A173284 -id:A173284 - cp's OEIS frontend

A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.

1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169, 63245985, 102334155

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals row sums of triangle A173284. - Gary W. Adamson, Feb 14 2010
The Kn21 sums (see A180662 for definition) of the 'Races with Ties' triangle A035317 produce this sequence. - Johannes W. Meijer, Jul 20 2011
a(n-1), for n >= 1, gives the number of compositions of n with relative prime parts, and parts not exceeding 2. See the row sums of triangle A030528 where for even n the leading 1 is missing. - Wolfdieter Lang, Jul 27 2023

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 33*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023
K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1, eq (1).
Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A062114, A173284, A059841, A014217, A180662, A035317.
Partial sums of A008346, first differences of A129696.
Cf. also A000032, A000045, A030528.
Cf. A001595, A054450, A066983, A074331, A168309.

List([0..40], n-> Fibonacci(n+2) -(1-(-1)^n)/2); # G. C. Greubel, Jul 10 2019

a052952 n = a052952_list !! n
a052952_list = 1 : 1 : zipWith (+)
   a059841_list (zipWith (+) a052952_list $ tail a052952_list)
-- Reinhard Zumkeller, Jan 06 2012

[Fibonacci(n+2)-(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Dec 02 2016

A052952 :=proc(n)
    option remember;
    local t1;
    if n <= 1 then
        return 1 ;
    fi:
    if n mod 2 = 1 then
        t1:=0
    else
        t1:=1;
    fi:
    procname(n-1)+procname(n-2)+t1;
end proc;
seq(A052952(n), n=0..40) ; # N. J. A. Sloane, May 25 2008

Table[Fibonacci[n+2] -(1-(-1)^n)/2, {n, 0, 40}] (* Vincenzo Librandi, Dec 02 2016 *)
Sum[(-1)^k*Fibonacci[Range[2,41], 1-k], {k,0,1}] (* G. C. Greubel, Oct 21 2019 *)
CoefficientList[Series[1/((1-x-x^2)*(1-x^2)),{x,0,40}],x] (* Harvey P. Dale, Sep 12 2020 *)

PARI
```
{a(n) = fibonacci(n+2) - n%2};
```

[fibonacci(n+2) -(1-(-1)^n)/2 for n in (0..40)] # G. C. Greubel, Jul 10 2019

G.f.: 1/((1-x-x^2)*(1-x^2)).
a(n) = A074331(n+1).
a(n) = A054450(n+1, 1) (second column of triangle).
a(n) = 2*a(n-2) + a(n-3) + 1, with a(0)=1, a(1)=1, a(2)=3.
a(n) = Sum_{alpha=RootOf(-1+z+z^2)} (3+alpha)*alpha^(-1-n)/3 - Sum_{beta=RootOf(-1+z^2)} beta^(-1-n)/2.
a(2*k) = Sum_{j=0..k} F(2*j+1) = F(2*(k+1)) for k >= 0; a(2*k-1) = Sum_{j=0..k} F(2*j) = F(2*k+1)-1 for k >= 1 (F = A000045, Fibonacci numbers).
a(n) = a(n-1) + a(n-2) + (1+(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1, k). - Paul Barry, Oct 23 2004
a(n) = floor(phi^(n+2) / sqrt(5)), where phi is the golden ratio: phi = (1+sqrt(5))/2. - Reinhard Zumkeller, Apr 19 2005
a(n) = Fibonacci(n+1) + a(n-2) with n>1, a(0)=a(1)=1. - Zerinvary Lajos, Mar 17 2008
a(n) = floor(Fibonacci(n+3)^2/Fibonacci(n+4)). - Gary Detlefs, Nov 29 2010
a(n) = (A001595(n+3) - A066983(n+4))/2. - Gary Detlefs, Dec 19 2010
a(4*n) = F(4*n+2); a(4*n+1) = F(4*n+3) - 1; a(4*n+2) = F(4*n+4); a(4*n+3) = F(4*n+5) - 1. - Johannes W. Meijer, Jul 20 2011
a(n+1) = a(n) + a(n-1) + A059841(n+1). - Reinhard Zumkeller, Jan 06 2012
a(n) = floor(|F((1+i)*(n+2))|), n >= 0, with the complex Fibonacci function F: C -> C, z -> F(z) with F(z) := (exp(log(phi)*z) - exp(i*Pi*z)*exp(-log(phi)*z))/(2*phi-1) with the modulus |z|, the imaginary unit i and the golden section phi:=(1+sqrt(5))/2. A Conjecture: For F(z) see, e.g., the T. Koshy reference. ch. 45, p. 523, where F is called f, given in A000045. - Wolfdieter Lang, Jul 24 2012
5*a(n) = (L(n+3)-1)*(L(n+4)+3) -14 -Sum_{k=0..n} L(k+1)*L(k+5) = (L(n+3)-1)*(L(n+4)+3) -L(2*n+7) +A168309(n), where L=A000032. - J. M. Bergot, Jun 13 2014
a(n) = floor(phi*Fibonacci(n+1)), where phi is the golden section. - Michel Dekking, Dec 02 2016
a(n) = -(-1)^n * a(-4-n) for all n in Z. - Michael Somos, Dec 03 2016
a(n) = Sum_{k=0..n} Sum_{i=0..n} C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = floor(1/(Sum_{k>=n+4} 1/Fibonacci(k))) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018
a(n) = floor(abs(chebyshevU(n/2, 3/2))). - Federico Provvedi, Feb 23 2022
E.g.f.: exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5 - sinh(x). - Stefano Spezia, Mar 09 2024

Additional formulas and more terms from Wolfdieter Lang, May 02 2000

Better description from Olivier Gérard, Jun 05 2001

A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

Original entry on oeis.org

1, 1, 4, 5, 12, 17, 33, 50, 88, 138, 232, 370, 609, 979, 1596, 2575, 4180, 6755, 10945, 17700, 28656, 46356, 75024, 121380, 196417, 317797, 514228, 832025, 1346268, 2178293, 3524577, 5702870, 9227464, 14930334, 24157816, 39088150, 63245985, 102334135

Offset: 0

Wolfdieter Lang, Apr 27 2000

Equals triangle A173284 * [1, 2, 3, ...]. - Gary W. Adamson, Mar 03 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-3,1,1).

