A102001 -id:A102001 - cp's OEIS frontend

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

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

A295781 T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0 or 2 king-move neighboring 1s.

Original entry on oeis.org

2, 3, 3, 5, 9, 5, 8, 19, 19, 8, 13, 49, 56, 49, 13, 21, 123, 198, 198, 123, 21, 34, 297, 665, 1059, 665, 297, 34, 55, 739, 2213, 5263, 5263, 2213, 739, 55, 89, 1825, 7479, 25529, 37897, 25529, 7479, 1825, 89, 144, 4491, 25105, 127731, 262707, 262707, 127731

Offset: 1

R. H. Hardin, Nov 27 2017

Table starts
..2....3.....5.......8.......13.........21..........34............55
..3....9....19......49......123........297.........739..........1825
..5...19....56.....198......665.......2213........7479.........25105
..8...49...198....1059.....5263......25529......127731........630988
.13..123...665....5263....37897.....262707.....1905471......13577504
.21..297..2213...25529...262707....2602065....27085621.....276224518
.34..739..7479..127731..1905471...27085621...409525082....6037613762
.55.1825.25105..630988.13577504..276224518..6037613762..128171125105
.89.4491.84326.3118368.96726819.2819029784.89072774496.2722986415966

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..0. .0..0..1..0. .0..0..0..1. .0..0..0..0
..0..1..0..0. .0..0..0..1. .0..0..0..0. .1..1..0..0. .1..0..1..0
..0..0..0..0. .0..0..0..0. .1..0..0..0. .1..0..0..1. .0..0..1..1
..1..1..0..0. .1..1..0..0. .1..1..0..1. .0..0..1..1. .1..0..0..0
..1..0..0..1. .0..1..0..0. .0..0..0..0. .0..0..0..0. .1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..311

Column 1 is A000045(n+2).

Column 2 is A102001(n+1).

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +2*a(n-2) +4*a(n-3)
k=3: a(n) = 2*a(n-1) +3*a(n-2) +6*a(n-3) -3*a(n-4) +2*a(n-5)
k=4: [order 10]
k=5: [order 22]
k=6: [order 43]
k=7: [order 91]

A279682 The maximum number of coins that can be processed in n weighings where all coins are real except for one LHR-coin.

Original entry on oeis.org

1, 3, 9, 19, 49, 123, 297, 707, 1697, 4043, 9561, 22547, 53073, 124571, 291721, 682083, 1592577, 3713643, 8650425, 20132275, 46818225, 108804923, 252718825, 586701827, 1361496929, 3158352139, 7324384281, 16981143379, 39360789521

Offset: 0

Tanya Khovanova and Konstantin Knop, Dec 16 2016

An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.
Also the number of outcomes of n weighings such that every even-numbered imbalance that is not the last one must be followed by a balance or every odd-numbered imbalance that is not the last one must be followed by a balance.
The first seven terms coincide with sequence A102001, which counts all the outcomes that don't have three imbalances in a row.
This sequence also counts the possible outcomes starting in the light or heavy state, and for the coins starting in the real state the possible number of outcomes is a subset for coins starting in the light state.

			Consider a(7): in addition to outcomes that do not have three imbalances in a row, we are not allowed to have any outcomes like <<=<=<<, in which the first (odd-numbered imbalance) and the fourth (even-numbered imbalance) are both followed by an imbalance. We can replace a less-than sign with a greater-than sign. That means a(7) = A102001(7) - 32 = 739 - 32 = 707.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (3,-1,1,-2,-8).

Cf. A279673, A279674, A279684, A279685.

I:=[1,3,9,19,49]; [n le 5 select I[n] else 3*Self(n-1)-Self(n-2)+Self(n-3)- 2*Self(n-4)-8*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Dec 18 2016

LinearRecurrence[{3, -1, 1, -2, -8}, {1, 3, 9, 19, 49}, 30]

Vec((1 + x^2 - 6*x^3)/((1 + x)*(1 - 2*x)*(1 - 2*x + x^2 - 4*x^3)) + O(x^40)) \\ Colin Barker, Dec 19 2016

a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) - 8*a(n-5).

G.f.: (1 + x^2 - 6*x^3)/((1 + x)*(1 - 2*x)*(1 - 2*x + x^2 - 4*x^3)). - Ilya Gutkovskiy, Dec 17 2016

A103770 A weighted tribonacci sequence, (1,3,9).

