A109499 -id:A109499 - cp's OEIS frontend

A015448 a(0) = 1, a(1) = 1, and a(n) = 4*a(n-1) + a(n-2) for n >= 2.

1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141, 701408733, 2971215073, 12586269025, 53316291173, 225851433717, 956722026041, 4052739537881, 17167680177565, 72723460248141, 308061521170129, 1304969544928657, 5527939700884757

Offset: 0

Olivier Gérard

If one deletes the leading 0 in A084326, takes the inverse binomial transform, and adds a(0)=1 in front, one obtains this sequence here. - Al Hakanson (hawkuu(AT)gmail.com), May 02 2009
For n >= 1, row sums of triangle
  m |k=0     1     2     3     4     5     6     7
====+=============================================
  0 |  1
  1 |  1     4
  2 |  1     4    16
  3 |  1     8    16    64
  4 |  1     8    48    64   256
  5 |  1    12    48   256   256  1024
  6 |  1    12    96   256  1280  1024  4096
  7 |  1    16    96   640  1280  6144  4096 16384
which is triangle for numbers 4^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
a(n) = a(n;-2) = 3^n*Sum_{k=0..n} binomial(n,k)*F(k+1)*(-2/3)^k, where a(n;d), n=0,1,...,d, denotes the delta-Fibonacci numbers defined in comments to A000045 (see also the papers of Witula et al.). We note that (see A033887) F(3n+1) = 3^n*a(n,2/3) = Sum_{k=0..n} binomial(n,k)*F(k-1)*(-2/3)^k, which implies F(3n+1) + 3^(-n)*a(n) = Sum_{k=0..n} binomial(n,k)*L(k)*(-2/3)^k, where L(k) denotes the k-th Lucas number. - Roman Witula, Jul 12 2012
a(n+1) is (for n >= 0) the number of length-n strings of 5 letters {0,1,2,3,4} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012
Starting with offset 1 the sequence is the INVERT transform of (1, 4, 4*3, 4*3^2, 4*3^3, ...); i.e., of A003946: (1, 4, 12, 36, 108, ...). - Gary W. Adamson, Aug 06 2016
a(n+1) equals the number of quinary sequences of length n such that no two consecutive terms differ by 3. - David Nacin, May 31 2017

T. D. Noe, Table of n, a(n) for n = 0..200
Joerg Arndt, Matters Computational (The Fxtbook), pp. 313-315.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876 (see Corollary 1 (vi)).
Gary Detlefs and Wolfdieter Lang, Improved Formula for the Multi-Section of the Linear Three-Term Recurrence Sequence, arXiv:2304.12937 [math.CO], 2023.
Amelia Gilson, Hadley Killen, Steven J. Miller, Nadia Razek, Joshua M. Siktar, and Liza Sulkin, Zeckendorf's Theorem Using Indices in an Arithmetic Progression, arXiv:2005.10396 [math.NT], 2020.
Edyta Hetmaniok, Bozena Piatek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Math. 15 (2017), 477-485.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Bahar Kuloğlu, Engin Özkan, and Marin Marin, Fibonacci and Lucas Polynomials in n-gon, An. Şt. Univ. Ovidius Constanţa (Romania 2023) Vol. 31, No 2, 127-140.
Roman Witula, Binomials transformation formulae of scaled Lucas numbers, Demonstratio Math. 46 (2013), 15-27.
R. Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A001076, A147722 (INVERT transform), A109499 (INVERTi transform), A154626 (Binomial transform), A086344 (inverse binomial transform), A003946, A049310.

Cf. A000032, A000045, A014445, A014448, A033887, A046854, A055830, A055870, A084326, A167808.

[Fibonacci(3*n-1): n in [0..40]]; // Vincenzo Librandi, Jul 04 2015

with(combinat): a:=n->fibonacci(n,4)-3*fibonacci(n-1,4): seq(a(n), n=1..23); # Zerinvary Lajos, Apr 04 2008

