A253368 -id:A253368 - cp's OEIS frontend

A004187 a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784, 34111385, 233802911, 1602508992, 10983760033, 75283811239, 516002918640, 3536736619241, 24241153416047, 166151337293088, 1138818207635569, 7805576116155895, 53500214605455696, 366695926122033977

Offset: 0

N. J. A. Sloane, R. K. Guy

Define the sequence T(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}= 0 . A004187 (with initial 0 omitted) is T(1,7).
This is a divisibility sequence.
For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 7's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
a(n) and b(n) := A056854(n) are the proper and improper nonnegative solutions of the Pell equation b(n)^2 - 5*(3*a(n))^2 = +4. see the cross-reference to A056854 below. - Wolfdieter Lang, Jun 26 2013
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,2,3,4,5,6}. - Milan Janjic, Jan 25 2015
The digital root is A253298, which shares its digital root with A253368. - Peter M. Chema, Jul 04 2016
Lim_{n->oo} a(n+1)/a(n) = 2 + 3*phi = 1+ A090550 = 6.854101...  - Wolfdieter Lang, Nov 16 2023

			a(2) = 7*a(1) - a(0) = 7*7 - 1 = 48. - _Michael B. Porter_, Jul 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 12
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
Zvonko Cerin, Some alternating sums of Lucas numbers, Centr. Eur. J. Math. vol 3 no 1 (2005) 1-13.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=7, q=-1.
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
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=9.
Index entries for sequences related to Chebyshev polynomials.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).
Index entries for Pisot sequences

Cf. A000027, A001906, A001353, A004254, A001109, A049685, A033888. a(n)=sqrt((A056854(n)^2 - 4)/45).

Second column of array A028412.

[Fibonacci(4*n)/3 : n in [0..30]]; // Vincenzo Librandi, Jun 07 2011

/* By definition: */ [n le 2 select n-1 else 7*Self(n-1)-Self(n-2): n in [1..23]]; // Bruno Berselli, Dec 24 2012

seq(combinat:-fibonacci(4*n)/3, n = 0 .. 30); # Robert Israel, Jan 26 2015

LinearRecurrence[{7,-1},{0,1},30] (* Harvey P. Dale, Jul 13 2011 *)
CoefficientList[Series[x/(1 - 7*x + x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 23 2012 *)

a[0]:0$ a[1]:1$ a[n]:=7*a[n-1] - a[n-2]$ A004187(n):=a[n]$ makelist(A004187(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

numlib::fibonacci(4*n)/3 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(4*n)/3 \\ Charles R Greathouse IV, Mar 09 2012

