A015579 -id:A015579 - cp's OEIS frontend

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

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
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

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

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 *)

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

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

A015580 Expansion of x/(1 - 9x - 4x^2).

Original entry on oeis.org

0, 1, 9, 85, 801, 7549, 71145, 670501, 6319089, 59553805, 561260601, 5289560629, 49851088065, 469818035101, 4427766668169, 41729172153925, 393273616058001, 3706379233137709, 34930507562471385, 329200084994793301, 3102522795203025249, 29239505496806400445

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 1, 2, 1, 3, 2, 48, 2, 6, 3, 10, 2, 42, 48, 6, 4, 24, 6,360, 3, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,4)

Cf. A015579, A099371.

[n le 2 select n-1 else 9*Self(n-1) + 4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2012

LinearRecurrence[{9, 4}, {0, 1}, 30] (* Vincenzo Librandi, Nov 14 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-9*x-4*x^2))) \\ G. C. Greubel, Jan 06 2018

[lucas_number1(n,9,-4) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

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

Extended by T. D. Noe, May 23 2011

A015581 a(n) = 9a(n-1) + 5a(n-2).

Original entry on oeis.org

0, 1, 9, 86, 819, 7801, 74304, 707741, 6741189, 64209406, 611590599, 5825362421, 55486214784, 528502745161, 5033955780369, 47948115749126, 456702820643979, 4350065964541441, 41434107784092864, 394657299879542981, 3759086237836351149, 35805062639924875246

Offset: 0

Olivier Gérard

A linear 2nd-order recurrence.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,5).

Cf. A015579, A099371.

[n le 2 select n-1 else 9*Self(n-1) + 5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

Join[{a=0,b=1},Table[c=9*b+5*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{9, 5}, {0, 1}, 30] (* Vincenzo Librandi, Nov 15 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-9*x-5*x^2))) \\ G. C. Greubel, Jan 06 2018

[lucas_number1(n,9,-5) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

G.f.: x/(1 - 9*x - 5*x^2). - R. J. Mathar, Dec 02 2007

Extended by T. D. Noe, May 23 2011

A015583 a(0) = 0, a(1) = 1; for n >= 2, a(n) = 9a(n-1) + 7a(n-2).

Original entry on oeis.org

0, 1, 9, 88, 855, 8311, 80784, 785233, 7632585, 74189896, 721137159, 7009563703, 68134033440, 662273246881, 6437397456009, 62572489832248, 608214190682295, 5911935144966391, 57464915639473584, 558567786770026993, 5429364490406558025, 52774254921049211176

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,7).

Cf. A015579, A099371.

a:=[0,1];; for n in [3..22] do a[n]:=9*a[n-1]+7*a[n-2]; od; a; # Muniru A Asiru, Jul 15 2018

[n le 2 select n-1 else 9*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

a:=proc(n) option remember: if n=0 then 0 elif n=1 then 1 else 9*procname(n-1)+7*procname(n-2) fi: end; seq(a(n),n=0..22); # Muniru A Asiru, Jul 15 2018

Join[{a=0,b=1},Table[c=9*b+7*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{9, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 15 2012 *)

x='x+O('x^30); concat([0], Vec(1/(1-9*x-7*x^2))) \\ G. C. Greubel, Jan 06 2018

[lucas_number1(n,9,-7) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

G.f.: 1/(1 - 9*x - 7*x^2). - Zerinvary Lajos, Dec 20 2008

Extended by T. D. Noe, May 23 2011

A015584 Expansion of g.f. x/(1 - 9x - 8x^2).

Original entry on oeis.org

0, 1, 9, 89, 873, 8569, 84105, 825497, 8102313, 79524793, 780541641, 7661073113, 75193991145, 738034505209, 7243862476041, 71099038326041, 697842244742697, 6849372509292601, 67227090541574985, 659838794948515673, 6476365878869240937, 63566003269411293817

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 1, 4, 1, 24, 4, 6, 1, 4, 24, 10, 4, 12, 6, 24, 1,144, 4, 15, 24, ... . - R. J. Mathar, Aug 10 2012

For n >= 2, the number of positive integers with n-1 decimal digits in which adjacent digits differ by at most 8. - Edwin Hermann, Apr 19 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,8).

Cf. A015579, A099371.

[n le 2 select n-1 else 9*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

LinearRecurrence[{9, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 15 2012 *)
CoefficientList[Series[x/(1-9x-8x^2),{x,0,30}],x] (* Harvey P. Dale, Sep 06 2022 *)

concat(0, Vec(x / (1-9*x-8*x^2) + O(x^30))) \\ Colin Barker, May 16 2017

[lucas_number1(n,9,-8) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

a(n) = 9*a(n-1) + 8*a(n-2).
a(n) = (-((9-sqrt(113))/2)^n + ((9+sqrt(113))/2)^n) / sqrt(113). - Colin Barker, May 16 2017
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(113)*x/2)/sqrt(113). - Stefano Spezia, Oct 25 2023

Extended by T. D. Noe, May 23 2011

A181353 a(n) = 9a(n-1) + 3a(n-2); a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 9, 84, 783, 7299, 68040, 634257, 5912433, 55114668, 513769311, 4789267803, 44644718160, 416170266849, 3879466556121, 36163709805636, 337111787919087, 3142497220688691, 29293810349955480, 273071784811665393, 2545527494354854977, 23728962803628690972

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 27 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,3).

Cf. A015579, A099371.