Fibonacci/@(3*Range[0,30]-1) (* Vladimir Joseph Stephan Orlovsky, Mar 01 2010 *)
LinearRecurrence[{4,1},{1,1},30] (* Harvey P. Dale, May 15 2019 *)

a[0]:1$
a[1]:1$
a[n]:=4*a[n-1]+a[n-2]$
A015448(n):=a[n]$
makelist(A015448(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n) = fibonacci(3*n-1); \\ Altug Alkan, Dec 10 2015

a(n) = Fibonacci(3n-1) = ( (1+sqrt(5))*(2-sqrt(5))^n - (1-sqrt(5))*(2+sqrt(5))^n )/ (2*sqrt(5)).
O.g.f.: (1-3*x)/(1-4*x-x^2). - Len Smiley, Dec 09 2001
a(n) = Sum_{k=0..n} 3^k*A055830(n,k). - Philippe Deléham, Oct 18 2006
a(n) = upper left term in the 2 X 2 matrix [1,2; 2,3]^n. - Gary W. Adamson, Mar 02 2008
[a(n), A001076(n)] = [1,4; 1,3]^n * [1,0]. - Gary W. Adamson, Mar 21 2008
a(n) = A167808(3*n-1) for n > 0. - Reinhard Zumkeller, Nov 12 2009
a(n) = Fibonacci(3n+1) mod Fibonacci(3n), n > 0.
a(n) = (A000032(3*n)-Fibonacci(3*n))/2 = (A014448(n)-A014445(n))/2.
For n >= 2, a(n) = F_n(4) + F_(n+1)(4), where F_n(x) is a Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} binomial(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
a(n) = A001076(n+1) - 3*A001076(n). - R. J. Mathar, Jul 12 2012
From Gary Detlefs and Wolfdieter Lang, Aug 20 2012: (Start)
a(n) = (5*F(n)^3 + 5*F(n-1)^3 + 3*(-1)^n*F(n-2))/2,
a(n) = (F(n+1)^3 + 2*F(n)^3 - F(n-2)^3)/2, n >= 0, with F(-1) = 1 and F(-2) = -1. Second line from first one with 3*(-1)^n* F(n-2) = F(n-1)^3 - 4*F(n-2)^3 - F(n-3)^3 (in Koshy's book, p. 89, 32. (with a - sign) and 33. For the Koshy reference see A000045) and the F^3 recurrence (see row n=4 of A055870, or Koshy p. 87, 1.). First line from the preceding R. J. Mathar formula with F(3*n) = 5*F(n)^3 + 3*(-1)^n*F(n) (Koshy p. 89, 46.) and the above mentioned formula, Koshy's 32. and 33., with n -> n+2 in order to eliminate F(n+1)^3. (End)
For n > 0, a(n) = L(n-1)*L(n)*F(n) + F(n+1)*(-1)^n with L(n)=A000032(n). - J. M. Bergot, Dec 10 2015
For n > 1, a(n)^2 is the denominator of continued fraction [4,4,...,4, 6, 4,4,...4], which has n-1 4's before, and n-1 4's after, the middle 6. - Greg Dresden, Sep 18 2019
From Gary Detlefs and Wolfdieter Lang, Mar 06 2023: (Start)
a(n) = A001076(n) + A001076(n-1), with A001076(-1) = 1. See the R. J. Mathar formula above.
a(n+1) = i^n*(S(n-1,-4*i) - i*S(n-2,-4*i)), for n >= 0, with i = sqrt(-1), and the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. From the simplified Fibonacci trisection formula for {F(3*n+2)}_{n>=0}. (End)
a(n) = Sum_{k=0..n} A046854(n-1,k)*4^k. - R. J. Mathar, Feb 10 2024
E.g.f.: exp(2*x)*(5*cosh(sqrt(5)*x) - sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Jun 03 2024

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

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

A015552 a(n) = 6a(n-1) + 7a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 6, 43, 300, 2101, 14706, 102943, 720600, 5044201, 35309406, 247165843, 1730160900, 12111126301, 84777884106, 593445188743, 4154116321200, 29078814248401, 203551699738806, 1424861898171643, 9974033287201500, 69818233010410501, 488727631072873506

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_8. Example: a(2)=6 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGH are: ACB, ADB, AEB, AFB, AGB and AHB. - Emeric Deutsch, Apr 01 2004
General form: k=7^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The ratio a(n+1)/a(n) converges to 7 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

			G.f. = x + 6*x^2 + 43*x^3 + 300*x^4 + 2101*x^5 + 14706*x^6 + 102943*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 (6,7).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(7^n/8): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(7^n/8),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=7^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(7^n - (-1)^n)/8, {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

