A078024 -id:A078024 - cp's OEIS frontend

A008346 a(n) = Fibonacci(n) + (-1)^n.

1, 0, 2, 1, 4, 4, 9, 12, 22, 33, 56, 88, 145, 232, 378, 609, 988, 1596, 2585, 4180, 6766, 10945, 17712, 28656, 46369, 75024, 121394, 196417, 317812, 514228, 832041, 1346268, 2178310, 3524577, 5702888, 9227464, 14930353, 24157816, 39088170, 63245985, 102334156

Offset: 0

N. J. A. Sloane

Diagonal sums of A059260. - Paul Barry, Oct 25 2004
The absolute value of the Euler characteristic of the Boolean complex of the Coxeter group A_n. - Bridget Tenner, Jun 04 2008
a(n) is the number of compositions (ordered partitions) of n into two sorts of 2's and one sort of 3's.  Example: the a(5)=4 compositions of 5 are 2+3, 2'+3, 3+2 and 3+2'. - Bob Selcoe, Jun 21 2013
Let r = 0.70980344286129... denote the rabbit constant A014565. The sequence 2^a(n) gives the simple continued fraction expansion of the constant r/2 = 0.35490172143064565732 ... = 1/(2^1 + 1/(2^0 + 1/(2^2 + 1/(2^1 + 1/(2^4 + 1/(2^4 + 1/(2^9 + 1/(2^12 + ... )))))))). Cf. A099925. - Peter Bala, Nov 06 2013
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [0, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
a(n) is the number of growing self-avoiding walks with n+3 edges on the grid graph of integer points (x,y) with x >= 0 and y in {0, 1} and with a trapped endpoint. - Jay Pantone, Jul 26 2024

			The Boolean complex of Coxeter group A_4 is homotopy equivalent to the wedge of 2 spheres S^3, which has Euler characteristic 1 - 2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Bilgici, Generalized order-k Pell-Padovan-like numbers by matrix methods, Pure and Applied Mathematics Journal, 2013; 2(6): 174-178.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, Predators and altruists arriving on jammed Riviera, arXiv:2401.01225 [math.CO], 2024. See p. 14.
N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 445
Jay Pantone, A. R. Klotz, and E. Sullivan, Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height., arXiv:2407.18205 [math.CO], 2024.
K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system, arXiv:0806.0906 [math.CO], 2008-2009.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A007492, A066983, A078024, A119282, A014565, A099925, A098600, A374297.

List([0..50], n-> Fibonacci(n) + (-1)^n); # G. C. Greubel, Jul 13 2019

[Fibonacci(n) + (-1)^n: n in [0..50]]; // Vincenzo Librandi, Apr 23 2011

with(combinat): f := n->fibonacci(n)+(-1)^n; seq(f(n), n=0..40);

Table[Fibonacci[n]+(-1)^n,{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
CoefficientList[Series[1/(1-2x^2-x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 10 2013 *)
LinearRecurrence[{0,2,1}, {1,0,2}, 51] (* Ray Chandler, Sep 08 2015 *)

a(n)=fibonacci(n)+(-1)^n \\ Charles R Greathouse IV, Feb 03 2014

[fibonacci(n)+(-1)^n for n in (0..50)] # G. C. Greubel, Jul 13 2019

G.f.: 1/(1 - 2*x^2 - x^3).
a(n) = 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} (-1)^(n-k-j)binomial(j, k). Diagonal sums of A059260. - Paul Barry, Sep 23 2004
From Paul Barry, Oct 04 2004: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^(3k-n).
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^k(1/2)^(n-2k). (End)
From Paul Barry, Oct 25 2004: (Start)
G.f.: 1/((1+x)*(1-x-x^2)).
a(n) = Sum_{k=0..n} binomial(n-k-1, k). (End)
a(n) = |1 + (-1)^(n-1)*Fibonacci(n-1)|. - Bridget Tenner, Jun 04 2008
a(n) = A000045(n) + A033999(n). - Michel Marcus, Nov 14 2013
a(n) = Fibonacci(n+1) - a(n-1), with a(0) = 1. - Franklin T. Adams-Watters, Mar 26 2014
a(n) = b(n+1) where b(n) = b(n-1) + b(n-2) + (-1)^(n+1), b(0) = 0, b(1) = 1. See also A098600.  - Richard R. Forberg, Aug 30 2014
a(n) = b(n+2) where b(n) = Sum_{k=1..n} b(n-k)*A000931(k+1), b(0) = 1. - J. Conrad, Apr 19 2017
a(n) = Sum_{j=n+1..2*n+1} F(j) mod Sum_{j=0..n} F(j) for n > 2 and F(j)=A000045(j). - Art Baker, Jan 20 2019

A112465 Riordan array (1/(1+x), x/(1-x)).

