk, Jan 23 2007

Cf. A000668, A076481.

Cf. A003463, A004061, A074479.

Do[f=(5^n-1)/4;If[PrimeQ[f],Print[{n,f}]],{n,1,1000}] (* Alexander Adamchuk, Jan 23 2007 *)
Select[(5^Range[300]-1)/4,PrimeQ] (* Harvey P. Dale, Dec 11 2016 *)

a(n) = (5^A004061(n) - 1)/4 = A003463[ A004061(n) ]. - Alexander Adamchuk, Jan 23 2007

A003464 INTERSECT A000040.

More terms from Alexander Adamchuk, Jan 23 2007

A218750 a(n) = (47^n - 1)/46.

Original entry on oeis.org

0, 1, 48, 2257, 106080, 4985761, 234330768, 11013546097, 517636666560, 24328923328321, 1143459396431088, 53742591632261137, 2525901806716273440, 118717384915664851681, 5579717091036248029008, 262246703278703657363377, 12325595054099071896078720

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 47 (A009991).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums
Index entries related to q-numbers
Index entries for linear recurrences with constant coefficients, signature (48,-47)

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009991.

[n le 2 select n-1 else 48*Self(n-1) - 47*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

Table[(47^n - 1)/46, {n, 0, 19}] (* Alonso del Arte, Nov 04 2012 *)
LinearRecurrence[{48, -47}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218750(n):=(47^n-1)/46$ makelist(A218750(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218750(n)=47^n\46
```

a(n) = floor(47^n/46).
G.f.: x/(47*x^2-48*x+1) = x/((1-x)*(1-47*x)). [Colin Barker, Nov 06 2012]
a(0)=0, a(n) = 47*a(n-1) + 1. - Vincenzo Librandi, Nov 08 2012
a(n) = 48*a(n-1) - 47*a(n-2). - Wesley Ivan Hurt, Jan 25 2022
E.g.f.: exp(24*x)*sinh(23*x)/23. - Elmo R. Oliveira, Aug 27 2024

A309062 Oblong numbers that are repdigits with length > 2 in more than two bases.

Original entry on oeis.org

61035156, 641431602, 38146972656, 70607384120, 953674316406, 5824521280620, 23841857910156, 51472783023662, 145655559307440, 463255047212960, 1838956877846660, 14901161193847656, 37523658824249780, 88453695801367260, 166354152295794960, 416972378738246240

Offset: 1

Michel Marcus, Jul 10 2019

All initial terms come from the b-file in A290869.
For the given terms, the number of bases are respectively 4, 3, 3, 4, 4, 4, 4, 3, 4, 3 and 4.
A003463(64), A003463(24) (confirmed) and A003463(36) are candidates for 5, 6 and 7 bases representations.
From Bernard Schott, Jul 24 2019: (Start)
The terms of this sequence are necessarily of the form (b^(2*q) - 1)/4 with q > 2 and b = 4*m+1 with m > 0, but when b = c^2 is an odd square (A016754), then some terms can also have the form (b^(2*q+1) - 1)/4 as a(8) and a(23). If these terms have representations in u bases, the values of (b, 2*q or 2*q+1, u) for the first eleven terms are respectively (5, 12, 4), (37, 6, 3), (5, 16, 3), (9, 12, 4), (5, 18, 4), (13, 12, 4), (5, 20, 4), (9, 15, 3), (17, 12, 4), (9, 16, 3) and (21, 12, 4).
For any b = 4*m+1 with m > 0 and r > 2, (b^(4*r) - 1)/4 is an oblong repdigit with length > 2 in at least bases b, b^2 and b^4; hence this sequence is infinite.
(End)
From Chai Wah Wu, Jul 24 2019: (Start)
Other values of (b, q, u) for which (b^(2*q) - 1)/4 is a term with representations in u bases:
(5, 12, 6), (5, 14, 4), (5, 15, 6), (9, 9, 4), (9, 10, 4), (13, 8, 3), (13, 9, 4), (17, 8, 3), (29, 6, 4), (33, 6, 4), (37, 6, 4), (41, 6, 4), (45, 6, 4).
(End)
From Bernard Schott, Jul 24 2019: (Start)
Theorem: if tau(2*q) = r > 4, (b^(2*q) - 1)/4 is a term that has exactly r-2 representations as repdigits with length > 2 in bases that are powers of b.