Cf. A054450, A049310, A000045, A052952.
Cf. A007382.
Cf. A173284.

BB:=1/(1-k^2)^2/(1-k-k^2): seq(coeff(series(BB, k, n+1), k, n), n=0..50); # Zerinvary Lajos, May 16 2007

LinearRecurrence[{1,3,-2,-3,1,1},{1,1,4,5,12,17},40] (* Harvey P. Dale, Oct 06 2024 *)

Vec(-1/((x-1)^2*(x+1)^2*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Jun 14 2015

a(n) = A054450(n+2, 2).
G.f.: Fib(x)/(1-x^2)^2, with Fib(x)=1/(1-x-x^2) = g.f. A000045 (Fibonacci numbers without 0).
a(2*k) = A027941(k)= F(2*k+3)-1; a(2*k+1)= F(2*(k+2))-(k+2)= A054452(k), k >= 0.
a(n-2) = Fibonacci(n+1) - binomial(n-floor(n/2), floor(n/2)), or a(n-2) = Sum_{i=0..floor(n/2)-1} binomial(n-i, i). - Jon Perry, Mar 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, k). - Paul Barry, Oct 23 2004

More terms from James Sellers, Apr 28 2000

A173285 A(x) satisfies: Fibonacci(x)/x = A(x)/A(x^2).

Original entry on oeis.org

1, 1, 3, 4, 10, 14, 28, 42, 80, 122, 216, 338, 582, 920, 1544, 2464, 4088, 6552, 10762, 17314, 28292, 45606, 74236, 119842, 194660, 314502, 510082, 824584, 1336210, 2160794, 3499468, 5660262, 9163818, 14824080, 23994450, 38818530, 62823742, 101642272

Offset: 0

Gary W. Adamson, Feb 14 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000045, A173284.

A173285 := proc(n) option remember; if n = 0 then 1; else add(procname(l)*combinat[fibonacci](n-2*l+1),l=0..n/2) ; end if; end proc:
seq(A173285(n),n=0..60) ; # R. J. Mathar, Apr 01 2010

a[n_] := a[n] = If[n == 0, 1, Sum[Fibonacci[n-2k+1] a[k], {k, 0, n/2}]];
a /@ Range[0, 40] (* Jean-François Alcover, Oct 02 2019 *)

a(n) = Sum_{k=0..n/2} A000045(n-2*k+1)*a(k). - R. J. Mathar, Apr 02 2010
Given M = triangle A173284, A173285 as a left-shifted vector = lim_{n->inf} M^n.
G.f.: Product_{k>=0} 1/(1 - x^(2^k) - x^(2^(k + 1))). - Ilya Gutkovskiy, Aug 30 2017
a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=1} 1/(1 - x^(2^k) - x^(2^(k+1))) = 2.6009165618094467356830434687244547021995030468423430186926... and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 08 2022

Division through x added to definition and sequence extended by R. J. Mathar, Apr 22 2010

A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

Original entry on oeis.org

1, 4, 7, 1, 10, 4, 13, 7, 1, 16, 10, 4, 19, 13, 7, 1, 22, 16, 10, 4, 25, 19, 13, 7, 1, 28, 22, 16, 10, 4, 31, 25, 19, 13, 7, 1, 34, 28, 22, 16, 10, 4, 37, 31, 25, 19, 13, 7, 1, 40, 34, 28, 22, 16, 10, 4, 43, 37, 31, 25, 19, 13, 7, 1, 46, 40, 34, 28, 22, 16, 10

Offset: 1

Ctibor O. Zizka, Mar 11 2012

OEIS contains a lot of similar sequences, for example A152204, A122196, A173284.
Row sums for this sequence gives A006578.
In general, by given triangle with (A-B,2*A-B,...,A*n-B,...) in every column, shifted down K-times, we have the row sum s(n)= A*(n*n+K*n+nmodK)/(2*K) - B*(n+nmodK)/K. In this sequence K=2,A=3,B=2, in A152204 K=2,A=2,B=1.
No triangle with primes in every column, shifted down by K>=2 in OEIS, no row sums of it in OEIS.
From Johannes W. Meijer, Sep 28 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A143971.
The alternating row sums equal A004524(n+2) + 2*A004524(n+1).
The antidiagonal sums equal A171452(n+1). (End)

			Triangle:
1
4
7,  1
10, 4
13, 7,  1
16, 10, 4
19, 13, 7,  1
22, 16, 10, 4
25, 19, 13, 7,  1
28, 22, 16, 10, 4
...

Cf. A008315, A011973, A102541, A122196, A122197, A128099, A152198, A152204, A173284, A207538.

Cf. (Related to triangle sums) A006578, A000217, A002620, A004524, A171452.

T := (n, k) -> 3*n - 6*k + 4: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..15); # Johannes W. Meijer, Sep 28 2013

From Johannes W. Meijer, Sep 28 2013: (Start)
T(n, k) = 3*n - 6*k + 4, n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A143971(n-k+1, k), n >= 1 and 1 <= k <= floor((n+1)/2). (End)

cp's OEIS Frontend

A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.

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A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

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A173285 A(x) satisfies: Fibonacci(x)/x = A(x)/A(x^2).

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A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

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