A015518 -id:A015518 - cp's OEIS frontend

A002532 a(n) = 2a(n-1) + 5a(n-2), a(0) = 0, a(1) = 1.

0, 1, 2, 9, 28, 101, 342, 1189, 4088, 14121, 48682, 167969, 579348, 1998541, 6893822, 23780349, 82029808, 282961361, 976071762, 3366950329, 11614259468, 40063270581, 138197838502, 476712029909, 1644413252328, 5672386654201, 19566839570042, 67495612411089

Offset: 0

N. J. A. Sloane

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 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - Cino Hilliard, Sep 25 2005
For n>=2, number of ordered partitions of n-1 into parts of sizes 1 and 2 where there are two types of 1 (singletons) and five types of 2 (twins). For example, the number of possible configurations of families of n-1 male (M) and female (F) offspring considering only single births and twins, where the birth order of M/F/pair-of-twins is considered and there are five types of twins; namely, both F (identical twins), both F (fraternal twins), both M (identical), both M (fraternal), or one F and one M - where birth order within a pair of twins itself is disregarded. In particular, for a(3)=9, two children could be either: (1) F, then M; (2) M, then F; (3) F,F; (4) M,M; (5) F,F identical twins; (6) F,F fraternal twins; (7) M,M identical twins; (8) M,M fraternal twins; or (9) M,F twins (emphasizing that birth order is irrelevant here when children are the same gender, when two children are within the same pair of twins and when pairs of twins have both the same gender(s) and identical-vs-fraternal characteristics). - Rick L. Shepherd, Sep 19 2004
Pisano period lengths: 1, 2, 3, 4, 4, 6, 24, 8, 3, 4, 120, 12, 56, 24, 12, 16, 288, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012

			G.f. = x + 2*x^2 + 9*x^3 + 28*x^4 + 101*x^5 + 342*x^6 + 1189*x^7 + ...

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12 [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,5).

Cf. A015581 (similar application, but no distinguishing identical vs. fraternal twins).

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

[Floor(((1+Sqrt(6))^n-(1-Sqrt(6))^n)/(2*Sqrt(6))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

[n le 2 select n-1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

A002532:=-z/(-1+2*z+5*z**2); # Conjectured by Simon Plouffe in his 1992 dissertation
# second program
seq(simplify(2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5)), n = 2..25); # Peter Bala, Jul 06 2025

Expand[Table[((1 + Sqrt[6])^n - (1 - Sqrt[6])^n)/(2Sqrt[6]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
a[n_]:=(MatrixPower[{{1,2},{1,-3}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{2,5},{0,1},30] (* Harvey P. Dale, Nov 03 2011 *)

Vec(1/(1-2*x-5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 17 2012

from sage.combinat.sloane_functions import recur_gen2; it = recur_gen2(0,1,2,5); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

[lucas_number1(n,2,-5) for n in range(0, 26)] # Zerinvary Lajos, Apr 22 2009

From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)
a(2*n+1) = 5*a(n)^2 + a(n+1)^2.
6*a(2*n+1) = 5*A002533(n)^2 + A002533(n+1)^2. (End)
From Paul Barry, Sep 20 2003: (Start)
G.f.: x/(1-2*x-5*x^2).
E.g.f.: exp(x)*sinh(sqrt(6)*x)/sqrt(6).
a(n) = ((1+sqrt(6))^n - (1-sqrt(6))^n)/(2*sqrt(6)). (End)
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*6^k. - Paul Barry, Sep 29 2004
G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
From Peter Bala, Jul 06 2025: (Start)
For n >= 0, a(n+1) = (2^n) * Sum_{k = 0..floor(n/2)} binomial(n-k, k) * (5/4)^k.
For n >= 2, a(n) = 2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5).
Sum_{n >= 1} (-5)^n/(a(n)*a(n+1)) = -(sqrt(6) - 1).
Sum_{n >= 1} 5^n/(a(n)*a(n+2)) = 5/4; Sum_{n >= 1} 5^n/(a(n)*a(n+4)) = 755/7056.
G.f. A(x) = x*exp(Sum_{n >= 1} a(2*n)/a(n)*x^n/n) = x + 2*x^2 + 9*x^3 + 28*x^4 + .... (End)

More terms from Rick L. Shepherd, Sep 19 2004

A015531 Linear 2nd order recurrence: a(n) = 4a(n-1) + 5a(n-2).