There exist cases where a term also has representation in another base that is not power of b. For instance a(2), see example, where base 3446 is not a perfect power of 37.
Conclusion: if m = (b^(2*q) - 1)/4 is a term and if beta"(m) is the number of representations of this term as repdigits with length > 2, then, beta"(m) >= tau(2*q) - 2. (End)

			From _Bernard Schott_, Jul 18 2019: (Start)
a(1) = 61035156 = 7812*7813 = 111111111111_5 = 666666_25 = (31,31,31)_125 = (156,156,156)_625.
a(2) = 641431602 = 25326*25327 = 999999_37 = (342,342,342)_1469 = (54,54,54)_3446.
(End)
a(11) = 1838956877846660 = 42883060*42883061 = 555555555555_21 = (110, 110, 110, 110, 110, 110)_441 = (2315, 2315, 2315, 2315)_9261 = (48620, 48620, 48620)_194481. - _Chai Wah Wu_, Jul 24 2019

Giovanni Resta, Table of n, a(n) for n = 1..26

Intersection of A002378 and A290869.

Cf. A326378 (similar in no base), A326384 (similar in one base), A326385 (similar in 2 bases).

isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
okrepu3(b, target, lim) = {my(k = 3, nb = 0, x); while ((x=(b^k-1)/(b-1)) <= target, if (x==target, nb++); k++); nb;}
dge3(n) = {my(d=divisors(n), nb=0, ndi, limi); for (i=1, #d, ndi = n/d[i]; limi = sqrtint(ndi); for (k=d[i]+1, limi, nb += okrepu3(k, ndi, limi););); nb;}
isok(n) = isoblong(n) && (dge3(n) >= 3);

a(11) from Chai Wah Wu, Jul 21 2019

a(12)-a(16) from Giovanni Resta, Jul 28 2019

A060458 Maximum value seen in the final n decimal digits of 2^j for all values of j.

Original entry on oeis.org

8, 96, 992, 9984, 99968, 999936, 9999872, 99999744, 999999488, 9999998976, 99999997952, 999999995904, 9999999991808, 99999999983616, 999999999967232, 9999999999934464, 99999999999868928, 999999999999737856, 9999999999999475712, 99999999999998951424, 999999999999997902848

Offset: 1

Labos Elemer, Apr 09 2001

Consider the final n decimal digits of 2^j for all values of j. They are periodic. Sequence gives maximal value seen in these n digits.

With f(n) = a(n+1) - a(n), the difference f(n) - a(n) is always 8*10^n meaning that a(n) becomes its own "first differences" sequence when each term is prefixed a digit '8'. For higher order differences, the prefix 8 becomes: 8*10^n*Sum_{k=0..m-1} 9^k where m is the order. - R. J. Cano, May 11 2014

			Maximum of the last 4 digits of powers of 2 is 9984=10000-16. It occurs at 2^254. 2^254 = 289480223.....01978282409984 (with 77 digits, last 4 ones are ...9984). The period length of the last-4-digit segment is A005054(4)=500. For n=4 period: amplitude=9984, phase=254.

Index entries for sequences related to final digits of numbers.
Index entries for linear recurrences with constant coefficients, signature (12,-20).

Cf. A000079, A003463, A005054, A060460, A016134.

[10^n-2^n : n in [1..20]]; // Wesley Ivan Hurt, Sep 25 2014

A060458:=n->10^n-2^n: seq(A060458(n), n=1..20); # Wesley Ivan Hurt, Sep 25 2014

RecurrenceTable[{a[n] == 12 a[n - 1] - 20 a[n - 2], a[0] == 0, a[1] == 8}, a[n], {n, 1, 20}]  (* Geoffrey Critzer, Dec 15 2011*)

a(n)=sum(j=0,n-1,2^(3*n-2*j)*binomial(n,j)) \\ R. J. Cano, May 15 2014

A060458(n)=(5^n-1)<M. F. Hasler, Oct 31 2014

[10^n - 2^n for n in range(1,19)] # Zerinvary Lajos, Jun 05 2009

a(n) = 10^n - 2^n = 2^n*(5^n - 1).
From Geoffrey Critzer, Dec 15 2011: (Start)
a(n) = 12*a(n-1) - 20*a(n-2).
O.g.f.: 1/(1-10*x) - 1/(1-2*x). (End)
a(n) = f(n,0) where f(x,y) = Sum_{j=0..x+y-1} (2^(3*x-2*j)*binomial(x,j)). - R. J. Cano, May 15 2014
