A082311 -id:A082311 - cp's OEIS frontend

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

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

A007613 a(n) = (8^n + 2*(-1)^n)/3.

Original entry on oeis.org

1, 2, 22, 170, 1366, 10922, 87382, 699050, 5592406, 44739242, 357913942, 2863311530, 22906492246, 183251937962, 1466015503702, 11728124029610, 93824992236886, 750599937895082, 6004799503160662, 48038396025285290, 384307168202282326, 3074457345618258602

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Also, the cogrowth sequence of C3 X C3 = ; that is, the number of words of length 3n that reduce to the identity. - Sean A. Irvine, Nov 04 2024

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
D. S. Clark, Proof without words, Math. Mag., 63 (1990), 29.
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A001045, A006566, A078008, A082311, A139459.

[(8^n + 2*(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 14 2011

LinearRecurrence[{7,8}, {1,2}, 41] (* G. C. Greubel, Apr 23 2023 *)

a(n)=(8^n + 2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 06 2011

Vec((5*x-1)/((x+1)*(8*x-1)) + O(x^50)) \\ Colin Barker, Sep 29 2014

[(8^n -4*(n%2) +2)/3 for n in range(41)] # G. C. Greubel, Apr 23 2023

a(n) = A078008(3*n). - Paul Barry, Nov 29 2003
From Paul Barry, Mar 24 2004: (Start)
a(n) = (A082311(n) + (-1)^n)/2.
a(n) = (A001045(3*n+1) + (-1)^n)/2. (End)
a(n) = Sum_{k=0..n} binomial(3*n, 3*k). - Paul Barry, Jan 13 2005
a(n) = 8*a(n-1) + 6*(-1)^n. - Paul Curtz, Nov 19 2007
From Colin Barker, Sep 29 2014: (Start)
a(n) = 7*a(n-1) + 8*a(n-2).
G.f.: (1-5*x) / ((1+x)*(1-8*x)). (End)
E.g.f.: (1/3)*(exp(8*x) + 2*exp(-x)). - G. C. Greubel, Apr 23 2023

More terms from Colin Barker, Sep 29 2014

A082365 A Jacobsthal number sequence.

Original entry on oeis.org

1, 11, 85, 683, 5461, 43691, 349525, 2796203, 22369621, 178956971, 1431655765, 11453246123, 91625968981, 733007751851, 5864062014805, 46912496118443, 375299968947541, 3002399751580331, 24019198012642645, 192153584101141163

Offset: 0

Paul Barry, Apr 09 2003

A trisection of A024495. - Paul Curtz, Nov 18 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A015565, A082311.

[4*8^n/3-(-1)^n/3: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

f[n_] := (4*8^n - (-1)^n)/3; Array[f, 20, 0] (* Robert G. Wilson v, Aug 13 2011 *)
LinearRecurrence[{7,8},{1,11},20] (* Harvey P. Dale, May 06 2012 *)

vector(30, n, n--; (4*8^n -(-1)^n)/3) \\ G. C. Greubel, Sep 16 2018

a(n) = (4*8^n -(-1)^n)/3.
a(n) = J(3*n+2) = A001045(3*n)/3.
a(n) = 4*A015565(n)+A015565(n+1).
From Philippe Deléham, Nov 19 2007: (Start)
a(0)=1, a(1)=11, a(n+1) = 7*a(n) + 8*a(n-1) for n>=1 .
G.f.: (1+4*x)/(1-7*x-8*x^2). (End)

A132400 Period 4: repeat [1, 5, 3, 1].

Original entry on oeis.org

1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1, 1, 5, 3, 1

Offset: 0

Paul Curtz, Nov 20 2007

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A082311.

