A203976 -id:A203976 - cp's OEIS frontend

A022998 If n is odd then n, otherwise 2n.

0, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128, 65, 132, 67

Offset: 0

N. J. A. Sloane

Also for n > 0: numerator of Sum_{i=1..n} 2/(i*(i+1)), denominator=A026741. - Reinhard Zumkeller, Jul 25 2002
For n > 2: a(n) = gcd(A143051((n-1)^2), A143051(1+(n-1)^2)) = A050873(A000290(n-1), A002522(n-1)). - Reinhard Zumkeller, Jul 20 2008
Partial sums give the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Multiples of 4 and odd numbers interleaved. - Omar E. Pol, Sep 25 2011
The Pisano period lengths modulo m appear to be A066043(m). - R. J. Mathar, Oct 08 2011
The partial sums a(n)/A026741(n+1) given by R. Zumkeller in a comment above are 2*n/(n+1) (telescopic sum), and thus converge to 2. - Wolfdieter Lang, Apr 09 2013
a(n) = numerator(H(n,1)), where H(n,1) = 2*n/(n+1) is the harmonic mean of 1 and n. a(n) = 2*n/gcd(2n, n+1) = 2*n/gcd(n+1,2). a(n) = A227041(n,1), n>=1. - Wolfdieter Lang, Jul 04 2013
a(n) = numerator of the mean (2n/(n+1), after reduction), of the compositions of n; denominator is given by A001792(n-1). - Clark Kimberling, Mar 11 2014
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of numerators. The sequence of denominators of the continued fraction convergents [1, 1, 3, 2, 5, 3, 7, ...] is A026741, also a strong divisibility sequence. Cf. A203976. - Peter Bala, May 19 2014
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized octagonal numbers. - Omar E. Pol, Jul 27 2018
a(n) is the number of petals of the Rhodonea curve r = a*cos(n*theta) or r = a*sin(n*theta). - Matt Westwood, Nov 19 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rose
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A026741, A059026, A203976, A345082.
Column 4 of A195151. - Omar E. Pol, Sep 25 2011
Cf. A000034, A001082 (partial sums).
Cf. A227041 (first column). - Wolfdieter Lang, Jul 04 2013
Row 2 of A349593. A385555, A385556, A385557, A385558, A385559, and A385560 are respectively rows 3, 4, 5-6, 7, 8, and 9-10.

a022998 n = a000034 (n + 1) * n
a022998_list = zipWith (*) [0..] $ tail a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[((-1)^n+3)*n/2: n in [0..70]]; // Vincenzo Librandi, Sep 17 2011

A022998 := proc(n) if type(n,'odd') then n ; else 2*n; end if; end proc: # R. J. Mathar, Mar 10 2011

Table[n (3 + (-1)^n)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 13 2013 *)
Table[If[OddQ[n],n,2n],{n,0,150}] (* or *) Riffle[ 2*Range[ 0,150,2], Range[ 1,150,2]] (* Harvey P. Dale, Feb 06 2017 *)

