A147704 -id:A147704 - cp's OEIS frontend

A147703 Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 13, 27, 20, 7, 1, 34, 80, 73, 35, 9, 1, 89, 234, 252, 151, 54, 11, 1, 233, 677, 837, 597, 269, 77, 13, 1, 610, 1941, 2702, 2225, 1199, 435, 104, 15, 1, 1597, 5523, 8533, 7943, 4956, 2158, 657, 135, 17, 1

Offset: 0

Paul Barry, Nov 10 2008

Equal to A062110*A007318 when A062110 is regarded as a triangle read by rows.

			Triangle begins
   1;
   1,   1;
   2,   3,   1;
   5,   9,   5,   1;
  13,  27,  20,   7,  1;
  34,  80,  73,  35,  9,  1;
  89, 234, 252, 151, 54, 11, 1;

Indranil Ghosh, Rows 0..100, flattened

Row sums are A006012. Diagonal sums are A147704.

Cf. A062110, A007318, A206800, A001519, A321620.

# The function RiordanSquare is defined in A321620:
RiordanSquare(1 / (1 - x / (1 - x / (1 - x))), 10); # Peter Luschny, Jan 26 2020

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 2*x)/(1 - (3 + y)*x + (1 + y)*x^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 11 2017 *)

Riordan array ((1-2x)/(1-3x+x^2), x(1-x)/(1-3x+x^2)).
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Dec 01 2008
G.f.: (1-2*x)/(1-(3+y)*x+(1+y)*x^2). - Philippe Deléham, Nov 26 2011
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), for n > 1. - Philippe Deléham, Feb 12 2012
The Riordan square of the odd indexed Fibonacci numbers A001519. - Peter Luschny, Jan 26 2020

A263519 T(n,k) = Number of (n+1) X (k+1) arrays of permutations of 0..(n+1)*(k+1)-1 filled by rows with each element moved a city block distance of 0 or 1, and rows and columns in increasing lexicographic order.

Original entry on oeis.org

3, 7, 8, 15, 35, 23, 29, 160, 208, 66, 53, 660, 2076, 1198, 190, 93, 2651, 18369, 25968, 7022, 547, 159, 10350, 158109, 489294, 331130, 41035, 1575, 267, 39807, 1317780, 9051857, 13332096, 4213002, 240237, 4535, 443, 151463, 10791350, 162207955

Offset: 1

R. H. Hardin, Oct 19 2015

Table starts
.....3.......7.........15............29...............53..................93
.....8......35........160...........660.............2651...............10350
....23.....208.......2076.........18369...........158109.............1317780
....66....1198......25968........489294..........9051857...........162207955
...190....7022.....331130......13332096........529329240.........20339400914
...547...41035....4213002.....362159570......30867389241.......2543460828164
..1575..240237...53712998....9866744449....1805523575884.....319022980139204
..4535.1406038..684799391..268827612021..105637731091773...40028581755172441
.13058.8230727.8732881192.7327820172316.6184312882582853.5025951440933512579

			Some solutions for n=3 k=4
..0..1..7..8..9....0..1..7..8..9....0..1..2..3..4....0..1..2..4..9
..6..5..2..3..4...10..5..2..3..4....5..6..8..7..9....5..7..6..3..8
.10.12.11.13.19...11..6.12.13.14...15.10.13.12.14...10.12.11.14.13
.15.17.16.18.14...15.16.17.19.18...16.11.18.17.19...15.16.17.18.19

R. H. Hardin, Table of n, a(n) for n = 1..144

Column 1 is A147704(n+1).

Row 1 is A192960.

Empirical for column k:
k=1: a(n) = 3*a(n-1) -a(n-3)
k=2: [order 10]
k=3: [order 35]
Empirical for row n:
n=1: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4)
n=2: [order 10]
n=3: [order 29]
n=4: [order 92]

A215885 a(n) = 3*a(n-1) - a(n-3), with a(0) = 3, a(1) = 3, and a(2) = 9.

Original entry on oeis.org

3, 3, 9, 24, 69, 198, 570, 1641, 4725, 13605, 39174, 112797, 324786, 935184, 2692755, 7753479, 22325253, 64283004, 185095533, 532961346, 1534601034, 4418707569, 12723161361, 36634883049, 105485941578, 303734663373, 874569107070, 2518221379632, 7250929475523

Offset: 0

Roman Witula, Aug 25 2012

The Berndt-type sequence number 5a for the argument 2Pi/9 defined by the first relation from the section "Formula". We see that a(n) is equal to the sum of the n-th negative powers of the c(j) := 2*cos(2*Pi*j/9), j=1,2,4 (the A215664(n) is equal to the respective n-th positive powers, further both sequences can be obtained from the two-sided recurrence relation: X(n+3) = 3*X(n+1) - X(n), n in Z, with X(-1) = X(0) = 3, and X(1) = 0).
From the last formula in Witula's comments to A215664 it follows that 2*(-1)^n*a(n) = A215664(n)^2 - A215664(2*n).
The following decomposition holds true: (X - c(1)^(-n))*(X - c(2)^(-n))*(X - c(4)^(-n)) = X^3 - a(n)*X^2 - (-1)^n*A215664(n)*X - (-1)^n.
For n >= 1, a(n) is the number of cyclic (0,1,2)-compositions of n that avoid the pattern 110 provided the positions of the parts of the composition on the circle are fixed. (Similar comments hold for the pattern 012 and for the pattern 001.) - Petros Hadjicostas, Sep 13 2017
See the Maple program by Edlin and Zeilberger for counting the q-ary cyclic compositions of n that avoid one or more patterns provided the positions of the parts of the composition are fixed on the circle. The program is located at D. Zeilberger's personal website (see links). For the sequence here, q=3 and the pattern is A=110. - Petros Hadjicostas, Sep 13 2017