&cat [[1, 5, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([1, 5, 3, 1]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 120, {1,5,3,1}] (* Harvey P. Dale, Sep 15 2014 *)

a(n)=[1,5,3,1][n%4+1] \\ Charles R Greathouse IV, Jun 02 2011

Final digits of A082311.
O.g.f.: (1+5*x+3*x^2+x^3)/((1-x)*(x+1)*(1+x^2)) = (5/2)/(1-x)-(1/2)/(x+1)+(2*x-1)/(1+x^2). - R. J. Mathar, Nov 28 2007
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = (5-cos(n*Pi)-2*cos(n*Pi/2)+4*sin(n*Pi/2)-I*sin(n*Pi))/2. (End)

A345455 a(n) = Sum_{k=0..n} binomial(5n+1,5k).

Original entry on oeis.org

1, 7, 474, 12393, 427351, 13333932, 430470899, 13733091643, 439924466026, 14072420067757, 450374698997499, 14411355379952868, 461170414282959151, 14757375158697584607, 472236871202375365274, 15111570273013075344193, 483570355262634763462351

Offset: 0

Seiichi Manyama, Jun 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21,353,-32).

Sum_{k=0..n} binomial(b*n+c,b*k): A082311 (b=3,c=1), A090407 (b=4,c=1), A070782 (b=5,c=0), this sequence (b=5,c=1), A345456 (b=5,c=2), A345457 (b=5,c=3), A345458 (b=5,c=4).

Cf. A139398.

a[n_] := Sum[Binomial[5*n + 1, 5*k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 20 2021 *)
LinearRecurrence[{21,353,-32},{1,7,474},20] (* Harvey P. Dale, Jul 20 2021 *)

a(n) = sum(k=0, n, binomial(5*n+1, 5*k));

my(N=20, x='x+O('x^N)); Vec((1-14*x-26*x^2)/((1-32*x)*(1+11*x-x^2)))

G.f.: (1 - 14*x - 26*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+1).
a(n) = 2^(5*n + 2)/10 + ((-475 + 213*sqrt(5))/phi^(5*n) - ( 65 - 33*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A087462 Generalized mod 3 multiplicative Jacobsthal sequence.

Original entry on oeis.org

1, 1, 1, 8, 5, 11, 64, 43, 85, 512, 341, 683, 4096, 2731, 5461, 32768, 21845, 43691, 262144, 174763, 349525, 2097152, 1398101, 2796203, 16777216, 11184811, 22369621, 134217728, 89478485, 178956971, 1073741824, 715827883, 1431655765, 8589934592, 5726623061

Offset: 0

Paul Barry, Sep 08 2003

2^n = a(n) + A087463(n) + A087464(n) provides a decomposition of Pascal's triangle.

Multiplicative analog of A078008.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,8).

Cf. A001045, A001018 (trisection), A082311 (trisection), A082365 (trisection).

Vec(-(4*x^5-2*x^4+x^3+x^2+x+1)/((x+1)*(2*x-1)*(x^2-x+1)*(4*x^2+2*x+1)) + O(x^100)) \\ Colin Barker, Nov 02 2015

a(n) = Sum_{k=0..n} if (mod(n*k, 3)=0, 1, 0) * C(n, k).
a(n) = 2^n-2/3*(1-cos(2*Pi*n/3))*(A001045(n)+2*A001045(n-1)+0^n).
From Colin Barker, Nov 02 2015: (Start)
a(n) = 7*a(n-3)+8*a(n-6) for n>5.
G.f.: -(4*x^5-2*x^4+x^3+x^2+x+1) / ((x+1)*(2*x-1)*(x^2-x+1)*(4*x^2+2*x+1)).
(End)

A092810 Binomial transform of a Jacobsthal trisection.

Original entry on oeis.org

1, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534

Offset: 0

Paul Barry, Mar 10 2004

Binomial transform of A082311.

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

Cf. A001045.