PARI
```
a(n)=if(n%2,n,2*n)
```

def A022998(n): return n if n&1 else n<<1 # Chai Wah Wu, Mar 05 2024

[n*(1+((n+1)%2)) for n in (0..80)] # G. C. Greubel, Jul 31 2022

Denominator of (n+1)*(n-1)*(2*n+1)/(2*n) (for n > 0).
a(n+1) = lcm(n, n+2)/n + lcm(n, n+2)/(n+2) for all n >= 1. - Asher Auel, Dec 15 2000
Multiplicative with a(2^e) = 2^(e+1), a(p^e) = p^e, p > 2.
G.f. x*(1 + 4*x + x^2)/(1-x^2)^2. - Ralf Stephan, Jun 10 2003
a(n) = 3*n/2 + n*(-1)^n/2 = n*(3 + (-1)^n)/2. - Paul Barry, Sep 04 2003
a(n) = A059029(n-1) + 1 = A043547(n+2) - 2.
a(n)*a(n+3) = -4 + a(n+1)*a(n+2).
a(n) = n*(((n+1) mod 2) + 1) = n^2 + 2*n - 2*n*floor((n+1)/2). - William A. Tedeschi, Feb 29 2008
a(n) = denominator((n+1)/(2*n)) for n >= 1; A026741(n+1) = numerator((n+1)/(2*n)) for n >= 1. - Johannes W. Meijer, Jun 18 2009
a(n) = 2*a(n-2) - a(n-4).
Dirichlet g.f. zeta(s-1)*(1+2^(1-s)). - R. J. Mathar, Mar 10 2011
a(n) = n * (2 - n mod 2) = n * A000034(n+1). - Reinhard Zumkeller, Mar 31 2012
a(n) = floor(2*n/(1 + (n mod 2))). - Wesley Ivan Hurt, Dec 13 2013
From Ilya Gutkovskiy, Mar 16 2017: (Start)
E.g.f.: x*(2*sinh(x) + cosh(x)).
It appears that a(n) is the period of the sequence k*(k + 1)/2 mod n. (End) [This is correct; see A349593. - Jianing Song, Jul 03 2025]
a(n) = Sum_{d | n} A345082(d). - Peter Bala, Jan 13 2024

More terms from Michael Somos, Aug 07 2000

A014217 a(n) = floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 29, 46, 76, 122, 199, 321, 521, 842, 1364, 2206, 3571, 5777, 9349, 15126, 24476, 39602, 64079, 103681, 167761, 271442, 439204, 710646, 1149851, 1860497, 3010349, 4870846, 7881196, 12752042, 20633239, 33385281, 54018521, 87403802

Offset: 0

Clark Kimberling

a(n) = floor(lim_{k->oo} Fibonacci(k)/Fibonacci(k-n)). - Jon Perry, Jun 10 2003
For n > 1, a(n) is the maximum element in the continued fraction for A000045(n)*phi. - Benoit Cloitre, Jun 19 2005
a(n) is also the curvature (rounded down) of the circle inscribed in the n-th kite arranged in a spiral, starting with a unit circle, as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 29 2013
a(n) is the n-th Lucas number (A000032) if n is odd, and a(n) is the n-th Lucas number minus 1 if n is even. (Mario Catalani's formula below expresses this fact.) This is related to the fact that the powers of phi approach the values of the Lucas numbers, the odd powers from above and the even powers from below. - Geoffrey Caveney, Apr 18 2014
a(n) is the sum of the last summands over all Arndt compositions of n (see the Checa link). - Daniel Checa, Dec 25 2023
a(n) is the number of (saturated or unsaturated) substituted N-heterocycles in chemistry (N = nitrogen). That means the number of matchings in a cycle graph when the two maximum matchings in every cycle with an even number of vertices are indistinguishable (because the corresponding resonance structures in the molecule are equivalent). - Stefan Schuster, Mar 20 2025

Alois P. Heinz, Table of n, a(n) for n = 0..4784 (first 301 terms from T. D. Noe)
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.
Daniel F. Checa, Arndt Compositions: Connections with Fibonacci Numbers, Statistics, and Generalizations, 2023. p. 29.
Daniel F. Checa and José L. Ramírez, Arndt Compositions: A Generating Functions Approach, Integers (2024) Vol. 24, A35. See p. 14.
G. Harman, One hundred years of normal numbers
Kival Ngaokrajang, Illustration for n = 0..7
Stefan Schuster and Tatjana Malycheva, Enumeration of saturated and unsaturated substituted N-heterocycles, arXiv:2309.02343 [q-bio.BM], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A000032, A000045, A001622, A020956, A022319, A052952, A057146, A062114, A128440, A153382, A169985, A169986, A203976, A226328.