Original entry on oeis.org

1, 1, 4, 16, 37, 121, 376, 1072, 3289, 9889, 29404, 88672, 265885, 796537, 2392240, 7174816, 21520369, 64574977, 193709428, 581117680, 1743420757, 5230158649, 15690480040, 47071742800, 141214610761, 423644159521, 1270933677004

Offset: 0

Paul Barry, Feb 15 2005

The weighted tribonacci (1,r,r^2) with g.f. 1/(1 - x - r*x^2 - r^2*x^3) has general term Sum_{k=0..n} T(n-k,k)r^k.
Correspondence: a(n) = b(n+2)*3^n, where b(n) is the sequence of the arithmetic means of the previous three terms defined by b(n) = (1/3)*(b(n-1) + b(n-2) + b(n-3)) with initial values b(0)=0, b(1)=0, b(2)=1; the g.f. for b(n) is B(x) := x^2/(1-(x^1+x^2+x^3)/3), so the g.f. A(x) for a(n) satisfies A(x) = B(3*x)/(3*x)^2. Because b(n) converges to the limit lim_{x->1} (1-x)*B(x) = (1/6)*(b(0) + 2*b(1) + 3*b(2)) = 1/2, it follows that a(n)/3^n also converges to 1/2. This correspondence is valid in general (with necessary changes) for weighted sequences of order (1, p, p^2, p^3, p^4, ..., p^(p-1)) with integer p > 0. Forming such sequences c(n) := c(n-1) + p^1*c(n-2) + ... + p^(p-1)*c(n-p) the limit of c(n)/p^n is 2/(p+1) (see also A001045). - Hieronymus Fischer, Feb 04 2006
a(n)/3^n equals the probability that n will occur as a partial sum in a randomly-generated infinite sequence of 1s, 2s and 3s. The limiting ratio is 1/2. - Bob Selcoe, Jul 05 2013
Number of compositions of n into one sort of 1's, three sorts of 2's, and nine sorts of 3's. - Joerg Arndt, Jul 06 2013
Using the Markov Chain {{0, 1, 0}, {0, 0, 1}, {1/3, 1/3, 1/3}} and raising it to the n-th power can generate this sequence when looking at the element in the third row and third column and reading the numerator. - Robert P. P. McKone, May 25 2021

Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
Index entries for linear recurrences with constant coefficients, signature (1,3,9).

Cf. A000073, A102001.
Cf. A071675, A027907.
Cf. A121262, A088137.

LinearRecurrence[{1, 3, 9}, {1, 1, 4}, {1, 27}] (* Robert P. P. McKone, May 25 2021 *)

G.f.: 1/(1 - x - 3*x^2 - 9*x^3).
a(n) = Sum_{k=0..n} T(n-k, k)*3^k, T(n, k) = trinomial coefficients (A027907).
a(n) = Sum_{k=0..n} 3^(n-k) * (Sum_{i=0..floor((n-k)/2)} C(n-k-i, i)*C(k, n-k-i)). - Paul Barry, Apr 26 2005
a(n)/3^n converges to 1/2. - Hieronymus Fischer, Feb 02 2006
a(n) = a(n-1) + 3*a(n-2) + 9*a(n-3), n >= 3; a(0)=1, a(1)=1, a(2)=4. - Hieronymus Fischer, Feb 04 2006
a(n) = 3^n + b(n) + b(n-1), with b(n) = (-1)^A121262(n+1)*A088137(n+1). - Ralf Stephan, May 20 2007

A103425 a(n) = 3a(n-1) + a(n-2) - 3a(n-3).

Original entry on oeis.org

1, 3, 5, 15, 41, 123, 365, 1095, 3281, 9843, 29525, 88575, 265721, 797163, 2391485, 7174455, 21523361, 64570083, 193710245, 581130735, 1743392201, 5230176603, 15690529805, 47071589415, 141214768241, 423644304723, 1270932914165

Offset: 0

Paul Barry, Feb 05 2005

Binomial transform of A103424.

This is a (3, 1, -3) weighted tribonacci sequence, cf. A102001. The current sequence contains primes, including 3, 5, 41, 21523361. Is there an (a, b, c) weighted tribonacci sequence with a, b, c relatively prime which is prime-free? The general linear third-order recurrence equation x(n) = a*x(n-1) + b*x(n-2) + c*x(n-3) has a solution in terms of roots of a cubic polynomial, see Weisstein. - Jonathan Vos Post, Feb 05 2005

Eric Weisstein's World of Mathematics, Linear Recurrence Equation.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

G.f.: (1-5x^2)/((1-x^2)(1-3x)).
E.g.f.: exp(x)(1+sinh(2x)).
a(n) = 1 + (3^n - (-1)^n)/2.

cp's OEIS Frontend

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

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A295781 T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0 or 2 king-move neighboring 1s.

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A279682 The maximum number of coins that can be processed in n weighings where all coins are real except for one LHR-coin.

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A103770 A weighted tribonacci sequence, (1,3,9).

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A103425 a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3).

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A103425 a(n) = 3a(n-1) + a(n-2) - 3a(n-3).