[1] cat [2*3^(2*n-1): n in [1..20]]; // Vincenzo Librandi, Jun 20 2015

Table[EulerPhi[9^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

Vec((1-3*x)/(1-9*x) + O(x^30)) \\ Michel Marcus, Jun 18 2015

G.f.: (1-3*x)/(1-9*x).
E.g.f.: 2*exp(9*x)/3 + 1/3.
a(n) = 2*9^n/3 + 0^n/3.
a(n) = A054878(2n+1) - A054878(2n-1) + 0^n/3 = A015518(2n+1) - A015518(2n-1) + 0^n/3.
a(n) = 2*3^(2*n-1), for n>0. - Gionata Neri, Jun 18 2015

A242563 a(n) = 2a(n-1) - a(n-3) + 2a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.

Original entry on oeis.org

0, 0, 2, 3, 6, 10, 21, 42, 86, 171, 342, 682, 1365, 2730, 5462, 10923, 21846, 43690, 87381, 174762, 349526, 699051, 1398102, 2796202, 5592405, 11184810, 22369622, 44739243, 89478486, 178956970, 357913941, 715827882, 1431655766, 2863311531, 5726623062, 11453246122

Offset: 0

Paul Curtz, May 17 2014

Generally, a(n) is an autosequence if its inverse binomial transform is (-1)^n*a(n). It is of the first kind if the main diagonal is 0's and the first two upper diagonals (just above the main one) are the same. It is of the second kind if the main diagonal is equal to the first upper diagonal multiplied by 2. If the first upper diagonal is an autosequence, the sequence is a super autosequence. Example: A113405. The first upper diagonal is A001045(n). Another super autosequence: 0, 0, 0 followed by A059633(n). The first upper diagonal is A000045(n).
Difference table of a(n):
   0,  0,  2, 3, 6, 10, 21, 42, ...
   0,  2,  1, 3, 4, 11, 21, 44, ...
   2, -1,  2, 1, 7, 10, 23, 41, ...
  -3,  3, -1, 6, 3, 13, 18, 45, ... .
This is an autosequence of the second kind. The main diagonal is 2*A001045(n) = A078008(n). More precisely it is a super autosequence, companion of A113405(n).
a(n+1) mod 10 = period 12: repeat 0, 2, 3, 6, 0, 1, 2, 6, 1, 2, 2, 5.
It is shifted A081374(n+1) mod 10 =
    period 12: repeat 1, 2, 2, 5, 0, 2, 3, 6, 0, 1, 2, 6.
a(n) mod 9 = period 18:
repeat 0, 0, 2, 3, 6, 1, 3, 6, 5, 0, 0, 7, 6, 3, 8, 6, 3, 4 = c(n).
c(n) + c(n+9) = 0, 0, 9, 9, 9, 9, 9, 9, 9.

			G.f. = 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 21*x^6 + 42*x^7 + 86*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A000032, 1/(n+1), A164555/A027642 (all autosequences of 2nd kind). A007283, A175805.

a[n_] := (m = Mod[n, 6]; 1/3*(2^n + (-1)^n + 1/120*(m-6)*(m+1)*(m^3-29*m+40))); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 19 2014, a non-recursive formula, after Mathematica's RSolve *)
LinearRecurrence[{2, 0, -1, 2}, {0, 0, 2, 3},50] (* G. C. Greubel, Feb 21 2017 *)

concat([0,0], Vec(x^2*(x-2)/((x+1)*(2*x-1)*(x^2-x+1)) + O(x^100))) \\ Colin Barker, May 18 2014

a(n+3) = 3*2^n - a(n), a(0)=a(1)=0, a(2)=2.
a(n) = 2*A113405(n+1) - A113405(n).
a(n+1) = 2*a(n) + period 6: repeat 0, 2, -1, 0, -2, 1. a(0)=0.
a(n) = 2^n - A081374(n+1).
a(n+3) = a(n+1) + A130755(n).
G.f.: x^2*(x-2) / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, May 18 2014
a(n) = A024495(n) + A131531(n).
a(n+6) = a(n) + 21*2^n, a(0)=a(1)=0, a(2)=2, a(3)=3, a(4)=6, a(5)=10.
a(n) = A001045(n) - A092220(n).
a(n+12) = a(n) + 1365*2^n. First 12 values in the Data. (A024495(n+12) = A024495(n) + 1365*2^n).
a(3n)   = A132805(n) = 3*A015565(n).
a(3n+1) = A132804(n) = 6*A015565(n).
a(3n+2) = A132397(n) = 2*A082311(n).
a(n) = 1/3*((-1)^n - 2*cos((n*Pi)/3) + 2^n). - Alexander R. Povolotsky,  Jun 02 2014

More terms from Colin Barker, May 18 2014

A090409 a(n) = (78^n + 2(-1)^n)/9.

