A003948 -id:A003948 - cp's OEIS frontend

A003945 Expansion of g.f. (1+x)/(1-2*x).

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 3.
Number of Hamiltonian cycles in K_3 X P_n.
Number of ternary words of length n avoiding aa, bb, cc.
For n > 0, row sums of A029635. - Paul Barry, Jan 30 2005
Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462. - Philippe Deléham, Jul 23 2005
Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009
Equals (n+1)-th row sums of triangle A161175. - Gary W. Adamson, Jun 05 2009
a(n) written in base 2: a(0) = 1, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
INVERTi transform of A003688. - Gary W. Adamson, Aug 05 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 42, 138, 162 and 168, lead to this sequence. For the corner squares these vectors lead to the companion sequence A083329. - Johannes W. Meijer, Aug 15 2010
A216022(a(n)) != 2 and A216059(a(n)) != 3. - Reinhard Zumkeller, Sep 01 2012
Number of length-n strings of 3 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012
Sums of pairs of rows of Pascal's triangle A007318, T(2n,k)+T(2n+1,k); Sum_{n>=1} A000290(n)/a(n) = 4. - John Molokach, Sep 26 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
F. Faase, Counting Hamiltonian cycles in product graphs
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304
Markus Kuba and Alois Panholzer, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012
C. Richard and U. Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to trees

Essentially same as A007283 (3*2^n) and A042950.
Cf. A001787, A001045, A161175.
Generating functions of the form (1+x)/(1-k*x) for k=1 to 12: A040000, A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952.
Generating functions of the form (1+x)/(1-k*x) for k=13 to 30: A170732, A170733, A170734, A170735, A170736, A170737, A170738, A170739, A170740, A170741, A170742, A170743, A170744, A170745, A170746, A170747, A170748.
Generating functions of the form (1+x)/(1-k*x) for k=31 to 50: A170749, A170750, A170751, A170752, A170753, A170754, A170755, A170756, A170757, A170758, A170759, A170760, A170761, A170762, A170763, A170764, A170765, A170766, A170767, A170768, A170769.
Cf. A003688.

k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;

Join[{1}, 3*2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[2^n+Floor[2^(n-1)], {n,0,30}] (* Martin Grymel, Oct 17 2012 *)
CoefficientList[Series[(1+x)/(1-2x),{x,0,40}],x] (* or *) LinearRecurrence[ {2},{1,3},40] (* Harvey P. Dale, May 04 2017 *)

a(n)=if(n,3<Charles R Greathouse IV, Jan 12 2012

a(0) = 1; for n > 0, a(n) = 3*2^(n-1).
a(n) = 2*a(n-1), n > 1; a(0)=1, a(1)=3.
More generally, the g.f. (1+x)/(1-k*x) produces the sequence [1, 1 + k, (1 + k)*k, (1 + k)*k^2, ..., (1+k)*k^(n-1), ...], with a(0) = 1, a(n) = (1+k)*k^(n-1) for n >= 1. Also a(n+1) = k*a(n) for n >= 1. - Zak Seidov and N. J. A. Sloane, Dec 05 2009
The g.f. (1+x)/(1-k*x) produces the sequence with closed form (in PARI notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver Lafont, Dec 05 2009
Binomial transform of A000034. a(n) = (3*2^n - 0^n)/2. - Paul Barry, Apr 29 2003
a(n) = Sum_{k=0..n} (n+k)*binomial(n, k)/n. - Paul Barry, Jan 30 2005
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 1. - Philippe Deléham, Jul 10 2005
Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry, Aug 29 2006
a(0) = 1, a(n) = 2 + Sum_{k=0..n-1} a(k) for n >= 1. - Joerg Arndt, Aug 15 2012
a(n) = 2^n + floor(2^(n-1)). - Martin Grymel, Oct 17 2012
E.g.f.: (3*exp(2*x) - 1)/2. - Stefano Spezia, Jan 31 2023

Edited by N. J. A. Sloane, Dec 04 2009

A003946 Expansion of (1+x)/(1-3*x).