First differences give A181716.

a014217 n = a014217_list !! n
a014217_list = 1 : 1 : zipWith (+)
   a000035_list (zipWith (+) a014217_list $ tail a014217_list)
-- Reinhard Zumkeller, Jan 06 2012

[Floor( ((1+Sqrt(5))/2)^n ): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

A014217 := proc(n)
    option remember;
    if n <= 3 then
        op(n+1,[1,1,2,4]) ;
    else
        procname(n-1)+2*procname(n-2)-procname(n-3)-procname(n-4) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013
#
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-1|2|1>>^n. <<1, 1, 2, 4>>)[1, 1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 12 2017

Table[Floor[GoldenRatio^n], {n, 0, 36}] (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)
LinearRecurrence[{1, 2, -1, -1}, {1, 1, 2, 4}, 40] (* Jean-François Alcover, Nov 05 2017 *)

my(x='x+O('x^44)); Vec((1-x^2+x^3)/((1+x)*(1-x)*(1-x-x^2))) \\ Joerg Arndt, Jul 10 2023

from sympy import floor, sqrt
def A014217(n): return floor(((1+sqrt(5))/2)**n) # Chai Wah Wu, Dec 17 2021

[floor(golden_ratio^n) for n in range(37)] # Danny Rorabaugh, Apr 19 2015

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
a(n) = a(n-1) + a(n-2) + (1-(-1)^n)/2 = a(n-1) + a(n-2) + A000035(n).
a(n) = A000032(n) - (1 + (-1)^n)/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 17 2003
G.f.: (1-x^2+x^3)/((1+x)*(1-x)*(1-x-x^2)). - R. J. Mathar, Sep 06 2008
a(2n-1) = (Fibonacci(4n+1)-2)/Fibonacci(2n+2). - Gary Detlefs, Feb 16 2011
a(n) = floor(Fibonacci(2n+3)/Fibonacci(n+3)). - Gary Detlefs, Feb 28 2011
a(2n) = Fibonacci(2*n-1) + Fibonacci(2*n+1) - 1. - Gary Detlefs, Mar 10 2011
a(n+2*k) - a(n) = A203976(k)*A000032(n+k) if k odd, a(n+2*k) - a(n) = A203976(k)*A000045(n+k) if k even, for k > 0. - Paul Curtz, Jun 05 2013
a(n) = A052952(n) - A052952(n-2) + A052952(n-3). - R. J. Mathar, Jun 13 2013
a(n+6) - a(n-6) = 40*A000045(n), case k=6 of my formula above. - Paul Curtz, Jun 13 2013
From Paul Curtz, Jun 17 2013: (Start)
a(n-3) + a(n+3) = A153382(n).
a(n-1) + a(n+2) = A022319(n). (End)
For k > 0, a(2k) = A169985(2k)-1 and a(2k+1) = A169985(2k+1) (which is equivalent to Catalani's 2003 formula). - Danny Rorabaugh, Apr 15 2015
a(n) = ((-1)^(1+n)-1)/2 + ((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n. - Colin Barker, Nov 05 2017
a(n) = floor(2*sinh(n*arccsch(2))). - Federico Provvedi, Feb 23 2022
E.g.f.: 2*exp(x/2)*cosh(sqrt(5)*x/2) - cosh(x). - Stefano Spezia, Jul 26 2022
a(n) = floor(Fibonacci(n)*phi) + Fibonacci(n-1) = A074331(n) + A000045(n-1) = A052952(n-1) + A000045(n-1). This is the case k=1 of the formula (also found in A128440): floor(k * phi^n) = floor(Fibonacci(n)*k*phi) + Fibonacci(n-1) * k. - Chunqing Liu, Oct 03 2023

Corrected by T. D. Noe, Nov 09 2006

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A054888 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle Pi/2.