Original entry on oeis.org

0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021, 15894571940104, 79472859700521, 397364298502604, 1986821492513021, 9934107462565104

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct vertices of the complete graph K_6. Example: a(2)=4 because the walks of length 2 between the vertices A and B of the complete graph ABCDEF are: ACB, ADB, AEB and AFB. - Emeric Deutsch, Apr 01 2004
General form: k=5^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=charpoly(A,1). - Milan Janjic, Jan 27 2010
Pisano period lengths: 1, 2, 6, 2, 2, 6, 6, 4, 18, 2, 10, 6, 4, 6, 6, 8, 16, 18, 18, 2,... - R. J. Mathar, Aug 10 2012
The ratio a(n+1)/a(n) converges to 5 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
For odd n, a(n) is congruent to 1 (mod 10). For even n > 0, a(n) is congruent to 4 (mod 10). - Iain Fox, Dec 30 2017

Iain Fox, Table of n, a(n) for n = 0..1431 (terms 0..1000 from Vincenzo Librandi)
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014.
Index entries for linear recurrences with constant coefficients, signature (4,5).

A083425 shifted right.

Cf. A033115 (partial sums), A213128.

[Round(5^n/6): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(5^n/6), n=0..25); # Mircea Merca, Dec 28 2010

LinearRecurrence[{4,5},{0,1},30] (* Harvey P. Dale, Jul 09 2017 *)

a(n)=5^n\/6 ; \\ Charles R Greathouse IV, Apr 14 2014

first(n) = Vec(x/((1 - 5*x)*(1 + x)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

[lucas_number1(n,4,-5) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009

From Paul Barry, Apr 20 2003: (Start)
a(n) = (5^n -(-1)^n)/6.
G.f.: x/((1-5*x)*(1+x)).
E.g.f.(exp(5*x)-exp(-x))/6. (End) (corrected by M. F. Hasler, Jan 29 2012)
a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*6^(k-1). - Paul Barry, May 13 2003
a(n) = 5^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = ((2+sqrt(9))^n - (2-sqrt(9))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = round(5^n/6). - Mircea Merca, Dec 28 2010
The logarithmic generating function 1/6*log((1+x)/(1-5*x)) = x + 4*x^2/2 + 21*x^3/3 + 104*x^4/4 + ... has compositional inverse 6/(5+exp(-6*x)) - 1, the e.g.f. for a signed version of A213128. - Peter Bala, Jun 24 2012
a(n) = (-1)^(n-1)*Sum_{k=0..(n-1)} A135278(n-1,k)*(-6)^k = (5^n - (-1)^n)/6 = (-1)^(n-1)*Sum_{k=0..(n-1)} (-5)^k. Equals (-1)^(n-1)*Phi(n,-5) when n is an odd prime, where Phi is the cyclotomic polynomial. - Tom Copeland, Apr 14 2014

A083099 a(n) = 2a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 10, 32, 124, 440, 1624, 5888, 21520, 78368, 285856, 1041920, 3798976, 13849472, 50492800, 184082432, 671121664, 2446737920, 8920205824, 32520839168, 118562913280, 432250861568, 1575879202816, 5745263575040

Offset: 0

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

a(n+1) = a(n) + A083098(n+1). A083098(n+1)/a(n) converges to sqrt(7).
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators 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
Pisano period lengths: 1, 1, 2, 1, 12, 2, 7, 1, 6, 12, 60, 2,168, 7, 12, 1,288, 6, 18, 12, ... - R. J. Mathar, Aug 10 2012
a(n) is divisible by 2^ceiling(n/2), see formula below. - Ralf Stephan, Dec 24 2013
Connect the center of a regular hexagon with side length 1 with its six vertices. a(n) is the number of paths of length n from the center to any of its vertices. Number of paths of length n from the center to itself is 6*a(n-1). - Jianing Song, Apr 20 2019

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Project Euler, Problem 752, sequence beta(n).
Index entries for linear recurrences with constant coefficients, signature (2,6).