Original entry on oeis.org

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 4.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001
a(n) is the number of nonreversing random walks of the length of n edges on a two-dimensional square lattice, all beginning at a fixed point P. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005
Binomial transform of {1, 3, 5, 11, 21, 43, ...}, see A001045. Binomial transform is {1, 5, 21, 85, 341, 1365, ...}, see A002450. - Philippe Deléham, Jul 22 2005
For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007
Equals row sums of triangle A143865. - Gary W. Adamson, Sep 04 2008
Equals INVERT transform of the odd integers = 1/(1 - x - 3x^2 - 5x^3 - ...). - Gary W. Adamson, Jul 27 2009
a(n) is the number of generalized compositions of n+1 when there are 2 *i-1 different types of the part i, (i=1,2,...). - Milan Janjic, Aug 26 2010
Number of length-n strings of 4 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012
The sequence is the INVERTi transform of A015448: (1, 5, 21, 89, 377, ...). - Gary W. Adamson, Aug 06 2016
Let D(m) = {d(m,i)}, i = 1..q, denote the set of the q divisors of a number m, and consider s1(m) and s2(m) the sums of the divisors that are congruent to 1 and 2 (mod 3) respectively. For n > 0, the sequence a(n) lists the numbers m such that s1(m) = 5 and s2(m) = 2. - Michel Lagneau, Feb 09 2017
a(n) is the number of quaternary sequences of length n such that no two consecutive terms have distance 2. - David Nacin, May 31 2017
Also the number of maximal cliques in the n-Sierpinski gasket graph. - Eric W. Weisstein, Dec 01 2017
Number of 3-permutations of n elements avoiding the patterns 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

			G.f. = 1 + 4*x + 12*x^2 + 36*x^3 + 108*x^4 + 324*x^5 + 972*x^6 + 2916*x^7 + ...

T. D. Noe, Table of n, a(n) for n = 0..200
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
John Elias, Illustration: Sierpinski Hexagrams
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 305
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 3.
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (3).
Index entries for sequences related to trees

Cf. A029653, A143865, column 4 in A265583, A015448.

[1] cat [4*3^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012

Maple
```
if n = 0 then 1 else 4*3^(n-1); fi;
```

Join[{1}, 4 3^Range[0, 30]] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
Join[{1}, NestList[3# &, 4, 30]] (* Harvey P. Dale, Nov 30 2011 *)
CoefficientList[Series[(1 + x)/(1 - 3 x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2012 *)
Join[{1}, LinearRecurrence[{3}, {4}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)

A003946[n]:=if n<1 then 1 else 4*3^(n-1)$
makelist(A003946[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

{a(n) = if( n<1, n==0, 4 * 3^(n-1))}; /* Michael Somos, Jun 18 2002 */

Vec((1+x)/(1-3*x) + O(x^100)) \\ Altug Alkan, Dec 07 2015

a(n) = floor(4*3^(n-1)). - Michael Somos, Jun 18 2002
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 2. - Philippe Deléham, Jul 10 2005
The Hankel transform of this sequence is [1,-4,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n + 1) = (((1 + sqrt(-11))/2)^n + ((1 - sqrt(-11))/2)^n)^2 - (((1 + sqrt(-11))/2)^n - ((1 - sqrt(-11))/2)^n)^2. - Raphie Frank, Dec 07 2015
From Mario C. Enriquez, Apr 01 2017: (Start)
(L(a(n+k)) - 1)/a(n) reduces to the form C/a(n-1), where n > 1, k >= 0, L(a(n)) is the a(n)-th Lucas number and C = (L(a(n+k)) - 1)/3.
(L(a(n+k)) - 1)/3 mod (L(a(n)) - 1)/3 = (L(a(n)) - 1)/3 - 1, where n >= 1, k >= 0 and L(a(n)) is the a(n)-th Lucas number. (End)
E.g.f.: (4*exp(3*x) - 1)/3. - Stefano Spezia, Jan 31 2025

Additional comments from Michael Somos, Jun 18 2002

Edited by N. J. A. Sloane, Dec 04 2009

A003947 Expansion of (1+x)/(1-4*x).

