A119282 -id:A119282 - cp's OEIS frontend

A181716 a(n) = a(n-1) + a(n-2) + (-1)^n, with a(0)=0 and a(1)=1.

0, 1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522, 87403802

Offset: 0

Robert G. Wilson v, Nov 07 2010

Aside from the first term, duplicate of A098600.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A000045, A008346, A119282.

First differences of A014217.

I:=[0, 1, 2]; [n le 3 select I[n] else 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 09 2012

[Lucas(n-1)+(-1)^n: n in [0..40]]; // G. C. Greubel, Mar 25 2024

a[0]= 0; a[1]= 1; a[n_]:= a[n]= a[n-1] +a[n-2] +(-1)^n; Array[a,38,0]
LinearRecurrence[{0,2,1},{0,1,2},40] (* Vincenzo Librandi, Jan 09 2012 *)

[lucas_number2(n-1,1,-1)+(-1)^n for n in range(41)] # G. C. Greubel, Mar 25 2024

a(n) = a(n-1) + a(n-2) + (-1)^n.
a(n) = 2*a(n-2) + a(n-3).
a(n) - A000045(n) = A008346(n-2).
G.f.: x*(1+2*x)/(1-2*x^2-x^3). - Colin Barker, Jan 09 2012
a(n) = A000032(n-1) + (-1)^n. - G. C. Greubel, Mar 25 2024
E.g.f.: exp(x/2)*(sqrt(5)*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2)) + exp(-x). - Stefano Spezia, Jun 18 2024

A141169 Triangle of Fibonacci numbers, read by rows: T(n,k) = A000045(k), 0<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 2, 3, 5, 0, 1, 1, 2, 3, 5, 8, 0, 1, 1, 2, 3, 5, 8, 13, 0, 1, 1, 2, 3, 5, 8, 13, 21, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 0, 1, 1, 2, 3, 5, 8, 13, 21

Offset: 0

Reinhard Zumkeller, Mar 21 2011

T(n,0) = A000004(n); T(n,n) = A000045(n);
central terms: T(2*n,n) = A000045(n);
sums of rows: Sum(T(n,k): 0<=k<=n) = A000071(n+2);
alternating sums of rows: Sum(T(n,k)*(-1)^k: 0<=k<=n) = A119282(n);
T(n,k) + T(n,n-k) = A094570(n,k).

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

Cf. A000045, A000004, A000071, A094570, A119282.

import Data.List (inits)
a141169 n k = a141169_tabl !! n !! k
a141169_row n = a141169_tabl !! n
a141169_tabl = tail $ inits a000045_list
a141169_list = concat $ a141169_tabl
-- Reinhard Zumkeller, Aug 24 2015, Mar 21 2011

A082987 a(n) = Sum_{k=0..n} 3^k*F(k) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

0, 3, 12, 66, 309, 1524, 7356, 35787, 173568, 842790, 4090485, 19856568, 96384072, 467861331, 2271040644, 11023873914, 53510987541, 259747827852, 1260842371428, 6120257564955, 29708354037720, 144207380197758

Offset: 0

Benoit Cloitre, May 29 2003

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,6,-9).

Cf. A014334, A082988, A119282.

LinearRecurrence[{4,6,-9},{0,3,12},30] (* Harvey P. Dale, Feb 03 2019 *)

a(n)=if(n<0,0,sum(k=0,n,fibonacci(k)*3^k))

a(0)=0, a(1)=3, a(2)=12, a(n)=4a(n-1)+6a(n-2)-9a(n-3).

