A046717 -id:A046717 - cp's OEIS frontend

A127620 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 6 with the steps (1,0), (0, 1), (2,0) and (0,2).

1, 1, 5, 22, 117, 654, 3843, 22882, 137443, 827998, 4995443, 30155494, 182083275, 1099560942, 6640309323, 40101959542, 242184540139, 1462610652718, 8833070227499, 53345145429670, 322164911643723, 1945636121710110

Offset: 0

Arvind Ayyer, Jan 20 2007

			a(2)=5 because we can reach (2,2) in the following ways:
(0,0),(1,0),(1,1),(2,1),(2,2)
(0,0),(2,0),(2,2)
(0,0),(1,0),(2,0),(2,2)
(0,0),(2,0),(2,1),(2,2)
(0,0),(1,0),(2,0),(2,1),(2,2)

Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points, arXiv:math/0610734 [math.CO], 2006-2007.
Index entries for linear recurrences with constant coefficients, signature (6, 5, -24, -28, -6, 8).

Cf. A000108, A046717, A122951, A127617, A127618, A127619.

b[n_, k_] := Boole[n >= 0 && k >= 0 && 0 <= n-k <= 6];
T[0, 0] = T[1, 1] = 1; T[n_, k_] /; b[n, k] == 1 := T[n, k] = b[n-1, k]* T[n-1, k] + b[n-2, k]*T[n-2, k] + b[n, k-1]*T[n, k-1] + b[n, k-2]*T[n, k-2]; T[, ] = 0;
a[n_] := T[n, n];
Table[a[n], {n, 0, 21}]
(* or: *)
LinearRecurrence[{6, 5, -24, -28, -6, 8}, {1, 1, 5, 22, 117, 654}, 22] (* Jean-François Alcover, Apr 02 2019 *)

G.f.: (1 - 5x - 6x^2 + 11x^3 + 12x^4 - 4x^5)/(1 - 6x - 5x^2 + 24x^3 + 28x^4 + 6x^5 - 8x^6). [corrected by Jean-François Alcover, Apr 02 2019]

A164907 a(n) = (3*3^n-(-1)^n)/2.

Original entry on oeis.org

1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Klaus Brockhaus, Aug 31 2009

Interleaving of A096053 and A083884 without initial term 1.
Partial sums are (essentially) in A080926.
First differences are (essentially) in A105723.
a(n)+a(n+1) = A008776(n+1) = A099856(n+1) = A110593(n+2).
Binomial transform of A056450. Inverse binomial transform of A164908.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Equals A046717 without initial term 1 and A080925 without initial term 0. Equals A084182 / 2 from second term onward.

Cf. A096053, A083884, A080926, A105723, A008776, A099856, A110593, A056450, A164908.

Magma
```
[ (3*3^n-(-1)^n)/2: n in [0..25] ];
```

A164907:=n->(3*3^n - (-1)^n)/2; seq(A164907(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Table[(3*3^n - (-1)^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 21 2014 *)
LinearRecurrence[{2,3},{1,5},50] (* Harvey P. Dale, Oct 31 2018 *)

a(n) = 2*a(n-1)+3*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1+3*x)/((1+x)*(1-3*x)).
a(n) = 3*a(n-1)+2*(-1)^n. - Carmine Suriano, Mar 21 2014

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A165553 a(n) = (3/2)*(1+(-3)^(n-1)).

Original entry on oeis.org

1, 3, -3, 15, -39, 123, -363, 1095, -3279, 9843, -29523, 88575, -265719, 797163, -2391483, 7174455, -21523359, 64570083, -193710243, 581130735, -1743392199, 5230176603, -15690529803, 47071589415, -141214768239

Offset: 0

Philippe Deléham, Sep 21 2009

a(n)/a(n-1) tends to -3.

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

Cf. A046717, A084247, A112555.

3/2*(1 + (-3)^(Range[0, 29] - 1)) (* or *)
LinearRecurrence[{-2, 3}, {1, 3}, 30] (* Paolo Xausa, Apr 22 2024 *)

a(0)=1, a(1)=3, a(n)=3*a(n-2)-2*a(n-1).
G.f.: (1+5x)/(1+2x-3x^2).
a(n)= Sum_{k=0..n} A112555(n,k)*2^(n-k).

A190943 a(n) = 8a(n-1) + 27a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 8, 91, 944, 10009, 105560, 1114723, 11767904, 124240753, 1311659432, 13847775787, 146197010960, 1543466033929, 16295047567352, 172033963454899, 1816237991957696, 19174820948943841, 202436993374408520, 2137216112616751867

Offset: 0

Bruno Berselli, May 24 2011

Bruno Berselli, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8, 27).

Cf. A000045, A046717, A015533 (for type of recurrence).

Cf. A015611, A190441 (for type of closed formula).