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -1, 1, 1, 1, 1, 0, 2, 2, 1, -1, 1, 2, 4, 3, 1, 1, 0, 3, 6, 7, 4, 1, -1, 1, 3, 9, 13, 11, 5, 1, 1, 0, 4, 12, 22, 24, 16, 6, 1, -1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 0, 5, 20, 50, 80, 86, 62, 29, 8, 1, -1, 1, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1, 0, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

Paul Barry, Sep 06 2005

Inverse is A112466. Note that C(n,k) = Sum_{j = 0..n-k} C(j+k-1, j).

			Triangle starts
   1;
  -1, 1;
   1, 0, 1;
  -1, 1, 1,  1;
   1, 0, 2,  2,  1;
  -1, 1, 2,  4,  3,  1;
   1, 0, 3,  6,  7,  4,  1;
  -1, 1, 3,  9, 13, 11,  5, 1;
   1, 0, 4, 12, 22, 24, 16, 6, 1;
Production matrix begins
  -1, 1;
   0, 1, 1;
   0, 0, 1, 1;
   0, 0, 0, 1, 1;
   0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 0, 1, 1; - _Paul Barry_, Apr 08 2011

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A059260, A108561, A112466, A112468.
Columns: A033999(n) (k=0), A000035(n) (k=1), A004526(n) (k=2), A002620(n-1) (k=3), A002623(n-4) (k=4), A001752(n-5) (k=5), A001753(n-6) (k=6), A001769(n-7) (k=7), A001779(n-8) (k=8), A001780(n-9) (k=9), A001781(n-10) (k=10), A001786(n-11) (k=11), A001808(n-12) (k=12).
Diagonals: A000012(n) (k=n), A023443(n) (k=n-1), A152947(n-1) (k=n-2), A283551(n-3) (k=n-3).
Main diagonal: A072547.
Sums: A078008 (row), A078024 (diagonal), A092220 (signed diagonal), A280560 (signed row).

a112465 n k = a112465_tabl !! n !! k
a112465_row n = a112465_tabl !! n
a112465_tabl = iterate f [1] where
   f xs'@(x:xs) = zipWith (+) ([-x] ++ xs ++ [0]) ([0] ++ xs')
-- Reinhard Zumkeller, Jan 03 2014

A112465:= func< n,k | (-1)^(n+k)*(&+[(-1)^j*Binomial(j+k-1,j): j in [0..n-k]]) >;
[A112465(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Apr 18 2025

T[n_, k_]:= Sum[Binomial[j+k-1, j]*(-1)^(n-k-j), {j, 0, n-k}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jul 23 2018 *)

def A112465(n,k): return (-1)^(n+k)*sum((-1)^j*binomial(j+k-1,j) for j in range(n-k+1))
print(flatten([[A112465(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 18 2025

Number triangle T(n, k) = Sum_{j=0..n-k} (-1)^(n-k-j)*C(j+k-1, j).
T(2*n, n) = A072547(n) (main diagonal). - Paul Barry, Apr 08 2011
From Reinhard Zumkeller, Jan 03 2014: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k), 0 < k < n, with T(n, 0) = (-1)^n and T(n, n) = 1.
T(n, k) = A108561(n, n-k). (End)
T(n, k) = T(n-1, k-1) + T(n-2, k) + T(n-2, k-1), T(0, 0) = 1, T(1, 0) = -1, T(1, 1) = 1, T(n, k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 + x + x^2/2! + x^3/3!) = -1 + 2*x^2/2! + 6*x^3/3! + 13*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014

A104004 Expansion of (1-x) * (1+x) / ((1-2x)(1-x-x^2)).

Original entry on oeis.org

1, 3, 7, 16, 35, 75, 158, 329, 679, 1392, 2839, 5767, 11678, 23589, 47555, 95720, 192427, 386451, 775486, 1555153, 3117071, 6245088, 12507887, 25044431, 50135230, 100345485, 200812363, 401821144, 803960099, 1608434427, 3217700894, 6436748057

Offset: 0

Creighton Dement, Feb 24 2005

A floretion-generated sequence relating to Fibonacci numbers and powers of 2. The sequence results from a particular transform of the sequence A000079*(-1)^n (powers of 2).

Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[ ( 5'i + .5i' + .5'ii' + .5e)*( + .5j' + .5'kk' + .5'ki' + .5e ) ], 1vesforseq = A000079(n+1)*(-1)^(n+1), ForType: 1A. Identity used: jesfor = jesrightfor + jesleftfor

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

Cf. A000045, A000079, A001911, A008466, A016777, A022958, A027934, A042950, A078024, A099036, A221719.