a(n) = 2^(n+2)*A003463(n). - R. J. Cano, Sep 25 2014
a(n) = 8*A016134(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: exp(2*x)*(exp(8*x) - 1). - Elmo R. Oliveira, Mar 26 2025

Edited by M. F. Hasler, Oct 31 2014

More terms from Elmo R. Oliveira, Mar 26 2025

A218726 a(n) = (23^n - 1)/22.

Original entry on oeis.org

0, 1, 24, 553, 12720, 292561, 6728904, 154764793, 3559590240, 81870575521, 1883023236984, 43309534450633, 996119292364560, 22910743724384881, 526947105660852264, 12119783430199602073, 278755018894590847680, 6411365434575589496641, 147461404995238558422744

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 23, q-integers for q=23: diagonal k=1 in triangle A022187.

Partial sums are in A014909. Also, the sequence is related to A014941 by A014941(n) = n*a(n) - Sum{a(i), i=0..n-1} for n > 0. - Bruno Berselli, Nov 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (24,-23).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A014909, A014941, A022187.

[n le 2 select n-1 else 24*Self(n-1)-23*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{24, -23}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(23^Range[0,20]-1)/22 (* Harvey P. Dale, Nov 09 2012 *)

A218726(n):=(23^n-1)/22$
makelist(A218726(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218726(n)=23^n\22
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-23*x)).
a(n) = floor(23^n/22).
a(n) = 24*a(n-1) - 23*a(n-2). (End)
E.g.f.: exp(12*x)*sinh(11*x)/11. - Elmo R. Oliveira, Aug 27 2024

A218732 a(n) = (29^n - 1)/28.

Original entry on oeis.org

0, 1, 30, 871, 25260, 732541, 21243690, 616067011, 17865943320, 518112356281, 15025258332150, 435732491632351, 12636242257338180, 366451025462807221, 10627079738421409410, 308185312414220872891, 8937374060012405313840, 259183847740359754101361

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 29 (A009973).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (30,-29).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009973.

[n le 2 select n-1 else 30*Self(n-1)-29*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{30, -29}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218732(n):=(29^n-1)/28$
makelist(A218732(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=29^n\28
```

a(n) = floor(29^n/28).
G.f.: x/((1-x)*(1-29*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = 30*a(n-1) - 29*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(15*x)*sinh(14*x)/14. - Elmo R. Oliveira, Aug 27 2024

cp's OEIS Frontend

A016218 Expansion of 1/((1-x)*(1-4*x)*(1-5*x)).

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A016256 Expansion of 1/((1-x)*(1-8*x)*(1-9*x)).

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A125831 a(n) = (5^n - 1)/2.

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A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-7*x^2).

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Magma

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A086122 Primes of the form (5^k-1)/4.

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Extensions

A218750 a(n) = (47^n - 1)/46.

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Magma

Mathematica

Maxima

PARI

Formula

A309062 Oblong numbers that are repdigits with length > 2 in more than two bases.

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Examples

A016218 Expansion of 1/((1-x)(1-4x)(1-5x)).

A016256 Expansion of 1/((1-x)(1-8x)(1-9x)).

A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-7x^2).