concat(0, Vec(x/(1-7*x+x^2) + O(x^99))) \\ Altug Alkan, Jul 03 2016

[lucas_number1(n,7,1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

[fibonacci(4*n)/3 for n in range(0, 21)] # Zerinvary Lajos, May 15 2009

G.f.: x/(1-7*x+x^2).
a(n) = F(4*n)/3 = A033888(n)/3, where F=A000045 (the Fibonacci sequence).
a(n) = S(2*n-1, sqrt(9))/sqrt(9) = S(n-1, 7); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n) = Sum_{i = 0..n-1} C(2*n-1-i, i)*5^(n-i-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
[A049685(n-1), a(n)] = [1,5; 1,6]^n * [1,0]. - Gary W. Adamson, Mar 21 2008
a(n) = A167816(4*n). - Reinhard Zumkeller, Nov 13 2009
a(n) = (((7+sqrt(45))/2)^n-((7-sqrt(45))/2)^n)/sqrt(45). - Noureddine Chair, Aug 31 2011
a(n+1) = Sum_{k = 0..n} A101950(n,k)*6^k. - Philippe Deléham, Feb 10 2012
a(n) = (A081072(n)/3)-1. - Martin Ettl, Nov 11 2012
From Peter Bala, Dec 23 2012: (Start)
Product {n >= 1} (1 + 1/a(n)) = (1/5)*(5 + 3*sqrt(5)).
Product {n >= 2} (1 - 1/a(n)) = (1/14)*(5 + 3*sqrt(5)). (End)
From Peter Bala, Apr 02 2015: (Start)
Sum_{n >= 1} a(n)*x^(2*n) = -A(x)*A(-x), where A(x) = Sum_{n >= 1} Fibonacci(2*n)* x^n.
1 + 5*Sum_{n >= 1} a(n)*x^(2*n) = F(x)*F(-x) = G(x)*G(-x), where F(x) = 1 + A(x) and G(x) = 1 + 5*A(x).
1 + Sum_{n >= 1} a(n)*x^(2*n) = H(x)*H(-x) = I(x)*I(-x), where H(x) = 1 + Sum_{n >= 1} Fibonacci(2*n + 3)*x^n and I(x) = 1 + x + x*Sum_{n >= 1} Fibonacci(2*n - 1)*x^n. (End)
E.g.f.: 2*exp(7*x/2)*sinh(3*sqrt(5)*x/2)/(3*sqrt(5)). - Ilya Gutkovskiy, Jul 03 2016
a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*9^k*binomial(n+k, 2*k+1). - Peter Bala, Jul 17 2023
a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*7^(n-2*k)*binomial(n-k, k). - Greg Dresden, Aug 03 2024
From Peter Bala, Jul 22 2025: (Start)
The following products telescope:
Product {n >= 2} (1 + (-1)^n/a(n)) = (3/14)*(3 + sqrt(5)).
Product {n >= 1} (1 - (-1)^n/a(n)) = (1/3)*(3 + sqrt(5)).
Product_{n >= 1} (a(2*n) + 1)/(a(2*n) - 1) = (3/5)*sqrt(5). (End)

Entry improved by comments from Michael Somos and Wolfdieter Lang, Aug 02 2000

A298271 Expansion of x/((1 - x)(1 - 322x + x^2)).

Original entry on oeis.org

0, 1, 323, 104006, 33489610, 10783550415, 3472269744021, 1118060074024348, 360011871566096036, 115922704584208899245, 37326750864243699460855, 12019097855581887017496066, 3870112182746503375934272398, 1246164103746518505163818216091

Offset: 0

Bruno Berselli, Jan 16 2018

Colin Barker, Table of n, a(n) for n = 0..399
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A000045, A253368, A298101.

CoefficientList[x/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]

makelist(coeff(taylor(x/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);

a(n)=([0,1,0; 0,0,1; 1,-323,323]^n*[0;1;323])[1,1] \\ Charles R Greathouse IV, Jan 18 2018

concat(0, Vec(x / ((1 - x)*(1 - 322*x + x^2)) + O(x^15))) \\ Colin Barker, Jan 19 2018

gf = x/((1-x)*(1-322*x+x^2))
print(taylor(gf, x, 0, 20).list())

G.f.: x/((1 - x)*(1 - 322*x + x^2)).
a(n) = a(-n-1) = 323*a(n-1) - 323*a(n-2) + a(n-3).
a(n) = (1/5760)*((2 + sqrt(5))^(4*n+2) + (2 + sqrt(5))^-(4*n+2) - 18).
a(n) = A298101(n) - A298101(n-1) + A298101(n-2) - A298101(n-3) + ..., hence:
a(n) + a(n-1) = A298101(n).
a(n) - a(n-1) = (1/144)*Fibonacci(12*n).
a(n) - a(n-2) = (1/8)*Fibonacci(12*n-6).
a(n)*a(n-2) = a(n-1)*(a(n-1) - 1).
Sum_{j>1} 1/a(j) = 161 - 72*sqrt(5) = A094214^12.
a(n) = A157459(n+1)/72. - Greg Dresden, Dec 02 2021

A298101 Expansion of x(1 + x)/((1 - x)(1 - 322*x + x^2)).

Original entry on oeis.org

0, 1, 324, 104329, 33593616, 10817040025, 3483053294436, 1121532343768369, 361129931640120384, 116282716455774995281, 37442673568827908360100, 12056424606446130716956921, 3882131280602085262951768464, 1250034215929265008539752488489

Offset: 0

Bruno Berselli, Jan 12 2018

16*k*a(n) provides infinitely many x-values solutions (x,y) of x*(5*x + k) = y^2.
This follows from the fact that 5*16*a(n) + 1 is a perfect square: more precisely, 80*a(n) + 1 = A023039(n)^2.
This is a divisibility sequence, that is a(n) divides a(m) if n divides m. It is the case P1 = 324, P2 = 644, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Jan 19 2018

Colin Barker, Table of n, a(n) for n = 0..300
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A023039, A049660, A225786, A253368, A298271.