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

Cf. A154245, A291008.

[n le 2 select n-1 else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

A083099 := proc(n)
    option remember;
    if n <= 1 then
        n;
    else
        2*procname(n-1)+6*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

CoefficientList[Series[x/(1-2x-6x^2), {x, 0, 25}], x] (* Adapted for offset 0 by Vincenzo Librandi, Feb 07 2014 *)
Expand[Table[((1 + Sqrt[7])^n - (1 - Sqrt[7])^n)7/(14Sqrt[7]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{2,6}, {0,1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)

a(n)=([0,1; 6,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x-6*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,2,-6) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

A083099=BinaryRecurrenceSequence(2,6,0,1)
[A083099(n) for n in range(41)] # G. C. Greubel, Jun 01 2023

G.f.: x/(1 - 2*x - 6*x^2).
From Paul Barry, Sep 29 2004: (Start)
E.g.f.: (d/dx)(exp(x)*sinh(sqrt(7)*x)/sqrt(7));
a(n-1) = Sum_{k=0..n} binomial(n, 2k+1)*7^k. (End)
Simplified formula: a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(28). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
G.f.: G(0)*x/(2*(1-x)), 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
a(2n) = 2^n * A154245(n), a(2n+1) = 2^n * (5*A154245(n) - 9*A154245(n-1)). - Ralf Stephan, Dec 24 2013
a(n) = Sum_{k=1,3,5,...<=n} binomial(n,k)*7^((k-1)/2). - Vladimir Shevelev, Feb 06 2014
a(n) = i^(n-1)*6^((n-1)/2)*ChebyshevU(n-1, -i/sqrt(6)). - G. C. Greubel, Jun 01 2023

A015519 a(n) = 2a(n-1) + 7a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 11, 36, 149, 550, 2143, 8136, 31273, 119498, 457907, 1752300, 6709949, 25685998, 98341639, 376485264, 1441362001, 5518120850, 21125775707, 80878397364, 309637224677, 1185423230902, 4538307034543, 17374576685400

Offset: 0

Olivier Gérard

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard, Sep 25 2005

Pisano period lengths: 1, 2, 8, 4, 24, 8, 3, 8, 24, 24, 15, 8, 168, 6, 24, 16, 16, 24, 120, 24, ... . - R. J. Mathar, Aug 10 2012

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

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

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

[ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

LinearRecurrence[{2,7},{0,1},30] (* Harvey P. Dale, Oct 09 2017 *)

a(n)=([0,1; 7,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

[lucas_number1(n,2,-7) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

From Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003: (Start)
a(n) = a(n-1) + A083100(n-2), n>1.
A083100(n)/a(n+1) converges to sqrt(8). (End)
From Paul Barry, Jul 17 2003: (Start)
G.f.: x/ ( 1-2*x-7*x^2 ).
a(n) = ((1+2*sqrt(2))^n-(1-2*sqrt(2))^n)*sqrt(2)/8. (End)
E.g.f.: exp(x)*sinh(2*sqrt(2)*x)/(2*sqrt(2)). - Paul Barry, Nov 20 2003
Second binomial transform is A000129(2n)/2 (A001109). - Paul Barry, Apr 21 2004
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*(7/2)^k*2^(n-k-1). - Paul Barry, Jul 17 2004
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*8^k. - Paul Barry, Sep 29 2004
G.f.: G(0)*x/(2*(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 26 2013

A002533 a(n) = 2a(n-1) + 5a(n-2), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 7, 19, 73, 241, 847, 2899, 10033, 34561, 119287, 411379, 1419193, 4895281, 16886527, 58249459, 200931553, 693110401, 2390878567, 8247309139, 28449011113, 98134567921, 338514191407, 1167701222419, 4027973401873, 13894452915841, 47928772841047, 165329810261299

Offset: 0

N. J. A. Sloane

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 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - Cino Hilliard, Sep 25 2005
a(n), n>0 = term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 2,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010
Pisano period lengths: 1, 1, 1, 4, 4, 1, 24, 4, 3, 4, 120, 4, 56, 24, 4, 8, 288, 3, 18, 4, ... - R. J. Mathar, Aug 10 2012
a(k*m) is divisible by a(m) if k is odd.  - Robert Israel, May 03 2024

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,5).