G.f.: 3*x / ((x-1)*(9*x^2+3*x-1)). - Colin Barker, Jun 26 2013

Offset changed to 0 by Seiichi Manyama, Oct 03 2023

A082988 a(n) = Sum_{k=0..n} 4^k*F(k) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

0, 4, 20, 148, 916, 6036, 38804, 251796, 1628052, 10540948, 68212628, 441505684, 2857424788, 18493790100, 119693957012, 774676469652, 5013809190804, 32450060277652, 210021188163476, 1359285717096340, 8797481879000980

Offset: 0

Benoit Cloitre, May 29 2003

More generally for any complex number z, the sequence a(n) = Sum_{k=0..n} z^k*F(k) satisfies the recurrence: a(0) = 0, a(1) = z, a(2) = z(z+1), for n > 2 a(n) = (z+1)*a(n-1)+z*(z-1)*a(n-2)-z^2*a(n-3).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,12,-16).

Cf. A000045, A014334, A082987, A119282.

LinearRecurrence[{5, 12, -16}, {0, 4, 20}, 21] (* Amiram Eldar, Apr 29 2025 *)

a(n)=if(n<0,0,sum(k=0,n,fibonacci(k)*4^k));

a(0) = 0, a(1) = 4, a(2) = 20, a(n) = 5a(n-1)+12a(n-2)-16a(n-3).

O.g.f.: 4*x/((x-1)*(16*x^2+4*x-1)). - R. J. Mathar, Dec 05 2007

A117724 Triangle T(n,k) = coefficient [x^n] of x^2/(1-(k+1)*x^2-x^3) for row n, and columns k = 0..n, read by rows.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 4, 9, 16, 25, 36, 49, 2, 4, 6, 8, 10, 12, 14, 16, 2, 9, 28, 65, 126, 217, 344, 513, 730, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 4, 22, 90, 268, 640, 1314, 2422, 4120, 6588, 10030, 14674

Offset: 0

Roger L. Bagula, Apr 13 2006

			The table starts:
  0;
  0,  0;
  1,  1,  1;
  0,  0,  0,  0;
  1,  2,  3,  4,   5;
  1,  1,  1,  1,   1,   1;
  1,  4,  9, 16,  25,  36,  49;
  2,  4,  6,  8,  10,  12,  14,  16;
  2,  9, 28, 65, 126, 217, 344, 513, 730;
  3, 12, 27, 48,  75, 108, 147, 192, 243, 300;

Nathaniel Johnston, Rows n = 0..50, flattened

Cf. A000931, A008346, A052931, A117716, A119282.

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A117724:= func< n, k | Coefficient(R!( x^2/(1-(k+1)*x^2-x^3) ), n) >;
[A117724(n, k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 23 2023

t:=taylor(x^2/(1-(k+1)*x^2-x^3), x, 15):
seq(seq(coeff(t,x,n), k=0..n),n=0..12); # Nathaniel Johnston, Apr 27 2011

T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x^2-x^3), {x,0,n+ 2}], x, n];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A117724(n, k):
    P. = PowerSeriesRing(QQ)
    return P( x^2/(1-(k+1)*x^2-x^3) ).list()[n]
flatten([[A117724(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 23 2023

T(n,k) = coefficient [x^n] ( x^2/(1-(k+1)*x^2-x^3) ).
T(n, 0) = A000931(n+1).
T(n, 1) = A008346(n-2) = (-1)^(n-1)*A119282(n-1).
T(n, 2) = A052931(n-2).

Sign in definition corrected, offset set to -1 by Assoc. Eds. of the OEIS, Jun 15 2010

Edited by G. C. Greubel, Jul 23 2023

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A181716 a(n) = a(n-1) + a(n-2) + (-1)^n, with a(0)=0 and a(1)=1.

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A141169 Triangle of Fibonacci numbers, read by rows: T(n,k) = A000045(k), 0<=k<=n.

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A082987 a(n) = Sum_{k=0..n} 3^k*F(k) where F(k) is the k-th Fibonacci number.

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A082988 a(n) = Sum_{k=0..n} 4^k*F(k) where F(k) is the k-th Fibonacci number.

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A117724 Triangle T(n,k) = coefficient [x^n] of x^2/(1-(k+1)*x^2-x^3) for row n, and columns k = 0..n, read by rows.

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