P:=proc(n) trunc(evalf(((2+sqrt(5))^(4*n)+(2-sqrt(5))^(4*n)-2)/320,1000));
end: seq(P(i),i=0..13); # Paolo P. Lava, Jan 18 2018

CoefficientList[x (1 + x)/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]

makelist(coeff(taylor(x*(1+x)/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);

first(n) = Vec(x*(1 + x)/((1 - x)*(1 - 322*x + x^2)) + O(x^n), -n) \\ Iain Fox, Jan 12 2018

gf = x*(1+x)/((1-x)*(1-322*x+x^2))
print(taylor(gf, x, 0, 20).list())

G.f.: x*(1 + x)/((1 - x)*(1 - 322*x + x^2)).
a(n) = a(-n) = ((2 + sqrt(5))^(4*n) + (2 - sqrt(5))^(4*n) - 2)/320.
a(n) = A225786(n)/48. This is the case k=3 of the first comment. Example: for n = 2, 16*3*a(2) = A225786(2) = 15552 and 15552*(5*15552+3) = 34776^2.
a(n) = A049660(n)^2.
a(n)*(80*a(n) + 1) = 81*A253368(n)^2 for n>0.
a(n)*a(n-2) = (a(n-1) - 1)^2.
a(n) = 322*a(n-1) - a(n-2) + 2.
a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3). - Iain Fox, Jan 12 2018
a(n) = A298271(n)+A298271(n-1). - R. J. Mathar, Nov 20 2020

A253298 Digital root for the following sequences, F(4n)/F(4); F(12n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

Original entry on oeis.org

1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9, 1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9, 1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9, 1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9

Offset: 1

Peter M. Chema, Dec 30 2014

Cyclical and palindromic in two parts with periodicity 18: {1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9}.
Digital root of the period is 9, its mean and median is 5, and its product is (9!)^2.
See A253368 for the initial motivation for this sequence.
From Peter M. Chema, Jul 04 2016: (Start)
A composite of three respective digital root sequences in alternation: a "halving sequence" of 1, 5, 7, 8, 4, 2, a "doubling sequence" of 7, 5, 1, 2, 4, 8, and a three-six-nine circuit of 3, 3, 9, 6, 6, 9.
Also the digital root of A000045(4n)/3 or A004187(n).  In general terms, sequences defined by Fib(x*n)/ Fib(x) where x=(8*a-4) all share the same digital root (e.g., F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20); F(28*n)/F(28); F(36*n)/F(36), etc.) (End)

Tom Barnett, Phi VBM Tori Array, YouTube video (see first two minutes).
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).

Cf. A010888, A253368, A070366 and A001370.

f[n_] := Mod[ Fibonacci[ 12n]/144, 9]; Array[f, 5*18] (* Robert G. Wilson v, Jan 23 2015 *)
LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1},{1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8},72] (* Ray Chandler, Aug 12 2015 *)

a(n) = A010888(A253368(n)).

G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + x^7 + 9*x^8 + 8*x^9 + 2*x^10 + 6*x^11 + 4*x^12 + 4*x^13 + 6*x^14 + 2*x^15 + 8*x^16 + 9*x^17)/(1 - x^18). - Vincenzo Librandi, Mar 28 2016

Edited. Numbers and name changed to fit A253368. Formula adapted. Cross reference added. - Wolfdieter Lang, Jan 28 2015

Name generalized by Peter M. Chema, Jul 04 2016

A305413 a(n) = Fibonacci(11*n)/89.

