A093040 -id:A093040 - cp's OEIS frontend

A094686 A Fibonacci convolution.

1, 0, 1, 2, 2, 4, 7, 10, 17, 28, 44, 72, 117, 188, 305, 494, 798, 1292, 2091, 3382, 5473, 8856, 14328, 23184, 37513, 60696, 98209, 158906, 257114, 416020, 673135, 1089154, 1762289, 2851444, 4613732, 7465176, 12078909, 19544084, 31622993, 51167078, 82790070

Offset: 0

Paul Barry, May 19 2004

Convolution of A000045 and A049347.
Diagonal sums of number triangle A116088. - Paul Barry, Feb 04 2006
Let (b(n)) be the p-INVERT of (1,1,0,0,0,0,0,0,...) using p(S) = 1 - S^2; then b(n) = a(n+1) for n >=0. See A292324. - Clark Kimberling, Sep 15 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016 (see 1st column of Table 1 p. 8).
Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 2].
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 19.
David Broadhurst, Multiple Deligne values: a data mine with empirically tamed denominators, arXiv:1409.7204 [hep-th], 2014 (see p. 10).
Leonard Rozendaal, Pisano word, tesselation, plane-filling fractal, Preprint, 2017.
Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Cf. A000045, A005252, A049347, A093040, A116088, A292324.

[(Fibonacci(n+1) +((n+2) mod 3) -1)/2: n in [0..40]]; // G. C. Greubel, Feb 09 2023

LinearRecurrence[{0,1,2,1}, {1,0,1,2}, 40] (* Jean-François Alcover, Sep 21 2017 *)

Vec(1/((1-x-x^2)*(1+x+x^2)) + O(x^50)) \\ Michel Marcus, Sep 27 2014

[(fibonacci(n+1) + (n+2)%3 - 1)/2 for n in range(41)] # G. C. Greubel, Feb 09 2023

G.f.: 1/((1-x-x^2)*(1+x+x^2)).
a(n) = 2*sqrt(3)*Sum_{k=0..n} Fibonacci(k+1)*cos((4*(n-k)+1)*Pi/6)/3.
a(n) = a(n-2) + 2*a(n-3) + a(n-4).
From Paul Barry, Jan 13 2005: (Start)
a(n) = A005252(n) - (-cos((2*n+1)*Pi/3)/2 - sqrt(3)*sin((2*n+1)*Pi/3)/6 + sqrt(3)*cos(Pi*n/3+Pi/6)/6 + sin((2*n+1)*Pi/6)/2).
a(n) = Sum_{k=0..floor(n/2)} if(mod(n-k, 2)=0, binomial(n-k, k), 0).
a(n) = A093040(n-1) - Fibonacci(n). (End)
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(1+(-1)^(n-k))/2. - Paul Barry, Sep 09 2005
From Paul Barry, Feb 04 2006: (Start)
a(n) = Sum_{k=0..floor(n/2)} C(2*k, n-2*k).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(3*k,n-k)/C(3*k,k). (End)
2*a(n) = A000045(n+1) + A049347(n). - R. J. Mathar, Feb 13 2020
a(n) = (1/2)*(A000045(n+1) + A049347(n)). - G. C. Greubel, Feb 09 2023

A375315 Expansion of (1 + x)/(1 - x^2*(1 + x)^3).

Original entry on oeis.org

1, 1, 1, 4, 7, 11, 23, 45, 81, 154, 296, 555, 1046, 1986, 3753, 7085, 13404, 25348, 47904, 90568, 171245, 323728, 612009, 1157071, 2187496, 4135527, 7818464, 14781237, 27944604, 52830706, 99879234, 188826693, 356986401, 674901117, 1275934888, 2412219633, 4560424135

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,1,3,3,1).

Cf. A093040, A375316.

Cf. A116090, A375317.

my(N=40, x='x+O('x^N)); Vec((1+x)/(1-x^2*(1+x)^3))

a(n) = sum(k=0, n\2, binomial(3*k+1, n-2*k));

a(n) = a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*k+1,n-2*k).
a(n) = A116090(n) + A116090(n-1).

A375373 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^2) ).

Original entry on oeis.org

1, -2, 4, -4, 6, -4, 9, -4, 16, 0, 28, 16, 57, 58, 132, 172, 322, 476, 817, 1272, 2112, 3360, 5496, 8832, 14353, 23158, 37540, 60668, 98238, 158876, 257145, 415988, 673168, 1089120, 1762324, 2851408, 4613769, 7465138, 12078948, 19544044, 31623034, 51167036

Offset: 0

Seiichi Manyama, Aug 13 2024

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,0,4,6,4,1).

Cf. A093040, A375372.