Original entry on oeis.org

1, 5, 15, 40, 105, 275, 720, 1885, 4935, 12920, 33825, 88555, 231840, 606965, 1589055, 4160200, 10891545, 28514435, 74651760, 195440845, 511670775, 1339571480, 3507043665, 9181559515, 24037634880, 62931345125

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

The layer sequence is the sequence of the cardinalities of the layers accumulating around a (finite-sided) polygon of the tessellation under successive side-reflections.

Reinhard Zumkeller, Table of n, a(n) for n = 0..999 (indices corrected to start at zero by _Sidney Cadot_, Jan 07 2022)
Paolo Dominici, Illustration
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045, A001906, A002878, A004277, A203976.

Cf. A054886, A054887, A054889, A054890.

a054888 n = a054888_list !! (n-1)
a054888_list = 1 : zipWith (+) (tail a002878_list) a002878_list
-- Reinhard Zumkeller, Jan 11 2012

[n eq 0 select 1 else 5*Fibonacci(2*n): n in [0..40]]; // G. C. Greubel, Feb 08 2023

LinearRecurrence[{3,-1},{1,5,15},30] (* Harvey P. Dale, Jan 15 2023 *)
Join[{1}, 5*Fibonacci[2*Range[40]]] (* G. C. Greubel, Feb 08 2023 *)

{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^2*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna, Feb 21 2012 */

[5*fibonacci(2*n) + int(n==0) for n in range (41)] # G. C. Greubel, Feb 08 2023

a(n) = 5*A001906(n) + [n=0].
G.f.: (1+x)^2/(1-3*x+x^2).
G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^2 * x^n/n ). - Paul D. Hanna, Feb 21 2012
a(n) = A001906(n-1) + 2*A001906(n) + A001906(n+1). - R. J. Mathar, Nov 28 2011
a(n) = A203976(A004277(n-1)). - Reinhard Zumkeller, Jan 11 2012
a(n) = 5*A000045(2*n) for n >= 1. - Robert Israel, Jun 01 2015
a(n) = A002878(n-1)+A002878(n). - R. J. Mathar, Jul 09 2024

Offset changed to 0 by N. J. A. Sloane, Jan 03 2022 at the suggestion of Michel Marcus

A005013 a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=1, a(3)=4. Alternates Fibonacci (A000045) and Lucas (A000032) sequences for even and odd n.

Original entry on oeis.org

0, 1, 1, 4, 3, 11, 8, 29, 21, 76, 55, 199, 144, 521, 377, 1364, 987, 3571, 2584, 9349, 6765, 24476, 17711, 64079, 46368, 167761, 121393, 439204, 317811, 1149851, 832040, 3010349, 2178309, 7881196, 5702887, 20633239, 14930352, 54018521, 39088169, 141422324

Offset: 0

N. J. A. Sloane

S(n,sqrt(5)), with the Chebyshev polynomials A049310, is an integer sequence in the real quadratic number field Q(sqrt(5)) with basis numbers <1,phi>, phi:=(1+sqrt(5))/2. S(n,sqrt(5)) = A(n) + 2*B(n)*phi, with A(n)= a(n+1)*(-1)^n and B(n)= A147600(n-1), n>=0, with A147600(-1):=0.
a(n) = p(n+1) where p(x) is the unique degree-(n-1) polynomial such that p(k) = Fibonacci(k) for k = 1, ..., n. - Michael Somos, Jan 08 2012
Row sums of A227431. - Richard R. Forberg, Jul 29 2013
This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 5 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence satisfies a linear recurrence of order four. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -5, 1, -5, ...] = 1/(1 - 1/(5 - 1/(1 - 1/(5 - ...)))) = (1/2)*(5 - sqrt(5)) begins [0/1, 1/1, 5/4, 4/3, 15/11, 11/8, 40/29, ...]; the denominators give the present sequence. The sequence of numerators [0, 1, 5, 4, 15, 11, 40, ...] is A203976. Cf. A108412 and A026741. - Peter Bala, May 19 2014
Define a binary operation o on the real numbers by x o y = x*sqrt(1 + y^2) + y*sqrt(1 + x^2). The operation o is commutative and associative with identity 0. We have (1/2)*a(2*n + 1) = 1/2 o 1/2 o ... o 1/2 (2*n + 1 terms) and (1/2)*sqrt(5)* a(2*n) = 1/2 o 1/2 o ... o 1/2 (2*n terms). Cf. A084068 and A049629. - Peter Bala, Mar 23 2018