Original entry on oeis.org

0, 1, 199, 39602, 7880997, 1568358005, 312111123992, 62111682032413, 12360536835574179, 2459808941961294034, 489514339987133086945, 97415813466381445596089, 19386236394149894806708656, 3857958458249295447980618633, 767753119428003944042949816623

Offset: 0

Vincenzo Librandi, Jun 05 2018

S. Falcon, Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence, Journal of Mathematics Research, Vol. 4, No. 2 (2012), 97-100.
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (199,1).

Cf. A000045, A167398.

Cf. similar sequences: F(3*n)/2 (A001076), F(4*n)/3 (A004187), F(5*n)/5 (A049666), F(6*n)/8 (A049660), F(7*n)/13 (A049667), F(8*n)/21 (A049668), F(9*n)/34 (A049669), F(10*n)/55 (A049670), F(11*n)/89 (this sequence), F(12*n)/144 (A253368).

Magma
```
[Fibonacci(11*n)/89: n in [0..30]];
```

Fibonacci[11 Range[0, 20]]/89
LinearRecurrence[{199,1},{0,1},20] (* Harvey P. Dale, Aug 03 2024 *)

a(n) = fibonacci(11*n)/89 \\ Felix Fröhlich, Jul 30 2019

G.f.: x/(1 - 199*x - x^2).
a(n) = 199*a(n-1) + a(n-2) for n>1, a(0)=0, a(1)=1.
a(n) = A167398(n)/89.
For n >= 1, a(n) equals the denominator of the continued fraction [199, 199, ..., 199] (with n copies of 199). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 29 2019

A319197 All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)3m) for n >= 3 and all m >= 0.

Original entry on oeis.org

1, 9, 161, 51841, 6989569, 53156199689438143, 5581524253378492696105796918365541568492478783, 89171045849445921581733341920411050611581102638589828325078491812417901966295041

Offset: 3

Wolfdieter Lang, Oct 09 2018

nonn

It appears that Fibonacci(2^(n-2)*3*m)/(2^n) is a nonnegative integer for n >= 3 and all m >= 0. The remaining factor for n >= 3 is given by A(n) := Product_{j=3..n} a(j). For n = 3 and 4 see A049660 and A253368, respectively. Thus the conjecture is that I(n; m) := Fibonacci(2^(n-2)*3*m) / ((2^n)*A(n)) is a nonnegative integer for all m >= 0. This is best possible for all m (because for m = 1 this becomes 1); special m may allow more factors. E.g., n = 3: 8 | Fibonacci(6*m), for all m >= 0, but for even m larger powers of 2 than 2^3 appear; for m = 0 any power of 2.

The factorizations of a(n) are: 1, 3^2, 7*23, 47*1103, 3167*2207, 4481*11862575248703, 127*383*769*1087*5662847*6803327*19073614849*186812208641, 885503*119809*1359361*1769526527*4698167634523379875583*74374487830388825730162393840383, ...

			n=3: I(3; m) = A049660(m), m >= 0.
n=4: I(4; m) = A253368(m), with A253368(0) := 0.
n = 5: I(5; m) =  F(24*m)/(2^5*9*161) = F(24*m)/(2^5*3^2*7*23) = [0, 1, 103682, 10749957123, 1114577054323204, ...]

Cf. A000045, A049660, A253368.

I(n; m) := F(2^(n-2)*3*m) / ((2^n)* Product_{j=3..n} a(j)) is conjectured to be a nonnegative integer for n >= 3 and all m >= 0, where F = A000045. There are no more factors > 1 for all m >= 0 because I(n, 1) = 1.

cp's OEIS Frontend

A004187 a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

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A298271 Expansion of x/((1 - x)*(1 - 322*x + x^2)).

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A298101 Expansion of x*(1 + x)/((1 - x)*(1 - 322*x + x^2)).

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A253298 Digital root for the following sequences, F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

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A305413 a(n) = Fibonacci(11*n)/89.

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A319197 All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)*3*m) for n >= 3 and all m >= 0.

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A298271 Expansion of x/((1 - x)(1 - 322x + x^2)).

A298101 Expansion of x(1 + x)/((1 - x)(1 - 322*x + x^2)).

A253298 Digital root for the following sequences, F(4n)/F(4); F(12n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

A319197 All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)3m) for n >= 3 and all m >= 0.