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080, 351843720888320

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 5.
For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 10 2007
Number of length-n strings of 5 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012
Create a rectangular prism with edges of lengths 2^(n-2), 2^(n-1), and 2^(n) starting at n=2; then the surface area = a(n). - J. M. Bergot, Aug 08 2013

T. D. Noe, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 306
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).
Index entries for sequences related to trees

Cf. A003948, A003949. Column 5 in A265583.

Concatenation([1], List([1..30], n-> 5*4^(n-1) )); # G. C. Greubel, Aug 10 2019

[1] cat [5*4^(n-1): n in [1..30]]; // G. C. Greubel, Aug 10 2019

k := 5; if n = 0 then 1 else k*(k-1)^(n-1); fi;

q = 5; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 5*4^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
LinearRecurrence[{4},{1,5},30] (* Harvey P. Dale, Apr 19 2015 *)

a(n)=5*4^n\4 \\ Charles R Greathouse IV, Sep 08 2011

[1]+[5*4^(n-1) for n in (1..30)] # G. C. Greubel, Aug 10 2019

Binomial transform of A060925. Its binomial transform is A003463 (without leading zero). - Paul Barry, May 19 2003
From Paul Barry, May 19 2003: (Start)
a(n) = (5*4^n - 0^n)/4.
G.f.: (1+x)/(1-4*x).
E.g.f.: (5*exp(4*x) - exp(0))/4. (End)
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 3. - Philippe Deléham, Jul 10 2005
a(n) = A146523(n)*A011782(n). - R. J. Mathar, Jul 08 2009
a(n) = 5*A000302(n-1), n>0.
a(n) = 4*a(n-1), n>1. - Vincenzo Librandi, Dec 31 2010
G.f.: 2+x- 2/G(0), where G(k)= 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013

Edited by N. J. A. Sloane, Dec 04 2009

A003464 a(n) = (6^n - 1)/5.

Original entry on oeis.org

0, 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, 72559411, 435356467, 2612138803, 15672832819, 94036996915, 564221981491, 3385331888947, 20311991333683, 121871948002099, 731231688012595, 4387390128075571

Offset: 0

N. J. A. Sloane

a(n) = A125118(n, 5) for n>4. - Reinhard Zumkeller, Nov 21 2006
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=7, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>1, a(n-1)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010
Repunits to base 6. A repunit consisting of zero 1's (empty string) gives the empty sum, i.e., 0 (only case where leading zero is shown, for convenience). - Daniel Forgues, Jul 08 2011
3*a(n) is the total number of holes in a certain triangle fractal (start with 6 triangles, 3 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015

			a(n) in base 6.................... a(n) in base 10:
0..................................0
1..................................1
11.................................7
111................................43
1111...............................259
11111..............................1555
111111.............................9331
1111111............................55987, etc. - _Philippe Deléham_, Mar 12 2014

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..1000
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 375
Kival Ngaokrajang, Illustration of initial terms
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.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
Eric Weisstein's World of Mathematics, Repunit.
Index entries for linear recurrences with constant coefficients, signature (7,-6).

[n le 2 select n-1 else 7*Self(n-1) - 6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012

a:=n->sum(6^(n-j),j=1..n): seq(a(n), n=1..21); # Zerinvary Lajos, Jan 04 2007
A003464:=1/(6*z-1)/(z-1); # conjectured by Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+6*a[n-2]+2 od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008