			G.f. = x + x^2 + 4*x^3 + 3*x^4 + 11*x^5 + 8*x^6 + 29*x^7 + 21*x^8 + 76*x^9 + ...
a(3) = 4 since p(x) = (x^2 - 3*x + 4) / 2 interpolates p(1) = 1, p(2) = 1, p(3) = 2, and p(4) = 4. - _Michael Somos_, Jan 08 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..2000 (terms 0..500 from T. D. Noe)
A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979. [Annotated scanned copy]
Seong Ju Kim, R. Stees and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
E. W. Weisstein, MathWorld: Lehmer Number
H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000032, A000045, A001906, A002878, A005247, A096140, A026741, A108412, A084068, A049629.

a:=[0,1,1,4];; for n in [5..40] do a[n]:=3*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Oct 21 2018

a005013 n = a005013_list !! n
a005013_list = alt a000045_list a000032_list where
   alt (f::fs) (:l:ls) = f : l : alt fs ls
-- Reinhard Zumkeller, Jan 10 2012

I:=[0,1,1,4]; [n le 4 select I[n]  else 3*Self(n-2) - Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 09 2016

with(combinat): A005013 := n-> if n mod 2 = 0 then fibonacci(n) else fibonacci(n+1)+fibonacci(n-1); fi;
A005013:=z*(z**2+z+1)/((z**2+z-1)*(z**2-z-1)); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(x + x^2 + x^3)/(1 - 3x^2 + x^4), {x, 0, 40}], x]
f[n_] = Product[(1 + 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; a = Table[f[n], {n, 0, 30}]; Round[a]; FullSimplify[ExpandAll[a]] (* Roger L. Bagula and Gary W. Adamson, Nov 26 2008 *)
LinearRecurrence[{0, 3, 0, -1}, {0, 1, 1, 4}, 100] (* G. C. Greubel, Feb 08 2016 *)

{a(n) = if( n%2, fibonacci(n+1) + fibonacci(n-1), fibonacci(n))}; /* Michael Somos, Jan 08 2012 */

{a(n) = if( n<0, -a(-n), subst( polinterpolate( vector( n, k, fibonacci(k))), x, n+1))}; /* Michael Somos, Jan 08 2012 */

a(1) = a(2) = 1, a(3) = 4, a(n) = (a(n-1) * a(n-2) - 1) / a(n-3), unless n=3. a(-n) = -a(n).
a(2n) = A001906(n), a(2n+1) = A002878(n). a(n)=F(n+1)+(-1)^(n+1)F(n-1). - Mario Catalani (mario.catalani(AT)unito.it), Sep 20 2002
G.f.: x*(1+x+x^2)/((1-x-x^2)*(1+x-x^2)).
a(n) = Product_{k=1..floor((n-1)/2)} (1 + 4*sin(k*Pi/n)^2). - Roger L. Bagula and Gary W. Adamson, Nov 26 2008
Binomial transform is A096140. - Michael Somos, Apr 13 2012
From Peter Bala, Apr 18 2014: (Start)
a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = (1/2)*(sqrt(5) + 1) and beta = (1/2)*(sqrt(5) - 1). Equivalently, a(n) = U(n-1, sqrt(5)/2) for n odd and a(n) = (1/sqrt(5))*U(n-1, sqrt(5)/2) for n even, where U(n,x) is the Chebyshev polynomial of the second kind. (End)
E.g.f.: (Phi/sqrt(5))*exp(-Phi*x)*(exp(x)-1)*(exp(sqrt(5)*x) - 1/(Phi)^2), where Phi = (1+sqrt(5))/2. - G. C. Greubel, Feb 08 2016
a(n) = (5^floor((n-1)/2)/2^(n-1))*Sum_{k=0..n-1} binomial(n-1,k)/5^floor(k/2). - Tony Foster III, Oct 21 2018
a(n) = hypergeom([(1 - n)/2, (n + 1) mod 2 - n/2], [1 - n], -4) for n >= 2. - Peter Luschny, Sep 03 2019