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

[(1/2)*Floor((1+Sqrt(6))^n+(1-Sqrt(6))^n): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

[n le 2 select 1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

A002533:=(-1+z)/(-1+2*z+5*z**2); # conjectured by Simon Plouffe in his 1992 dissertation

f[n_] := Simplify[((1 + Sqrt[6])^n + (1 - Sqrt[6])^n)/2]; Array[f, 28, 0] (* Or *)
LinearRecurrence[{2, 5}, {1, 1}, 28] (* Or *)
Table[ MatrixPower[{{1, 2}, {3, 1}}, n][[1, 1]], {n, 0, 25}]
(* Robert G. Wilson v, Sep 18 2013 *)

a(n)=([0,1; 5,2]^n*[1;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

x='x+O('x^30); Vec((1-x)/(1-2*x-5*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number2(n,2,-5)/2 for n in range(0, 21)] # Zerinvary Lajos, Apr 30 2009

a(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
From Mario Catalani (mario.catalani(AT)unito.it), May 03 2003: (Start)
G.f.: (1-x)/(1-2*x-5*x^2).
a(n) = (1/2)*((1+sqrt(6))^n + (1-sqrt(6))^n).
a(n)/A083694(n) converges to sqrt(3/2).
a(n)/A083695(n) converges to sqrt(2/3).
a(n) = a(n-1) + 3*A083694(n-1).
a(n) = a(n-1) + 2*A083695(n-1), n>0. (End)
Binomial transform of expansion of cosh(sqrt(6)*x) (A000400, with interpolated zeros). E.g.f.: exp(x)*cosh(sqrt(6)*x) - Paul Barry, May 09 2003
From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)
a(2*n+1) = 2*a(n)*a(n+1) - (-5)^n.
a(n)^2 - 6*A002532(n)^2 = (-5)^n. (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 6^k. - Paul Barry, Jul 25 2004
a(n) = Sum_{k=0..n} A098158(n,k)*6^(n-k). - Philippe Deléham, Dec 26 2007
If p(1)=1, and p(I)=6, for i>1, and if A is the Hessenberg matrix of order n defined by: A(i,j) = p(j-i+1) for i<=j, A(i,j)=-1 for 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*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

Original entry on oeis.org

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
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 central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): 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[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - 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)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*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)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A014983 a(n) = (1 - (-3)^n)/4.

Original entry on oeis.org

0, 1, -2, 7, -20, 61, -182, 547, -1640, 4921, -14762, 44287, -132860, 398581, -1195742, 3587227, -10761680, 32285041, -96855122, 290565367, -871696100, 2615088301, -7845264902, 23535794707, -70607384120, 211822152361, -635466457082, 1906399371247

Offset: 0

Olivier Gérard

q-integers for q=-3.
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^n*charpoly(A,0). - Milan Janjic, Jan 27 2010
Pisano period lengths:  1, 2, 1, 4, 4, 2, 3, 8, 1, 4, 10, 4, 6, 6, 4, 16, 16, 2, 9, 4, ... - R. J. Mathar, Aug 10 2012

T. D. Noe, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 927
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
Index entries for linear recurrences with constant coefficients, signature (-2,3)

Cf. A077925, A014985, A014986, A014987, A014989, A014990, A014991, A014992, A014993, A014994. - Zerinvary Lajos, Dec 16 2008

[(1-(-3)^n)/4: n in [0..30]]; // G. C. Greubel, May 26 2018

a:=n->sum ((-3)^j, j=0..n): seq(a(n), n=-1..25); # Zerinvary Lajos, Dec 16 2008

nn = 25; CoefficientList[Series[x/((1 - x)*(1 + 3*x)), {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
Table[(1 - (-3)^n)/4, {n, 0, 27}] (* Michael De Vlieger, Nov 23 2016 *)

