A015580 -id:A015580

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A189800

a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

+10
4

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,8).

FORMULA

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

MATHEMATICA

LinearRecurrence[{6, 8}, {0, 1}, 50]

CoefficientList[Series[-(x/(-1+6 x+8 x^2)), {x, 0, 50}], x] (* Harvey P. Dale, Jul 26 2011 *)

PROG

(Magma) I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

(PARI) a(n)=([0, 1; 8, 6]^n*[0; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:

c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10

1..A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446

2..A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102

3..A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528

4..A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226

5..A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250

6..A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005

7..A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568

8..A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953

9..A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585

10.A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, May 24 2011

STATUS

approved

A287830

Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 7.

+10
0

1, 10, 94, 886, 8350, 78694, 741646, 6989590, 65872894, 620814406, 5850821230, 55140648694, 519669123166, 4897584703270, 46156938822094, 435002788211926, 4099652849195710, 38636886795609094, 364130592557264686, 3431722880197818550, 32342028292009425694

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

In general, the number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 5+k for k in {0,1,2,3,4} is given by a(n) = 9*a(n-1) + 2*k*a(n-2), a(0)=1, a(1)=10.

LINKS

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (9,4).

FORMULA

a(n) = 9*a(n-1) + 4*a(n-2), a(0)=1, a(1)=10.

G.f.: (-1 - x)/(-1 + 9*x + 4*x^2).

a(n) = ((1 - 11/sqrt(97))/2)*((9 - sqrt(97))/2)^n + ((1 + 11/sqrt(97))/2)*((9 + sqrt(97))/2)^n.

a(n) = A015580(n)+A015580(n+1). - R. J. Mathar, Oct 20 2019

MATHEMATICA

LinearRecurrence[{9, 4}, {1, 10}, 30]

PROG

(Python)

def a(n):

.if n in [0, 1]:

..return [1, 10][n]

.return 9*a(n-1)+4*a(n-2)