{a(n) = if ( n<0, 0, (7^n - (-1)^n) / 8)};

[lucas_number1(n,6,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 7*a(n-2).
From Emeric Deutsch, Apr 01 2004: (Start)
G.f.: x/(1-6*x-7*x^2).
a(n) = 7^(n-1) - a(n-1). (End)
a(n) = (7^n - (-1)^n)/8. - Rolf Pleisch, Jul 06 2009
a(n) = round(7^n/8). - Mircea Merca, Dec 28 2010
E.g.f. exp(3*x)*sinh(4*x)/4. - Elmo R. Oliveira, Aug 17 2024

A015540 a(n) = 5a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 5, 31, 185, 1111, 6665, 39991, 239945, 1439671, 8638025, 51828151, 310968905, 1865813431, 11194880585, 67169283511, 403015701065, 2418094206391, 14508565238345, 87051391430071, 522308348580425, 3133850091482551, 18803100548895305, 112818603293371831

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct vertices of the complete graph K_7. Example: a(2)=5 because the walks of length 2 between the vertices A and B of the complete graph ABCDEFG are ACB, ADB, AEB, AFB and AGB. - Emeric Deutsch, Apr 01 2004
General form: k=6^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500. -Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Pisano period lengths: 1, 1, 2, 2, 2, 2, 14, 2, 2, 2, 10, 2, 12, 14, 2, 2, 16, 2, 18, 2, ... - R. J. Mathar, Aug 10 2012
Sum_{i=0..m} (-1)^(m+i)*6^i, for m >= 0, gives all terms after 0. - Bruno Berselli, Aug 28 2013
The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. Also A053524, A080424, A051958. - Felix P. Muga II, Mar 09 2014

			G.f. = x + 5*x^2 + 31*x^3 + 185*x^4 + 1111*x^5 + 6665*x^6 + 39991*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (5,6).

Partial sums are in A033116. Cf. A014987.

[Floor(6^n/7-(-1)^n/7): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(6^n/7),n=0..25); # Mircea Merca, Dec 28 2010

k=0; lst={k}; Do[k = 6^n-k; AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[x / ((1 - 6 x) (1 + x)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{5,6},{0,1},30] (* Harvey P. Dale, May 12 2015 *)

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

a(n) = round(6^n/7); \\ Altug Alkan, Sep 05 2018

[lucas_number1(n,5,-6) for n in range(21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 6*a(n-2).
From Paul Barry, Apr 20 2003: (Start)
a(n) = (6^n - (-1)^n)/7.
G.f.: x/((1-6*x)*(1+x)).
E.g.f.: (exp(6*x) - exp(-x))/7. (End)
a(n) = 6^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004
a(n+1) = Sum_{k=0..n} binomial(n-k, k)*5^(n-2*k)*6^k. - Paul Barry, Jul 29 2004
a(n) = round(6^n/7). - Mircea Merca, Dec 28 2010
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-7)^k = (6^n - (-1)^n)/7 = (-1)^(n-1)*Sum_{k=0..n-1} (-6)^k. Equals (-1)^(n-1)*Phi(n,-6), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - Tom Copeland, Apr 14 2014

A109500 Number of closed walks of length n on the complete graph on 6 nodes from a given node.

Original entry on oeis.org

1, 0, 5, 20, 105, 520, 2605, 13020, 65105, 325520, 1627605, 8138020, 40690105, 203450520, 1017252605, 5086263020, 25431315105, 127156575520, 635782877605, 3178914388020, 15894571940105, 79472859700520

Offset: 0

Mitch Harris, Jun 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Christopher R. Kitching, Henri Kauhanen, Jordan Abbott, Deepthi Gopal, Ricardo Bermúdez-Otero, and Tobias Galla, Estimating transmission noise on networks from stationary local order, arXiv:2405.12023 [cond-mat.stat-mech], 2024. See p. 48.
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A109502.

Cf. sequences with the same recurrence form: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(5^n + 5*(-1)^n)/6: n in [0..30]]; // G. C. Greubel, Dec 30 2017

k=0;lst={k};Do[k=5^n-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[(1 - 4*x)/(1 - 4*x - 5*x^2), {x, 0, 50}], x] (* or *) Table[(5^n + 5*(-1)^n)/6, {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=0, 30, print1((5^n + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Dec 30 2017

G.f.: (1 - 4*x)/(1 - 4*x - 5*x^2).
a(n) = (5^n + 5*(-1)^n)/6.
a(n) = 5^(n-1) - a(n-1), a(0) = 1. - Jon E. Schoenfield, Feb 08 2015

Corrected by Franklin T. Adams-Watters, Sep 18 2006

Edited by Jon E. Schoenfield, Feb 08 2015

A015585 a(n) = 9a(n-1) + 10a(n-2).