Additional comments from Michael Somos, Jun 01 2000

A108412 Expansion of (1 + x + x^2)/(1 - 4x^2 + x^4).

Original entry on oeis.org

1, 1, 5, 4, 19, 15, 71, 56, 265, 209, 989, 780, 3691, 2911, 13775, 10864, 51409, 40545, 191861, 151316, 716035, 564719, 2672279, 2107560, 9973081, 7865521, 37220045, 29354524, 138907099, 109552575, 518408351, 408855776, 1934726305

Offset: 0

Ralf Stephan, Jun 05 2005

This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 6 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence satisfies a linear recurrence of order four. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -6, 1, -6, ...] = 1/(1 - 1/(6 - 1/(1 - 1/(6 - ...)))) = 3 - sqrt(3) begins [0/1, 1/1, 6/5, 5/4, 24/19, 19/15, 90/71,...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [1, 6, 5, 24, 19, 90,...] is also a strong divisibility sequence. Cf. A005013 and A203976. - Peter Bala, May 19 2014
From Peter Bala, Mar 25 2018: (Start)
The following remarks assume an offset of 1.
Define a binary operation o on the real numbers by x o y = x*sqrt(1 + (1/2)*y^2) + y*sqrt(1 + (1/2)*x^2). The operation o is commutative and associative with identity 0. We have a(2*n + 1) = 1 o 1 o ... o 1 (2*n + 1 terms) and sqrt(6)*a(2*n) = (1 o 1 o ... o 1) (2*n terms). Cf. A005013 and A084068. For example, 1 o 1 = sqrt(6) and 1 o 1 o 1 = sqrt(6) o 1 = 5 = a(3).
From the obvious identity ( 1 o 1 o ... o 1 (2*n terms) ) o ( 1 o 1 o ... o 1 (2*m terms) ) = 1 o 1 o ... o 1 (2*n + 2*m terms) we find the relation a(2*n+2*m) = a(2*n)*sqrt(1 + 3*a(2*m)^2) + a(2*m)*sqrt(1 + 3*a(2*n)^2).
Similarly, from a(2*n+1) o a(2*m+1) = sqrt(6)*a(2*n+2*m+2) we find sqrt(6)*a(2*n+2*m+2) = a(2*n+1)*sqrt(1 + (1/2)*a(2*m+1)^2) + a(2*m+1)*sqrt(1 + (1/2)*a(2*n+1)^2). (End)

			G.f. = 1 + x + 5*x^2 + 4*x^3 + 19*x^4 + 15*x^5 + 71*x^6 + 56*x^7 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
P. Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
E. W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A001353, A001834.

Cf. A026741, A005013, A084068.

a := proc (n) if `mod`(n, 2) = 1 then 1/sqrt(2)*( ((sqrt(6) + sqrt(2))/2 )^n - ( (sqrt(6) - sqrt(2))/2 )^n) else 1/sqrt(12)*( ((sqrt(6) + sqrt(2))/2 )^n - ( (sqrt(6) - sqrt(2))/2 )^n) end if;
end proc:
seq(simplify(a(n)), n = 1..30); # Peter Bala, Mar 25 2018

CoefficientList[Series[(1+x+x^2)/(1-4x^2+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,4,0,-1},{1,1,5,4},40] (* Harvey P. Dale, Nov 15 2012 *)