PARI
```
a(n)=(1-(-3)^n)/4
```

[gaussian_binomial(n,1,-3) for n in range(0,27)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + (-3)^(n-1).
G.f.: x/((1-x)*(1+3*x)).
a(n) = -(-1)^n*A015518(n).
a(n) = the (1, 2)-th element of M^n, where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - Simone Severini, Nov 25 2004
a(0)=0, a(1)=1, a(n) = -2*a(n-1) + 3*a(n-2) for n>1. - Philippe Deléham, Sep 19 2009
From Sergei N. Gladkovskii, Apr 29 2012: (Start)
G.f. A(x) = G(0)/4; G(k) = 1 - 1/(3^(2*k) - 3*x*3^(4*k)/(3*x*3^(2*k) + 1/(1 + 1/(3*3^(2*k) - 3^(3)*x*3^(4*k)/(3^2*x*3^(2*k) - 1/G(k+1)))))); (continued fraction, 3rd kind, 6-step).
E.g.f. E(x) = G(0)/4; G(k) = 1 - 1/(9^k - 3*x*81^k/(3*x*9^k + (2*k+1)/(1 + 1/(3*9^k - 27*x*81^k/(9*x*9^k - (2*k+2)/G(k+1)))))); (continued fraction, 3rd kind, 6-step). (End)
a(n) = A084222(n) - 1. - Filip Zaludek, Nov 19 2016
E.g.f.: sinh(x)*cosh(x)*exp(-x). - Ilya Gutkovskiy, Nov 20 2016

A015565 a(n) = 7a(n-1) + 8a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 7, 57, 455, 3641, 29127, 233017, 1864135, 14913081, 119304647, 954437177, 7635497415, 61083979321, 488671834567, 3909374676537, 31274997412295, 250199979298361, 2001599834386887, 16012798675095097, 128102389400760775, 1024819115206086201, 8198552921648689607

Offset: 0

Olivier Gérard

A linear 2nd order recurrence. A Jacobsthal number sequence.
Binomial transform of A053573 (preceded by zero). - Paul Barry, Apr 09 2003
Second binomial transform of A080424. Binomial transform of A053573, with leading zero. Binomial transform is 0,1,9,81,729,....(9^n - 0^n)/9. Second binomial transform is 0,1,11,111,1111,... (A002275: repunits). - Paul Barry, Mar 14 2004
Number of walks of length n between any two distinct nodes of the complete graph K_9. Example: a(2)=7 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHI are: ACB, ADB, AEB, AFB, AGB, AHB and AIB. - Emeric Deutsch, Apr 01 2004
Unsigned version of A014990. - Philippe Deléham, Feb 13 2007
General form: k=8^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The ratio a(n+1)/a(n) converges to 8 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

			G.f. = x + 7*x^2 + 57*x^3 + 455*x^4 + 3641*x^5 + 29127*x^6 + 233017*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Dale Gerdemann, Fractal generated from (7,8) recursion, YouTube Video, Dec 5, 2014.
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A002275, A014990, A053573, A080424, A082311, A082365.
Cf. A062160, A099322, A106566.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(8^n/9): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(8^n/9),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=8^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{7,8},{0,1},30] (* Harvey P. Dale, Mar 04 2016 *)

x='x+O('x^30); concat([0], Vec(x/(1-7*x-8*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,7,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

From Paul Barry, Apr 09 2003: (Start)
a(n) = (8^n - (-1)^n)/9.
a(n) = J(3*n)/3 = A001045(3*n)/3. (End)
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 8^(n-1) - a(n-1).
G.f.: x/(1-7*x-8*x^2). (End)
a(n) = Sum_{k = 0..n} A106566(n,k)*A099322(k). - Philippe Deléham, Oct 30 2008
a(n) = round(8^n/9). - Mircea Merca, Dec 28 2010
From Peter Bala, May 31 2024: (Start)
G.f: A(x) = x/(1 - x^2) o x/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A054878.
The black diamond product A(x) o A(x) is the g.f. for the number of walks of length n between any two distinct nodes of the complete graph K_81.
Row 8 of A062160. (End)
E.g.f.: exp(-x)*(exp(9*x) - 1)/9. - Elmo R. Oliveira, Aug 17 2024