[3*2^n-Fibonacci(n+3): n in [0..40]]; // Vincenzo Librandi, Aug 18 2017

with (combinat):a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=fibonacci(n-1)+2*a[n-1] od: seq(a[n], n=1..26); # Zerinvary Lajos, Mar 17 2008

LinearRecurrence[{3, -1, -2}, {1, 3, 7}, 80] (* Vincenzo Librandi, Aug 18 2017 *)
CoefficientList[Series[(1-x)(1+x)/((2x-1)(x^2+x-1)),{x,0,40}],x] (* Harvey P. Dale, Oct 12 2024 *)
A104004[n_]:= 3*2^n -Fibonacci[n+3]; (* G. C. Greubel, Jun 05 2025 *)

def A104004(n): return 3*2**n - fibonacci(n+3) # G. C. Greubel, Jun 05 2025

4*a(n) = A008466(n+3) + A027973(n) (FAMP result).
Suggestions made by Superseeker: (Start)
a(n+2) - a(n+1) - a(n) = A042950(n+1).
Coefficients of g.f.*(1-x)/(1+x) match A099036.
Coefficients of g.f./(1+x) match A027934.
Coefficients of g.f./(1-x^2) match A008466. (End)
a(n) = A101220(3, 2, n+1) - A101220(3, 2, n). - Ross La Haye, Aug 05 2005
a(n) = 3*2^n - Fibonacci(n+3) = A221719(n) + 1. - Ralf Stephan, May 20 2007, Hugo Pfoertner, Mar 06 2024
a(n) = (3*2^n - (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5)))) / sqrt(5)). - Colin Barker, Aug 18 2017
From G. C. Greubel, Jun 05 2025: (Start)
Sum_{k=0..n} A022958(k+1)*a(n-k) = A001911(n+1).
Sum_{k=0..n} (-1)^k*A016777(k)*a(n-k) = A078024(n).
E.g.f.: 3*exp(2*x) - (2/sqrt(5))*exp(x/2)*( 2*sinh(sqrt(5)*x/2) + sqrt(5)*cosh(sqrt(5)*x/2) ). (End)

A112469 Partial sums of (-1)^n*Fibonacci(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, -3, 10, -11, 23, -32, 57, -87, 146, -231, 379, -608, 989, -1595, 2586, -4179, 6767, -10944, 17713, -28655, 46370, -75023, 121395, -196416, 317813, -514227, 832042, -1346267, 2178311, -3524576, 5702889, -9227463, 14930354, -24157815, 39088171, -63245984, 102334157

Offset: 0

Paul Barry, Sep 06 2005

Diagonal sums of Riordan array (1/(1-x), x/(1+x)), A112468.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (0,2,-1).

Cf. A000045, A078024, A112468.

A112469:= func< n | 2 + (-1)^n*Fibonacci(n-2) >;
[A112469(n): n in [0..40]]; // G. C. Greubel, Apr 17 2025

a[0]:=1:a[1]:=1:a[2]:=2:a[3]:=1:for n from 4 to 50 do a[n]:=2*a[n-2]-a[n-3] od: seq(a[n], n=0..42); # Zerinvary Lajos, Apr 04 2008

Accumulate[Table[(-1)^n Fibonacci[n-1],{n,0,50}]] (* Harvey P. Dale, Nov 05 2011 *)
Table[2 +(-1)^n*Fibonacci[n-2], {n,0,50}] (* G. C. Greubel, Apr 17 2025 *)

def A112469(n): return 2+(-1)^n*fibonacci(n-2)
print([A112469(n) for n in range(41)]) # G. C. Greubel, Apr 17 2025

G.f.: (1+x)/((1-x)*(1+x-x^2)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} C(n-k-j-1, n-2k-j)*(-1)^(n-j).
From G. C. Greubel, Apr 17 2025: (Start)
a(n) = 2 + (-1)^n*Fibonacci(n-2).
E.g.f.: 2*exp(x) - exp(-x/2)*( cosh(sqrt(5)*x/2) + (3/sqrt(5))*sinh(sqrt(5)*x/2) ). (End)

cp's OEIS Frontend

A008346 a(n) = Fibonacci(n) + (-1)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A112465 Riordan array (1/(1+x), x/(1-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

SageMath

Formula

A104004 Expansion of (1-x) * (1+x) / ((1-2*x)*(1-x-x^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A112469 Partial sums of (-1)^n*Fibonacci(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A104004 Expansion of (1-x) * (1+x) / ((1-2x)(1-x-x^2)).