(6^Range[20]-1)/5 (* Harvey P. Dale, Dec 14 2010 *)
LinearRecurrence[{7, -6}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A003464(n):=floor((6^n-1)/5)$  makelist(A003464(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
for(n=1,10,print1((6^n-1)/5,","));
```

[lucas_number1(n,7,6) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

[gaussian_binomial(n,1,6) for n in range(1,22)] # Zerinvary Lajos, May 28 2009

Binomial transform of A003948. If preceded by 0, then binomial transform of powers of 5, A000351 (preceded by 0). - Paul Barry, Mar 28 2003
a(n) = Sum_{k=1..n} C(n, k)*5^(k-1).
E.g.f.: (exp(6*x) - exp(x))/5. - Paul Barry, Mar 28 2003
G.f.: x/((1-x)*(1-6*x)). - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 06 2005
a(n) = 6*a(n-1) + 1 with a(1)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = 7*a(n-1) - 6*a(n-2). - Vincenzo Librandi, Nov 08 2012

More terms from Reinhard Zumkeller, Nov 21 2006

G.f. corrected by Philippe Deléham, Mar 11 2014

A003950 Expansion of g.f.: (1+x)/(1-7*x).

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582792, 37980492079544, 265863444556808, 1861044111897656, 13027308783283592, 91191161482985144, 638338130380896008

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 8.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001.
For n>=1, a(n) equals the number of words of length n on the alphabet {0,1,...,7} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]
a(n) is the number of octonary sequences of length n such that no two consecutive terms have distance 4. - David Nacin, May 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 309
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (7).
Index entries for sequences related to trees

Cf. A003948, A003949, A003951.

k:=8;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

[1] cat [8*7^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012

k:=8; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 8*7^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-7*x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

a(n)=if(n,8*7^(n-1),1) \\ Charles R Greathouse IV, Mar 22 2016

k=8; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 6. - Philippe Deléham, Jul 10 2005
From Philippe Deléham, Nov 21 2007: (Start)
a(n) = 8*7^(n-1) for n>=1, a(0)=1 .
G.f.: (1+x)/(1-7x).
The Hankel transform of this sequence is [1,-8,0,0,0,0,0,0,0,0,...]. (End)
a(0)=1, a(1)=8, a(n) = 7*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (8*exp(7*x) - 1)/7. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

A003949 Expansion of g.f. (1+x)/(1-6*x).

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148672, 3291294892032, 19747769352192, 118486616113152, 710919696678912, 4265518180073472, 25593109080440832, 153558654482644992

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 7.
For n >= 1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6,7} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6,7} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 10 2007
For n >= 1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5,6} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 308
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (6).
Index entries for sequences related to trees

Cf. A003947, A003948, A003950, A003951.

k:=7;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=7; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1+x)/(1-6*x))); // Marius A. Burtea, Jan 20 2020

k:=7; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

q = 7; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* or *) Join[{1}, 7*6^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-6*x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)
LinearRecurrence[{6},{1,7},30] (* or *) Join[{1},NestList[6#&,7,30]] (* Harvey P. Dale, May 03 2025 *)

a(n)=if(n,7*6^(n-1),1) \\ Charles R Greathouse IV, Mar 22 2016

k=7; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

G.f.: (1+x)/(1-6*x).
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 5. - Philippe Deléham, Jul 10 2005
a(0)=1; for n > 0, a(n) = 7*6^(n-1). - Vincenzo Librandi, Nov 18 2010
a(0)=1, a(1)=7, a(n) = 6*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (7*exp(6*x) - 1)/6. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

A003952 Expansion of g.f.: (1+x)/(1-9*x).

Original entry on oeis.org

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596090, 2824295364810, 25418658283290, 228767924549610, 2058911320946490, 18530201888518410, 166771816996665690, 1500946352969991210, 13508517176729920890

Offset: 0

N. J. A. Sloane, Mar 15 1996

Coordination sequence for infinite tree with valency 10.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001
Except 1, all terms are in A033583. - Vincenzo Librandi, May 26 2014