A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 14, 28, 57, 114, 228, 455, 910, 1820, 3641, 7282, 14564, 29127, 58254, 116508, 233017, 466034, 932068, 1864135, 3728270, 7456540, 14913081, 29826162, 59652324, 119304647, 238609294, 477218588, 954437177, 1908874354, 3817748708

Offset: 0

Paul Barry, Oct 28 2005

A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.
Paul Curtz, Aug 05 2007, observes that the inverse binomial transform of 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,... gives the same sequence up to signs. That is, the extended sequence is an eigensequence for the inverse binomial transform (an autosequence).
The round() function enables the closed (non-recurrence) formula to take a very simple form: see Formula section. This can be generalized without loss of simplicity to a(n) = round(b^n/c), where b and c are very small, incommensurate integers (c may also be an integer fraction). Particular choices of small integers for b and c produce a number of well-known sequences which are usually defined by a recurrence - see Cross Reference. - Ross Drewe, Sep 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

From Ross Drewe, Sep 03 2009: (Start)
Other sequences a(n) = round(b^n / c), where b and c are very small integers:
A001045 b = 2; c = 3
A007910 b = 2; c = 5
A016029 b = 2; c = 5/3
A077947 b = 2; c = 7
abs(A078043) b = 2; c = 7/3
A007051 b = 3; c = 2
A015518 b = 3; c = 4
A034478 b = 5; c = 2
A003463 b = 5; c = 4
A015531 b = 5; c = 6
(End)

[Round(2^n/9): n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

A010892 := proc(n) op((n mod 6)+1,[1,1,0,-1,-1,0]) ; end proc:
A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # R. J. Mathar, Dec 17 2010

CoefficientList[Series[x^3/(1-2x+x^3-2x^4),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,-1,2},{0,0,0,1},40] (* Harvey P. Dale, Apr 30 2011 *)

a(n)=2^n\/9 \\ Charles R Greathouse IV, Jun 05 2011

def A113405(n): return ((1<Chai Wah Wu, Apr 17 2025

a(n) = 2a(n-1) - a(n-3) + 2a(n-4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A001045(k).
a(n) = Sum_{k=0..n} binomial((n+k)/2,k)*A001045((n-k)/2)*(1+(-1)^(n-k))/2.
a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - Paul Curtz, Nov 30 2007
From Paul Curtz, Dec 16 2007: (Start)
a(n+1) - 2a(n) = A131531(n).
a(n) + a(n+3) = 2^n. (End)
a(n) = round(2^n/9). - Ross Drewe, Sep 03 2009
9*a(n) = 2^n + (-1)^n - 3*A010892(n). - R. J. Mathar, Mar 24 2018

Edited by N. J. A. Sloane, Dec 13 2007

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

Original entry on oeis.org

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

cp's OEIS Frontend

A002532 a(n) = 2*a(n-1) + 5*a(n-2), a(0) = 0, a(1) = 1.

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A015531 Linear 2nd order recurrence: a(n) = 4*a(n-1) + 5*a(n-2).

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A083099 a(n) = 2*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.

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A015519 a(n) = 2*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.

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A002533 a(n) = 2*a(n-1) + 5*a(n-2), with a(0) = a(1) = 1.

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A002532 a(n) = 2a(n-1) + 5a(n-2), a(0) = 0, a(1) = 1.

A015531 Linear 2nd order recurrence: a(n) = 4a(n-1) + 5a(n-2).

A083099 a(n) = 2a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

A015519 a(n) = 2a(n-1) + 7a(n-2), with a(0) = 0, a(1) = 1.

A002533 a(n) = 2a(n-1) + 5a(n-2), with a(0) = a(1) = 1.

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

A015565 a(n) = 7a(n-1) + 8a(n-2), a(0) = 0, a(1) = 1.

A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

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