Original entry on oeis.org

0, 1, 9, 91, 909, 9091, 90909, 909091, 9090909, 90909091, 909090909, 9090909091, 90909090909, 909090909091, 9090909090909, 90909090909091, 909090909090909, 9090909090909091, 90909090909090909, 909090909090909091, 9090909090909090909, 90909090909090909091

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_11. Example: a(2)=9 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJK are: ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB and AKB. - Emeric Deutsch, Apr 01 2004
Beginning with n=1 and a(1)=1, these are the positive integers whose balanced base-10 representations (A097150) are the first n digits of 1,-1,1,-1,.... Also, a(n) = (-1)^(n-1)*A014992(n) = |A014992(n)| for n >= 1. - Rick L. Shepherd, Jul 30 2004
General form: k=10^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Index entries for linear recurrences with constant coefficients, signature (9,10).

Cf. A014992 (q-integers for q=-10), A097150.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Cf. A094028, A098610, A098611.

[Round(10^n/11): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

k=0;lst={k};Do[k=10^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{9,10},{0,1},30] (* Harvey P. Dale, Aug 08 2014 *)

a(n)=10^n\/11 \\ Charles R Greathouse IV, Jun 24 2011

[lucas_number1(n,9,-10) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

[abs(gaussian_binomial(n,1,-10)) for n in range(0,19)] # Zerinvary Lajos, May 28 2009

a(n) = 9*a(n-1) + 10*a(n-2).
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 10^(n-1) - a(n-1).
G.f.: x/(1 - 9x - 10x^2). (End)
From Henry Bottomley, Sep 17 2004: (Start)
a(n) = round(10^n/11).
a(n) = (10^n - (-1)^n)/11.
a(n) = A098611(n)/11 = 9*A094028(n+1)/A098610(n). (End)
E.g.f.: exp(-x)*(exp(11*x) - 1)/11. - Elmo R. Oliveira, Aug 17 2024

Extended by T. D. Noe, May 23 2011

A109501 Number of closed walks of length n on the complete graph on 7 nodes from a given node.

Original entry on oeis.org

1, 0, 6, 30, 186, 1110, 6666, 39990, 239946, 1439670, 8638026, 51828150, 310968906, 1865813430, 11194880586, 67169283510, 403015701066, 2418094206390, 14508565238346, 87051391430070, 522308348580426, 3133850091482550, 18803100548895306, 112818603293371830

Offset: 0

Mitch Harris, Jun 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Christopher R. Kitching, Henri Kauhanen, Jordan Abbott, Deepthi Gopal, Ricardo Bermúdez-Otero, and Tobias Galla, Estimating transmission noise on networks from stationary local order, arXiv:2405.12023 [cond-mat.stat-mech], 2024. See p. 48.
Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A109502.

Cf. sequences with the same recurrence form: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A015540. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(6^n + 6*(-1)^n)/7: n in [0..30]]; // G. C. Greubel, Dec 30 2017

k=0;lst={k};Do[k=6^n-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[(1 - 5*x)/(1 - 5*x - 6*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{5,6}, {1,0}, 30] (* G. C. Greubel, Dec 30 2017 *)

for(n=0,30, print1((6^n + 6*(-1)^n)/7, ", ")) \\ G. C. Greubel, Dec 30 2017

G.f.: (1 - 5*x)/(1 - 5*x - 6*x^2).
a(n) = (6^n + 6*(-1)^n)/7.
a(n) = 6^(n-1) - a(n-1), a(0) = 1. - Jon E. Schoenfield, Feb 09 2015
a(n) = 5*a(n-1) + 6*a(n-2). - G. C. Greubel, Dec 30 2017
E.g.f.: exp(-x)*(exp(7*x) + 6)/7. - Elmo R. Oliveira, Aug 17 2024