I:=[0,1]; [n le 2 select I[n] else 9*Self(n-1) + 3*Self(n-2): n in [1..30]];

Join[{a=0,b=1},Table[c=9*b+3*a;a=b;b=c,{n,60}]]
LinearRecurrence[{9,3}, {0,1}, 30] (* G. C. Greubel, Jan 24 2018 *)

x='x+O('x^30); concat([0], Vec(x/(1-9*x-3*x^2))) \\ G. C. Greubel, Jan 24 2018

a(n) = ((9+sqrt(93))^n - (9-sqrt(93))^n)/(2^n*sqrt(93)). - Rolf Pleisch, May 14 2011
G.f.: x/(1 - 9*x - 3*x^2). - Philippe Deléham, Nov 21 2011
a(n+1) = Sum_{k=0..n} A099097(n,k)*3^k. - Philippe Deléham, Nov 21 2011
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(93)*x/2)/sqrt(93). - Stefano Spezia, Apr 06 2023

A015589 Expansion of x/(1 - 10x - 7x^2).

Original entry on oeis.org

0, 1, 10, 107, 1140, 12149, 129470, 1379743, 14703720, 156695401, 1669880050, 17795668307, 189645843420, 2021028112349, 21537802027430, 229525217060743, 2446016784799440, 26066844367419601, 277790561167792090, 2960373522249858107, 31548269150673125700

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,7).

Cf. A015579, A099371.

[n le 2 select n-1 else 10*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

Join[{a=0,b=1},Table[c=10*b+7*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
CoefficientList[Series[x/(1-10x-7x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {10,7},{0,1},31] (* Harvey P. Dale, Nov 26 2011 *)

x='x+O('x^30); concat([0], Vec(x/(1-10*x-7*x^2))) \\ G. C. Greubel, Jan 06 2018

[lucas_number1(n,10,-7) for n in range(0, 18)] # Zerinvary Lajos, Apr 26 2009

G.f.: x/(1 - 10*x - 7*x^2).
a(n) = 10*a(n-1) + 7*a(n-2).
a(n) = (8+5*sqrt(2))*((5+4*sqrt(2))^n-(5-4*sqrt(2))^n)/(16*(5+4*sqrt(2))). - Wesley Ivan Hurt, Aug 04 2025

Extended by T. D. Noe, May 23 2011

A153191 a(n) = 9a(n-1) + 6a(n-2); a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 87, 837, 8055, 77517, 745983, 7178949, 69086439, 664851645, 6398183439, 61572760821, 592543948023, 5702332097133, 54876252562335, 528100265643813, 5082159906168327, 48908040749377821, 470665326181410351

Offset: 0

Zerinvary Lajos, Dec 20 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, 6).

Cf. A015579, A099371.

I:=[0,1]; [n le 2 select I[n] else 9*Self(n-1) + 6*Self(n-2): n in [1..25]]; // G. C. Greubel, Jan 24 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=9*a[n-1]+6*a[n-2]od: seq(a[n], n=0..33);

LinearRecurrence[{9,6}, {0,1}, 25] (* G. C. Greubel, Jan 24 2018 *)

x='x+O('x^25); concat([0], Vec(x/(1-9*x-6*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,9,-6) for n in range(0, 25)]#  Zerinvary Lajos, Apr 26 2009

G.f.: x/(1 - 9*x - 6*x^2).

Formula corrected by Philippe Deléham, Dec 20 2008

Edited by N. J. A. Sloane, Dec 21 2008

A287829 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 6.

Original entry on oeis.org

1, 10, 92, 848, 7816, 72040, 663992, 6120008, 56408056, 519912520, 4792028792, 44168084168, 407096815096, 3752207504200, 34584061167992, 318760965520328, 2938016812018936, 27079673239211080, 249593092776937592, 2300497181470860488, 21203660818791619576

Offset: 0

David Nacin, Jun 02 2017

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.

Index entries for linear recurrences with constant coefficients, signature (9,2).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

Mathematica
```
LinearRecurrence[{9, 2}, {1, 10}, 30]
```

def a(n):
 if n in [0, 1]:
  return [1, 10][n]
 return 9*a(n-1)+2*a(n-2)

a(n) = 9*a(n-1) + 2*a(n-2), a(0)=1, a(1)=10.
G.f.: (-1 - x)/(-1 + 9*x + 2*x^2).
a(n) = ((1 - 11/sqrt(89))/2)*((9 - sqrt(89))/2)^n + ((1 + 11/sqrt(89))/2)*((9 + sqrt(89))/2)^n.
a(n) = A015579(n)+A015579(n+1). - R. J. Mathar, Oct 20 2019

cp's OEIS Frontend

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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A015580 Expansion of x/(1 - 9*x - 4*x^2).

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A015581 a(n) = 9*a(n-1) + 5*a(n-2).

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A015583 a(0) = 0, a(1) = 1; for n >= 2, a(n) = 9*a(n-1) + 7*a(n-2).

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A015584 Expansion of g.f. x/(1 - 9*x - 8*x^2).

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A181353 a(n) = 9*a(n-1) + 3*a(n-2); a(0) = 0, a(1) = 1.

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A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A015580 Expansion of x/(1 - 9x - 4x^2).

A015581 a(n) = 9a(n-1) + 5a(n-2).

A015583 a(0) = 0, a(1) = 1; for n >= 2, a(n) = 9a(n-1) + 7a(n-2).

A015584 Expansion of g.f. x/(1 - 9x - 8x^2).

A181353 a(n) = 9a(n-1) + 3a(n-2); a(0) = 0, a(1) = 1.

A015589 Expansion of x/(1 - 10x - 7x^2).

A153191 a(n) = 9a(n-1) + 6a(n-2); a(0)=0, a(1)=1.