CoefficientList[Series[1/((1+x)^2(1-x^2(1+x)^2)),{x,0,50}],x] (* or *) LinearRecurrence[{-2,0,4,6,4,1},{1,-2,4,-4,6,-4},50] (* Harvey P. Dale, Dec 11 2024 *)

my(N=50, x='x+O('x^N)); Vec(1/((1+x)^2*(1-x^2*(1+x)^2)))

a(n) = sum(k=0, n\2, binomial(2*k-2, n-2*k));

a(n) = -2*a(n-1) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k-2,n-2*k).
2*a(n) = 2*(-1)^n*(n+1) +A212804(n)-A057078(n). - R. J. Mathar, Aug 14 2024

A178534 Triangle T(n,k) read by rows. T(n,1) = A000045(n+1), k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 3, 1, 1, 1, 13, 5, 3, 1, 1, 1, 21, 8, 4, 2, 1, 1, 1, 34, 13, 6, 4, 2, 1, 1, 1, 55, 21, 11, 6, 3, 2, 1, 1, 1, 89, 34, 17, 9, 6, 3, 2, 1, 1, 1, 144, 55, 27, 15, 9, 5, 3, 2, 1, 1, 1, 233, 89, 45, 25, 14, 9, 5, 3, 2, 1, 1, 1, 377, 144, 72, 40, 23, 14, 8, 5, 3, 2, 1, 1, 1

Offset: 1

Mats Granvik, May 29 2010

			Table begins:
   1;
   2,  1;
   3,  1,  1;
   5,  2,  1,  1;
   8,  3,  1,  1,  1;
  13,  5,  3,  1,  1,  1;
  21,  8,  4,  2,  1,  1,  1;
  34, 13,  6,  4,  2,  1,  1,  1;
  55, 21, 11,  6,  3,  2,  1,  1,  1;
  89, 34, 17,  9,  6,  3,  2,  1,  1,  1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Cf. 1st column=A000045(n+1), 2nd=A000045, 3rd=A093040, 4th=A006498. Matrix inverse of A178535.

Cf. A051731, A129713.

A178534 := proc(n, k)
    option remember;
    if k= 1 then
        combinat[fibonacci](n+1) ;
    elif k > n then
        0 ;
    else
        add(procname(n-i, k-1), i=1..k-1)-add(procname(n-i, k), i=1..k-1) ;
    end if;
end proc:
seq(seq(A178534(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Oct 28 2010

T[n_, 1] := Fibonacci[n+1];
T[n_, k_] := T[n, k] = If[k > n, 0, Sum[T[n-i, k-1], {i, 1, k-1}] - Sum[T[n-i, k], {i, 1, k-1}]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 23 2024 *)

T(n,k)=(n % k==0) + sum(j=1,n\k,fibonacci(n-j*k)) \\ Andrew Howroyd, Feb 23 2024

from sympy.core.cache import cacheit
from sympy import fibonacci
@cacheit
def A(n, k): return fibonacci(n + 1) if k==1 else 0 if k>n else sum([A(n - i, k - 1) for i in range(1, k)]) - sum([A(n - i, k) for i in range(1, k)])
for n in range(1, 13): print([A(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Sep 15 2017

T(n,1) = A000045(n+1), k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) - Sum_{i=1..k-1} T(n-i,k).
T(n,k) = A129713*A051731. - Mats Granvik, Oct 22 2010
From R. J. Mathar, Sep 16 2017: (Start)
G.f. 3rd column: x^3*(1+x)/((1-x-x^2)*(1+x+x^2)).
G.f. 4th column: x^4/((1-x-x^2)*(1+x^2)) =x^4*(1+x)/((1-x-x^2)*(1+x+x^2+x^3)).
G.f. 5th column: x^5*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4)).
G.f. 6th column: x^6/((1-x-x^2)*(1+x+x^2)*(1-x+x^2)) = x^6*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5)).
G.f. 7th column: x^7*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6)).
G.f. 8th column: x^8/((1-x-x^2)*(1+x^2)*(1+x^4)) = x^8*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6+x^7)).
Conjecture (by extrapolating): G.f. k-th column: x^k*(1-x^2)/((1-x-x^2)*(1-x^k)).
G.f.: (1-x^2)/(1-x-x^2)*Sum_{i>=1} (x*y)^i/(1-x^i) = (1-x^2)/(1-x-x^2)*A051731(x,y). (End)
T(n,k) = A051731(n,k) + Sum_{j=1..floor(n/k)} Fibonacci(n-j*k). - Andrew Howroyd, Feb 23 2024

A375316 Expansion of (1 + x)/(1 - x^2*(1 + x)^4).

