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A001610 a(n) = a(n-1) + a(n-2) + 1.
(Formerly M0764 N0291)
22
0, 2, 3, 6, 10, 17, 28, 46, 75, 122, 198, 321, 520, 842, 1363, 2206, 3570, 5777, 9348, 15126, 24475, 39602, 64078, 103681, 167760, 271442, 439203, 710646, 1149850, 1860497, 3010348, 4870846, 7881195, 12752042, 20633238, 33385281, 54018520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For prime p, p divides a(p-1). - T. D. Noe, Apr 11 2009

Generalization. If a(0,x)=0, a(1,x)=2 and, for n>=2, a(n,x)=a(n-1,x)+x*a(n-2,x)+1, then we obtain a sequence of polynomials Q_n(x)=a(n,x) of degree floor((n-1)/2), such that p is prime iff all coefficients of Q_(p-1)(x) are multiple of p (sf. A174625). Thus a(n) is the sum of coefficients of Q_(n-1)(x). - Vladimir Shevelev, Apr 23 2010

Odd numbers n such that n divides a(n-1) are in A005845. - Zak Seidov, May 04 2010; comment edited by N. J. A. Sloane, Aug 10 2010

a(n) = number of ways to modify a circular arrangement of n objects by swapping one or more adjacent pairs. E.g., for 1234, new arrangements are 2134, 2143, 1324, 4321, 1243, 4231 (taking 4 and 1 to be adjacent) and a(4) = 6. - Toby Gottfried, Aug 21 2011

REFERENCES

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

T. D. Noe, Table of n, a(n) for n=0..500

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv preprint, 2015.

F. Hazama, Spectra of graphs attached to the space of melodies, Discrete Math., 311 (2011), 2368-2383. See Table 2.1.

Rui Liu and Feng-Zhen Zhao, On the Sums of Reciprocal Hyperfibonacci Numbers and Hyperlucas Numbers, Journal of Integer Sequences, Vol. 15 (2012), #12.4.5. - From N. J. A. Sloane, Oct 05 2012

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence a_{1,s}, arXiv:0903.2514 [math.NT], 2009-2011.

N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences< a>, JIS 12 (2009) 09.4.5, Example 10.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

V. Shevelev, On divisibility of C(n-i-1,i-1) by i, Int. J. of Number Theory, 3 (2007), no.1, 119-139. [From Vladimir Shevelev, Apr 23 2010]

J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.

Li-Na Zheng, Rui Liu, and Feng-Zhen Zhao, On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers, Journal of Integer Sequences, Vol. 17 (2014), #14.1.4.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

FORMULA

a(n) = A000204(n)-1 = A000032(n+1)-1 = A000071(n+1)+A000045(n).

a(n) = F(n)+F(n+2)-1 where F(n) is the n-th Fibonacci number. - Zerinvary Lajos, Jan 31 2008

a(n) = A014217(n+1)- A000035(n+1). - Paul Curtz, Sep 21 2008

a(n) = -1+(1/2)*[1/2+(1/2)*sqrt(5)]^n+(1/2)*[1/2+(1/2)*sqrt(5)]^n*sqrt(5)-(1/2)*sqrt(5)*[1/2-(1/2) *sqrt(5)]^n+(1/2)*[1/2-(1/2)*sqrt(5)]^n, with n>=0. - Paolo P. Lava, Sep 29 2008

a(n) = Sum{i=1,...,floor((n+1)/2)}((n+1)/i)*C(n-i,i-1). In more general case of polynomials Q_n(x)=a(n,x) (see our comment) we have Q_n(x)=Sum{i=1,...,floor((n+1)/2)}((n+1)/i)*C(n-i,i-1)*x^(i-1). - Vladimir Shevelev, Apr 23 2010

a(n) = sum(lucas(k),k=0..n-1), Lucas(n)=A000032(n). - Gary Detlefs, Dec 07 2010

G.f.: x*(2-x)/((1-x-x^2)*(1-x)) = (2*x-x^2)/(1-2*x+x^3). - George F. Johnson, Jan 28 2013

a(0)=0, a(1)=2, a(2)=3; for n>=3, a(n) = 2*a(n-1)-a(n-3). - George F. Johnson, Jan 28 2013

For n > 1, a(n) = A048162(n+1) + 3. - Toby Gottfried, Apr 13 2013

For n > 0, a(n) = A014217(n-1) + A014217(n). - Paolo P. Lava, Mar 20 2015

MAPLE

A001610:=-z*(-2+z)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation

with(combinat): seq(fibonacci(n)+fibonacci(n+2)-1, n=0..36); # Zerinvary Lajos, Jan 31 2008

g:=(1+z^2)/(1-z-z^2): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-1, n=1..37); # Zerinvary Lajos, Jan 09 2009

MATHEMATICA

t={0, 2}; Do[AppendTo[t, t[[-1]] + t[[-2]] + 1], {n, 2, 36}]; t

RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + 1, a[0] == 0, a[1] == 2}, a, {n, 0, 36}] (* Robert G. Wilson v, Apr 13 2013 *)

CoefficientList[Series[x (2 - x) / ((1 - x - x^2) (1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)

PROG

(Haskell)

a001610 n = a001610_list !! n

a001610_list =

   0 : 2 : map (+ 1) (zipWith (+) a001610_list (tail a001610_list))

-- Reinhard Zumkeller, Aug 21 2011

(MAGMA) I:=[0, 2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)+1: n in [1..40]]; // Vincenzo Librandi, Mar 20 2015

CROSSREFS

Cf. A001610, A000032, A000204, A174625, A000071.

Sequence in context: A026647 A026669 A023614 * A238777 A245437 A135431

Adjacent sequences:  A001607 A001608 A001609 * A001611 A001612 A001613

KEYWORD

nonn,easy,hear

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Henry Bottomley, Jul 06 2000

STATUS

approved

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Last modified August 25 07:47 EDT 2016. Contains 275792 sequences.