Original entry on oeis.org

1, 6, 50, 398, 3186, 25486, 203890, 1631118, 13048946, 104391566, 835132530, 6681060238, 53448481906, 427587855246, 3420702841970, 27365622735758, 218924981886066, 1751399855088526, 14011198840708210, 112089590725665678, 896716725805325426, 7173733806442603406

Offset: 0

Paul Barry, Nov 29 2003

Index entries for linear recurrences with constant coefficients, signature (7,8).

First differences of A015565.

Cf. A007613, A082311, A082365.

LinearRecurrence[{7,8},{1,6},20] (* Harvey P. Dale, Aug 15 2016 *)

a(n) = Sum_{j=0..2} Sum_{k=0..n} C(3*n+j, 3*k)/3.
a(n) = (A007613(n) + A082311(n) + A082365(n))/3.
G.f.: (-1+x)/((1+x)*(8*x-1)). - R. J. Mathar, Dec 10 2014
From Elmo R. Oliveira, Aug 18 2024: (Start)
E.g.f.: exp(-x)*(7*exp(9*x) + 2)/9.
a(n) = 7*a(n-1) + 8*a(n-2) for n > 1. (End)

a(20)-a(21) from Elmo R. Oliveira, Aug 18 2024

A132397 Second trisection of A024494.

Original entry on oeis.org

2, 10, 86, 682, 5462, 43690, 349526, 2796202, 22369622, 178956970, 1431655766, 11453246122, 91625968982, 733007751850, 5864062014806, 46912496118442, 375299968947542, 3002399751580330, 24019198012642646, 192153584101141162

Offset: 0

Paul Curtz, Nov 20 2007

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A082311, A132805, A078008.

CoefficientList[Series[2(2x-1)/((x+1)(8x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{7,8},{2,10},30] (* Harvey P. Dale, Oct 14 2015 *)

Vec(2*(1 - 2*x) / ((1 + x)*(1 - 8*x)) + O(x^20)) \\ Colin Barker, Jun 05 2020

O.g.f.: 2(2x-1)/((x+1)(8x-1)). a(n) = 2*A082311(n). - R. J. Mathar, Jan 13 2008
a(0)=2, a(1)=10, a(n) = 7*a(n-1)+8*a(n-2). - Harvey P. Dale, Oct 14 2015
From Oboifeng Dira, Jun 05 2020: (Start)
a(n) = A078008(3*n+2). Third trisection of A078008.
a(n) = Sum_{k=0..n} binomial(3*n+2,3*k+1).
(End)
a(n) = 2*((-1)^n + 2^(1+3*n)) / 3 for n>1. - Colin Barker, Jun 05 2020

More terms from R. J. Mathar, Jan 13 2008

cp's OEIS Frontend

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

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A007613 a(n) = (8^n + 2*(-1)^n)/3.

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A082365 A Jacobsthal number sequence.

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A132400 Period 4: repeat [1, 5, 3, 1].

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A345455 a(n) = Sum_{k=0..n} binomial(5*n+1,5*k).

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A087462 Generalized mod 3 multiplicative Jacobsthal sequence.

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A092810 Binomial transform of a Jacobsthal trisection.

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

A345455 a(n) = Sum_{k=0..n} binomial(5n+1,5k).

A242563 a(n) = 2a(n-1) - a(n-3) + 2a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.

A090409 a(n) = (78^n + 2(-1)^n)/9.