Original entry on oeis.org

1, 1, 1, 5, 11, 19, 42, 98, 205, 429, 936, 2024, 4316, 9260, 19949, 42841, 91917, 197485, 424331, 911255, 1957086, 4203998, 9029949, 19394681, 41657808, 89478064, 192189304, 412801176, 886657081, 1904452689, 4090567673, 8786123349, 18871714923, 40534539675

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,1,4,6,4,1).

Cf. A093040, A375315.

Cf. A375314.

my(N=40, x='x+O('x^N)); Vec((1+x)/(1-x^2*(1+x)^4))

a(n) = sum(k=0, n\2, binomial(4*k+1, n-2*k));

a(n) = a(n-2) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(4*k+1,n-2*k).
a(n) = A375314(n) + A375314(n-1).

A375372 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^2) ).

Original entry on oeis.org

1, -1, 2, 0, 2, 2, 5, 5, 12, 16, 28, 44, 73, 115, 190, 304, 494, 798, 1293, 2089, 3384, 5472, 8856, 14328, 23185, 37511, 60698, 98208, 158906, 257114, 416021, 673133, 1089156, 1762288, 2851444, 4613732, 7465177, 12078907, 19544086, 31622992, 51167078, 82790070

Offset: 0

Seiichi Manyama, Aug 13 2024

Index entries for linear recurrences with constant coefficients, signature (-1,1,3,3,1).

Cf. A093040, A375373.

my(N=50, x='x+O('x^N)); Vec(1/((1+x)*(1-x^2*(1+x)^2)))

a(n) = sum(k=0, n\2, binomial(2*k-1, n-2*k));

a(n) = -a(n-1) + a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k-1,n-2*k).
a(n) = A375373(n) + A375373(n-1).
2*a(n) = 2*(-1)^n + A000045(n) + A057078(n+1). - R. J. Mathar, Aug 14 2024

A113066 Expansion of (1 + x)^2/((1 + x + x^2)(1 + 3x + x^2)).

Original entry on oeis.org

1, -2, 4, -10, 27, -72, 189, -494, 1292, -3382, 8855, -23184, 60697, -158906, 416020, -1089154, 2851443, -7465176, 19544085, -51167078, 133957148, -350704366, 918155951, -2403763488, 6293134513, -16475640050, 43133785636, -112925716858, 295643364939, -774004377960

Offset: 0

Creighton Dement, Oct 13 2005

Binomial transform gives signed version of A093040.
The positive sequence has g.f. (1 - x)^2/((1 - x + x^2)(1 - 3*x + x^2)) and a(n) = Sum_{k=0..n} binomial(n+k+1, n-k)*(1+(-1)^k)/2. - Paul Barry, Jul 06 2009
Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'

C. Dement, Floretion Integer Sequences (work in progress).

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (-4,-5,-4,-1).

Cf. A113067, A113068, A093040, A109961, A001906.

a:=[1,-2,4,-10];; for n in [5..35] do a[n]:=-4*a[n-1]-5*a[n-2]-4*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 11 2018

I:=[1,-2,4,-10]; [n le 4 select I[n] else -4*Self(n-1)-5*Self(n-2)- 4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 12 2018

seq(coeff(series((1+x)^2/((1+x+x^2)*(1+3*x+x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Sep 11 2018

LinearRecurrence[{-4, -5, -4, -1}, {1, -2, 4, -10}, 40] (* Vincenzo Librandi, Sep 12 2018 *)
CoefficientList[Series[(1 + x)^2/((1 + x + x^2)*(1 + 3 x + x^2)), {x, 0, 50}], x] (* Stefano Spezia, Sep 12 2018 *)

x='x+O('x^99); Vec((1+x)^2/((1+x+x^2)*(1+3*x+x^2))) \\ Altug Alkan, Sep 11 2018

a(n) + a(n+1) = (-1)^(n+1)*A109961(n+1).
a(n) + a(n+1) + a(n+2) = (-1)^n*A001906(n+2) = (-1)^n*F(2*n+4).
a(n) = A049347(n)/2 + (-1)^n*A001906(n+1)/2. - R. J. Mathar, Nov 10 2009
Lim_{n -> inf} a(n)/a(n-1) = -(1 + A001622). - A.H.M. Smeets, Sep 11 2018
a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4). - Muniru A Asiru, Sep 11 2018

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A094686 A Fibonacci convolution.

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A375315 Expansion of (1 + x)/(1 - x^2*(1 + x)^3).

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A375373 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^2) ).

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A178534 Triangle T(n,k) read by rows. T(n,1) = A000045(n+1), k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).

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A375316 Expansion of (1 + x)/(1 - x^2*(1 + x)^4).

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A375372 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^2) ).

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A113066 Expansion of (1 + x)^2/((1 + x + x^2)*(1 + 3*x + x^2)).

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A113066 Expansion of (1 + x)^2/((1 + x + x^2)(1 + 3x + x^2)).