A101000 -id:A101000 - cp's OEIS frontend

A130624 Binomial transform of A101000.

0, 1, 5, 12, 23, 43, 84, 169, 341, 684, 1367, 2731, 5460, 10921, 21845, 43692, 87383, 174763, 349524, 699049, 1398101, 2796204, 5592407, 11184811, 22369620, 44739241, 89478485, 178956972, 357913943, 715827883, 1431655764, 2863311529, 5726623061, 11453246124

Offset: 0

Paul Curtz, Jun 18 2007

nonn

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

Cf. A101000, A119910, A130625 (first differences), A130626 (second differences).

m:=32; S:=[[0, 1, 3][(n-1) mod 3 +1]: n in [1..m]]; [&+[Binomial(i-1, k-1)*S[k]: k in [1..i]]: i in [1..m]]; /* Klaus Brockhaus, Jun 21 2007 */

I:=[0,1,5]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Nov 15 2018

LinearRecurrence[{3,-3,2},{0,1,5},40] (* Harvey P. Dale, Mar 05 2013 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(2^n) + a[n-1] - a[n-2]}, a, {n, 50}] (* Vincenzo Librandi, Nov 15 2018 *)

{m=32; v=concat([0, 1, 5], vector(m-3)); for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} /* Klaus Brockhaus, Jun 21 2007 */

a(0)=0, a(1)=1, a(2)=5; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(0)=0; a(n+1) = 2*a(n) + A119910(n).
G.f.: x*(1 + 2*x)/((1 - 2*x)*(1 - x + x^2)).
a(n) = 2^n + a(n-1) - a(n-2). - Jon Maiga, Nov 14 2018

Edited and extended by Klaus Brockhaus, Jun 21 2007

A288913 a(n) = Lucas(4*n + 3).

Original entry on oeis.org

4, 29, 199, 1364, 9349, 64079, 439204, 3010349, 20633239, 141422324, 969323029, 6643838879, 45537549124, 312119004989, 2139295485799, 14662949395604, 100501350283429, 688846502588399, 4721424167835364, 32361122672259149, 221806434537978679, 1520283919093591604

Offset: 0

Bruno Berselli, Jun 19 2017

a(n) mod 4 gives A101000.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).

Cf. A000032, A000045, A004187, A101000.
Cf. A033891: fourth quadrisection of A000045.
Partial sums are in A081007 (after 0).
Positive terms of A098149, and subsequence of A001350, A002878, A016897, A093960, A068397.
Quadrisection of A000032: A056854 (first), A056914 (second), A246453 (third, without 11), this sequence (fourth).

[Lucas(4*n + 3): n in [0..30]]; // G. C. Greubel, Dec 22 2017

LucasL[4 Range[0, 21] + 3]
LinearRecurrence[{7,-1}, {4,29}, 30] (* G. C. Greubel, Dec 22 2017 *)

Vec((4 + x)/(1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 2017

from sympy import lucas
def a(n):  return lucas(4*n + 3)
print([a(n) for n in range(22)]) # Michael S. Branicky, Apr 29 2021

def L():
    x, y = -1, 4
    while True:
        yield y
        x, y = y, 7*y - x
r = L(); [next(r) for  in (0..21)] # _Peter Luschny, Jun 20 2017

G.f.: (4 + x)/(1 - 7*x + x^2).
a(n) = 7*a(n-1) - a(n-2) for n>1, with a(0)=4, a(1)=29.
a(n) = ((sqrt(5) + 1)^(4*n + 3) - (sqrt(5) - 1)^(4*n + 3))/(8*16^n).
a(n) = Fibonacci(4*n+4) + Fibonacci(4*n+2).
a(n) = 4*A004187(n+1) + A004187(n).
a(n) = 5*A003482(n) + 4 = 5*A081016(n) - 1.
a(n) = A002878(2*n+1) = A093960(2*n+3) = A001350(4*n+3) = A068397(4*n+3).
a(n+1)*a(n+k) - a(n)*a(n+k+1) = 15*Fibonacci(4*k). Example: for k=6, a(n+1)*a(n+6) - a(n)*a(n+7) = 15*Fibonacci(24) = 695520.

A102002 Weighted tribonacci (1,2,4), companion to A102001.

Original entry on oeis.org

1, 7, 13, 31, 85, 199, 493, 1231, 3013, 7447, 18397, 45343, 111925, 276199, 681421, 1681519, 4149157, 10237879, 25262269, 62334655, 153810709, 379529095, 936489133, 2310790159, 5701884805, 14069421655, 34716351901, 85662734431, 211373124853, 521564001319

Offset: 1

Gary W. Adamson, Dec 23 2004

a(n)/a(n-1) tends to 2.46750385...an eigenvalue of M and a root of the characteristic polynomial x^3 - x^2 - 2x - 4. A102001 is generated from [1 1 1 / 2 0 0 / 0 2 0] but has the same characteristic polynomial and recursive multipliers (1,2,4). A101000 uses the recursive multipliers (1,2,4,8).

			a(6) = 199 = 85 + 2*31 + 4*13 = a(5) + 2*a(4) + 4*a(3).
a(6) = 199 since M^6 * [1 1 1] = [85 199 493] = [a(5) a(6) a(7)].

Index entries for linear recurrences with constant coefficients, signature (1,2,4).

Cf. A102000, A102001.

LinearRecurrence[{1,2,4}, {1,7,13}, 50] (* Harvey P. Dale, Apr 28 2012 *)

from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(1,1,1,1,2,4)
[next(it) for i in range(32)]
# Zerinvary Lajos, Jun 25 2008

a(n) = a(n-1) + 2*a(n-2) + 4*a(n-3), a>3. a(n) = center term in M^n * [1 1 1], where M = the 3X3 matrix [0 1 0 / 0 0 1 / 4 2 1]; M^n * [1 1 1] = [a(n-1) a(n) a(n+1)].

G.f.: -x*(4*x^2+6*x+1)/(4*x^3+2*x^2+x-1). [Harvey P. Dale, Apr 28 2012]

More terms from Harvey P. Dale, Apr 28 2012

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A130624 Binomial transform of A101000.

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A288913 a(n) = Lucas(4*n + 3).

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A102002 Weighted tribonacci (1,2,4), companion to A102001.

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