Corrected by Franklin T. Adams-Watters, Sep 18 2006

Edited by Jon E. Schoenfield, Feb 09 2015

A015577 a(n+1) = 8a(n) + 9a(n-1), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 8, 73, 656, 5905, 53144, 478297, 4304672, 38742049, 348678440, 3138105961, 28242953648, 254186582833, 2287679245496, 20589113209465, 185302018885184, 1667718169966657, 15009463529699912, 135085171767299209, 1215766545905692880, 10941898913151235921

Offset: 0

Olivier Gérard

Binomial transform is A011557, with a leading zero. - Paul Barry, Jul 09 2003
Number of walks of length n between any two distinct nodes of the complete graph K_10. Example: a(2) = 8 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJ are: ACB, ADB, AEB, AFB, AGB, AHB, AIB and AJB. - Emeric Deutsch, Apr 01 2004
The ratio a(n+1)/a(n) converges to 9 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (8,9).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(9^n/10): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(9^n/10),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=9^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(9^n - (-1)^n)/10, {n,0,30}] (* or *) LinearRecurrence[{8,9}, {0,1}, 30] (* G. C. Greubel, Jan 06 2018 *)

a[0]:0$
a[n]:=9^(n-1)-a[n-1]$
A015577(n):=a[n]$
makelist(A015577(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A015577_vec(N=20)=Vec(O(x^N)+1/(1-8*x-9*x^2), -N-1) \\ M. F. Hasler, Jun 14 2008, edited Oct 25 2019

for(n=0,30, print1((9^n - (-1)^n)/10, ", ")) \\ G. C. Greubel, Jan 06 2018

apply( {A015577(n)=9^n\/10}, [0..25]) \\ M. F. Hasler, Oct 25 2019

[lucas_number1(n,8,-9) for n in range(0, 19)] # Zerinvary Lajos, Apr 25 2009

From Paul Barry, Jul 09 2003: (Start)
G.f.: x/((1+x)*(1-9*x)).
E.g.f. exp(4*x)*sinh(5*x)/5.
a(n) = (9^n - (-1)^n)/10. (End)
a(n) = 9^(n-1)-a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = round(9^n/10). - Mircea Merca, Dec 28 2010

Extended by T. D. Noe, May 23 2011

A093134 A Jacobsthal trisection.

Original entry on oeis.org

1, 0, 8, 56, 456, 3640, 29128, 233016, 1864136, 14913080, 119304648, 954437176, 7635497416, 61083979320, 488671834568, 3909374676536, 31274997412296, 250199979298360, 2001599834386888, 16012798675095096, 128102389400760776, 1024819115206086200, 8198552921648689608

Offset: 0

Paul Barry, Mar 23 2004

Counts closed walks at a vertex of the complete graph on 9 nodes K_9.

Second binomial transform is A047855.

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

Other sequences with a(n+1) = 8^n - a(n) are A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Cf. A047855.

[(8^n/9+8*(-1)^n/9): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

k=0;lst={1, k};Do[k=8^n-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(8^n + 8*(-1)^n)/9, {n,0,30}] (* or *) LinearRecurrence[{7,8}, {1,0}, 30] (* G. C. Greubel, Jan 06 2018 *)

for(n=0,30, print1((8^n + 8*(-1)^n)/9, ", ")) \\ G. C. Greubel, Jan 06 2018

G.f.: (1-7*x)/(1 - 7*x - 8*x^2).
a(n) = (8^n + 8*(-1)^n)/9.
a(n) = 8*A001045(3*n-3)/3.
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(-x)*(exp(9*x) + 8)/9.
a(n) = 7*a(n-1) + 8*a(n-2) for n > 1. (End)

cp's OEIS Frontend

A015448 a(0) = 1, a(1) = 1, and a(n) = 4*a(n-1) + a(n-2) for n >= 2.

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

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

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A015552 a(n) = 6*a(n-1) + 7*a(n-2), a(0) = 0, a(1) = 1.

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A015540 a(n) = 5*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.

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A109500 Number of closed walks of length n on the complete graph on 6 nodes from a given node.

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

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

A015552 a(n) = 6a(n-1) + 7a(n-2), a(0) = 0, a(1) = 1.

A015540 a(n) = 5a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

A015585 a(n) = 9a(n-1) + 10a(n-2).

A015577 a(n+1) = 8a(n) + 9a(n-1), a(0) = 0, a(1) = 1.