{a(n) = my( w = quadgen(24)); simplify( polchebyshev( n, 2, w/2) / if( n%2, w, 1))}; /* Michael Somos, Feb 10 2015 */

a(0)=a(1)=1, a(2)=5, a(n)a(n+3) - a(n+1)a(n+2) = -1.
a(0)=1, a(1)=1, a(2)=5, a(3)=4, a(n) = 4*a(n-2)-a(n-4). - Harvey P. Dale, Nov 15 2012
a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = (1/2)*(sqrt(6) + sqrt(2)) (A188887) and beta = (1/2)*(sqrt(6) - sqrt(2)) (A101263). Equivalently, a(n) = U(n-1,sqrt(6)/2) for n odd and a(n) = (1/sqrt(6))*U(n-1,sqrt(6)/2) for n even, where U(n,x) is the Chebyshev polynomial of the second kind. - Peter Bala, Apr 18 2014
a(2*n) = A001834(n). a(2*n + 1) = A001353(n+1). - Michael Somos, Feb 10 2015
a(n) = -a(-2-n) for all n in Z. - Michael Somos, Feb 10 2015

A330050 a(n) = 2((-1)^n - 1)(F(n) - 1) - (3(-1)^n + 7)/2 F(n+1) + 5*F(n+1)^2.

Original entry on oeis.org

0, 3, 10, 35, 100, 288, 780, 2115, 5610, 14883, 39160, 103040, 270280, 708963, 1857450, 4866435, 12744060, 33373728, 87382900, 228795875, 599019850, 1568318403, 4105974960, 10749749760, 28143378000, 73680759363, 192899171530, 505017737315, 1322154751060

Offset: 0

Michael Somos, Nov 29 2019

			G.f. = 3*x + 10*x^2 + 35*x^3 + 100*x^4 + 288*x^5 + 780*x^6 + 2115*x^7 + ...

V. Kotesovec, Why is this product equal to zero, when the correct result is 2+GoldenRatio, Mathematica StackExchange, Sep 22 2019.
Index entries for linear recurrences with constant coefficients, signature (4,-1,-11,11,1,-4,1).

Cf. A203976, A329421, A330051.

a[ n_] := 2((-1)^n - 1)(Fibonacci[n] - 1) - (3(-1)^n + 7)/2 Fibonacci[n + 1] + 5 Fibonacci[n + 1]^2;
LinearRecurrence[{4,-1,-11,11,1,-4,1},{0,3,10,35,100,288,780},30] (* Harvey P. Dale, Feb 10 2024 *)

{a(n) = n = abs(n+1)-1; polcoeff( x * O(x^n) + (3*x - 2*x^2 - 2*x^3 + 3*x^4) / (1 - 4*x + x^2 + 11*x^3 - 11*x^4 - x^5 + 4*x^6 - x^7), n)};

G.f.: (3*x - 2*x^2 - 2*x^3 + 3*x^4) / (1 - 4*x + x^2 + 11*x^3 - 11*x^4 - x^5 + 4*x^6 - x^7).
a(n) = 2 + L(2*n+2) - F(n+4) - (L(-n+2) + F(n+1))/2 where F=A000045, L=A000032.
a(n) + b(n) * sqrt(5) = F(2*n+2) * Product_{k=2..n} 1 / (1 - q^k/(1 - q^(2*k))) where q = (sqrt(5)-1)/2 and b=A330051.
a(n) = A203976(floor(n/2)+1) * A329421(n).
a(n) = a(-2-n) for all n in Z.

cp's OEIS Frontend

A022998 If n is odd then n, otherwise 2n.

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A014217 a(n) = floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio.

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A054888 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle Pi/2.

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A005013 a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=1, a(3)=4. Alternates Fibonacci (A000045) and Lucas (A000032) sequences for even and odd n.

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A108412 Expansion of (1 + x + x^2)/(1 - 4x^2 + x^4).

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A330050 a(n) = 2((-1)^n - 1)(F(n) - 1) - (3(-1)^n + 7)/2 F(n+1) + 5*F(n+1)^2.