For n>=1, a(n) equals the number of words of length n on alphabet {0,1,...,9} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]
a(n) is the number of sequences over the alphabet {0,1,...,9} of length n such that no two consecutive terms have distance 5. - David Nacin, May 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 311
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (9).
Index entries for sequences related to trees

Cf. A003949, A003950, A003951.

k:=10;; Concatenation([1], List([1..20], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

[(10*9^n-0^n)/9: n in [0..20] ]; // Vincenzo Librandi, Aug 19 2011

k:= 10; seq(`if`(n = 0, 1, k*(k-1)^(n-1)), n = 0..20); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 10*9^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[9#&,10,20]] (* Harvey P. Dale, Sep 01 2021 *)

a(n)=10*9^n\9 \\ Charles R Greathouse IV, Sep 08 2011

k=10; [1]+[k*(k-1)^(n-1) for n in (1..20)] # G. C. Greubel, Sep 24 2019

a(n) = (10*9^n - 0^n)/9. Binomial transform is A000042. - Paul Barry, Jan 29 2004
G.f.: (1+x)/(1-9*x). - Philippe Deléham, Jan 31 2004
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 8. - Philippe Deléham, Jul 10 2005
The Hankel transform of this sequence is: [1,-10,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
E.g.f.: (10*exp(9*x) - 1)/9. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

A003954 Expansion of g.f.: (1+x)/(1-11*x).

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520652, 414272545727172, 4556998002998892, 50126978032987812, 551396758362865932, 6065364341991525252, 66719007761906777772, 733909085380974555492

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 12.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001.
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,11} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..900
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 313.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (11).
Index entries for sequences related to trees.

Cf. A003946, A003948, A003950, A003952, A003953, A003954, A029653.

Concatenation([1], List([1..20], n-> 12*11^(n-1) )); # G. C. Greubel, Sep 23 2019

[1] cat [12*11^(n-1): n in [1..20]]; // Vincenzo Librandi, Dec 11 2012

k:=12; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 12*11^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-11x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

a(n)=12*11^n\11 \\ Charles R Greathouse IV, Aug 14 2015

[1]+[12*11^(n-1) for n in (1..20)] # G. C. Greubel, Sep 23 2019

a(n) = Sum_{k=0..n} A029653(n,k)*x^k for x = 10. - Philippe Deléham, Jul 10 2005
G.f.: (1+x)/(1-11*x). The Hankel transform of this sequence is [1,-12,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(0) = 1; for n>0, a(n) = 12*11^(n-1). - Vincenzo Librandi, Nov 18 2010
a(0) = 1, a(1)=12, a(n) = 11*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (12*exp(11*x) - 1)/11. - Elmo R. Oliveira, Mar 24 2025

Edited by N. J. A. Sloane, Dec 04 2009

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.

A108981 a(n) = 3a(n-1) + 4a(n-2), a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 19, 77, 307, 1229, 4915, 19661, 78643, 314573, 1258291, 5033165, 20132659, 80530637, 322122547, 1288490189, 5153960755, 20615843021, 82463372083, 329853488333, 1319413953331, 5277655813325, 21110623253299

Offset: 0

Philippe Deléham, Jul 23 2005

The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Apr 15 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-2, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = charpoly(A,2). - Milan Janjic, Jan 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A003948, A037481.

I:=[1,5]; [n le 2 select I[n] else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 17 2014

LinearRecurrence[{3,4},{1,5},30] (* Harvey P. Dale, Feb 16 2014 *)

Vec((1+2*x)/(1+x)/(1-4*x)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

def A108981(n): return ((4<<(m:=n+1<<1))|2)//5-((1<>1 # Chai Wah Wu, Apr 22 2025

Inverse binomial transform of A003948.
a(2n) = 4a(2n-1) - 1; a(2n+1) = 4a(2n) + 1.
O.g.f.: (1+2*x)/((1+x)(1-4*x)). - R. J. Mathar, Apr 02 2008
Sum_{k=0..n} a(k) = A037481(n+1). - Philippe Deléham, Apr 15 2008

Corrected by T. D. Noe, Nov 07 2006

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 14 2008

cp's OEIS Frontend

A003945 Expansion of g.f. (1+x)/(1-2*x).

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A003946 Expansion of (1+x)/(1-3*x).

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A003947 Expansion of (1+x)/(1-4*x).

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A003464 a(n) = (6^n - 1)/5.

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A003950 Expansion of g.f.: (1+x)/(1-7*x).

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