			For n=3, we have a(3) = 3^3 - 3 = 24 ternary cyclic compositions of n=3 (with fixed positions on the circle for the parts) that avoid 110 because we have to exclude 110, 101, and 011. - _Petros Hadjicostas_, Sep 13 2017

Michael De Vlieger, Table of n, a(n) for n = 0..1999
A. E. Edlin and D. Zeilberger, The Goulden-Jackson cluster method for cyclic words, Adv. Appl. Math. 25 (2000), 228-232.
A. E. Edlin and D. Zeilberger, Maple program; Local copy
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012), 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A130880, A147704, A215664, A215665, A215666, A274018.

Mathematica
```
LinearRecurrence[{3,0,-1}, {3,3,9}, 50]
```

my(x='x+O('x^30)); Vec(3*(1-2*x)/(1-3*x+x^3)) \\ Altug Alkan, Sep 13 2017

a(n) = 3*A147704(n).
a(n) = c(1)^(-n) + c(2)^(-n) + c(4)^(-n) = (-c(1)*c(2))^n + (-c(1)*c(4))^n + (-c(2)*c(4))^n, where c(j) := 2*cos(2*Pi*j/9).
G.f.: Sum_{n>=0} a(n)*x^n = 3-3*x*(x^2-1)/(1-3*x+x^3) = 3*(1-2*x)/(1-3*x+x^3).
G.f. of Edlin and Zeilberger (2000): 1+Sum_{n>=1} a(n)*x^n = 1-3*x*(x^2-1)/(1-3*x+x^3) = (1-2*x^3)/(1-3*x+x^3). - Petros Hadjicostas, Sep 13 2017
a(n) = ceiling(r^n) for n >= 1, where r = 1/A130880 is the largest root of x^3 - 3*x^2 + 1. - Tamas Lengyel, Feb 20 2022

A188128 Expansion of (4-6x-6x^2+x^3)/((1+x)(1-3x+x^3)).

Original entry on oeis.org

4, 2, 10, 23, 70, 197, 571, 1640, 4726, 13604, 39175, 112796, 324787, 935183, 2692756, 7753478, 22325254, 64283003, 185095534, 532961345, 1534601035, 4418707568, 12723161362, 36634883048, 105485941579, 303734663372, 874569107071

Offset: 0

L. Edson Jeffery, Apr 05 2011

Let A_{9,3} = [0,0,0,1; 0,0,1,1; 0,1,1,1; 1,1,1,1], a unit-primitive matrix (see [Jeffery]). Then a(n) = Trace([A_{9,3}]^n).

L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (2, 3, -1, -1).

CoefficientList[Series[(4-6x-6x^2+x^3)/((1+x)(1-3x+x^3)), {x,0,30}],x] (* or *) LinearRecurrence[{2,3,-1,-1},{4,2,10,23},30] (* Harvey P. Dale, Apr 22 2011 *)

G.f.: (4-6*x-6*x^2+x^3)/((1+x)*(1-3*x+x^3)).
a(n) = 2*a(n-1)+3*a(n-2)-a(n-3)-a(n-4), {a(m)}={4,2,10,23}, m=0,1,2,3.
a(n) = Sum_{k=1..4} ((x_k)^3-2*(x_k))^n, x_k=2*(-1)^(k-1)*cos(k*Pi/9).
a(n) = (-1)^n+(1+2*cos(Pi/9))^n+(1-cos(Pi/9)+sqrt(3)*sin(Pi/9))^n + (1-cos(Pi/9)-sqrt(3)*sin(Pi/9))^n. - L. Edson Jeffery, Dec 15 2011
a(n) = (-1)^n + 3*A147704(n). - R. J. Mathar, Oct 08 2016

A383078 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 6.

Original entry on oeis.org

9, 16, 50, 140, 407, 1168, 3367, 9691, 27908, 80354, 231374, 666211, 1918282, 5523469, 15904199, 45794312, 131859470, 379674208, 1093228315, 3147825472, 9063802211, 26098178315, 75146709476, 216376326214, 623030800330, 1793945691511, 5165460748322, 14873351444633

Offset: 1

Edwin Hermann, Apr 15 2025

Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A052268, A100062, A383070, A383071, A383072, A383077.

Cf. A147704.

G.f.: x*(2*x^4+x^3-9*x^2-2*x+9)/((x+1)*(x^3-3*x+1)). - Alois P. Heinz, Apr 15 2025

More terms from Alois P. Heinz, Apr 15 2025

cp's OEIS Frontend

A147703 Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

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Formula

A263519 T(n,k) = Number of (n+1) X (k+1) arrays of permutations of 0..(n+1)*(k+1)-1 filled by rows with each element moved a city block distance of 0 or 1, and rows and columns in increasing lexicographic order.

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Formula

A215885 a(n) = 3*a(n-1) - a(n-3), with a(0) = 3, a(1) = 3, and a(2) = 9.

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A188128 Expansion of (4-6*x-6*x^2+x^3)/((1+x)*(1-3*x+x^3)).

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Comments

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Programs

Mathematica

Formula

A383078 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 6.

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Author

Keywords

Links

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Formula

Extensions

A188128 Expansion of (4-6x-6x^2+x^3)/((1+x)(1-3x+x^3)).