[n le 2 select n-1 else 8*Self(n-1)+27*Self(n-2): n in [1..17]];

a = {0, 1}; Do[AppendTo[a, 8 a[[-1]] + 27 a[[-2]]], {18}]; a (* Bruno Berselli, Dec 26 2012 *)
CoefficientList[Series[x / (1 - 8 x - 27 x^2), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 19 2013 *)

a[0]:0$ a[1]:1$ a[n]:=8*a[n-1]+27*a[n-2]$ makelist(a[n], n, 0, 17);

x='x+O('x^30); concat([0], Vec(x/(1-8*x-27*x^2))) \\ G. C. Greubel, Dec 30 2017

G.f.: x/(1-8*x-27*x^2).

a(n) = ((4+sqrt(43))^n - (4-sqrt(43))^n)/(2*sqrt(43)).

A326347 Number of unordered pairs of 4-colorings of an n-cycle that differ in the coloring of exactly one vertex.

Original entry on oeis.org

36, 240, 780, 2952, 10164, 35040, 118044, 393720, 1299012, 4251600, 13817388, 44641128, 143488980, 459165120, 1463588412, 4649045976, 14721978468, 46490458800, 146444944716, 460255541064, 1443528741876, 4518872583840, 14121476823900, 44059007691192

Offset: 3

Andrew Howroyd, Sep 11 2019

Andrew Howroyd, Table of n, a(n) for n = 3..200
Eric Weisstein's World of Mathematics, Cycle Graph
Index entries for linear recurrences with constant coefficients, signature (4,2,-12,-9).

Cf. A046717, A218034, A309380.

PARI
```
a(n) = 6*n*(3^(n-2) + (-1)^n);
```

a(n) = n*(3*A218034(n-2) + A218034(n-1)).
a(n) = 6*n*(3^(n-2) + (-1)^n).
a(n) = 12*n*A046717(n-2).
a(n) = 4*a(n-1) + 2*a(n-2) - 12*a(n-3) - 9*a(n-4) for n > 6.
G.f.: 12*x^3*(3 + 8*x - 21*x^2 - 18*x^3)/((1 + x)^2*(1 - 3*x)^2).

A375248 Expansion of (1 - x)/(1 - 2x - 3x^2)^(7/2).

Original entry on oeis.org

1, 6, 35, 168, 756, 3192, 12936, 50688, 193479, 722722, 2651649, 9581936, 34176324, 120526056, 420852204, 1456709328, 5002984791, 17062825626, 57827993685, 194871361608, 653285629920, 2179701604080, 7241015510820, 23958512912880, 78978801164445

Offset: 0

Seiichi Manyama, Aug 07 2024

nonn

First differences of A374506.
Cf. A046717, A109188, A371408.
Cf. A014531, A375253.

a[n_]:=(1+n)(2+n)(3+n)(4+n)(5+n)Hypergeometric2F1[(1-n)/2,-n/2,3,4]/120; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

my(N=30, x='x+O('x^N)); Vec((1-x)/(1-2*x-3*x^2)^(7/2))

a(n) = (binomial(n+5,3)/10) * Sum_{k=0..floor(n/2)} binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = (binomial(n+5,3)/10) * A014531(n+1).
a(n) = ((n+5)/(n*(n+4))) * ((2*n+3)*a(n-1) + 3*(n+4)*a(n-2)).
a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*hypergeom([(1-n)/2, -n/2], [3], 4)/120. - Stefano Spezia, Aug 07 2024

A378796 Number of minimal edge cuts in the n-sun graph.

Original entry on oeis.org

1, 6, 15, 44, 125, 370, 1099, 3288, 9849, 29534, 88583, 265732, 797173, 2391498, 7174467, 21523376, 64570097, 193710262, 581130751, 1743392220, 5230176621, 15690529826, 47071589435, 141214768264, 423644304745, 1270932914190, 3812798742519, 11438396227508, 34315188682469, 102945566047354

Offset: 1

Eric W. Weisstein, Dec 07 2024

The sequence has been extended to n=1 using the formula. - Andrew Howroyd, Dec 12 2024

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Minimal Edge Cut.
Eric Weisstein's World of Mathematics, Sun Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Cf. A046717, A377770.

Table[((-1)^n + 3^n + 2 n - 2)/2, {n, 20}]
LinearRecurrence[{4, -2, -4, 3}, {1, 6, 15, 44}, 20]
CoefficientList[Series[(-1 - 2 x + 7 x^2)/((-1 + x)^2 (-1 + 2 x + 3 x^2)), {x, 0, 20}], x]

a(n) = (3^n + (-1)^n)/2 + n - 1 \\ Andrew Howroyd, Dec 12 2024

a(n) = (3^n + (-1)^n)/2 + n - 1 = A046717(n) + n - 1. - Andrew Howroyd, Dec 12 2024
G.f.: x*(-1-2*x+7*x^2)/((-1+x)^2*(-1+2*x+3*x^2)). - Eric W. Weisstein, Dec 18 2024
E.g.f.: exp(x)*(cosh(2*x) - 1 + x). - Stefano Spezia, Dec 19 2024

a(1)-a(2) prepended and a(7) onwards from Andrew Howroyd, Dec 12 2024

A087213 Expansion of (1+x-4x^2) / ((1+x)(1-4*x^2)).

Original entry on oeis.org

1, 0, 0, 4, -4, 20, -20, 84, -84, 340, -340, 1364, -1364, 5460, -5460, 21844, -21844, 87380, -87380, 349524, -349524, 1398100, -1398100, 5592404, -5592404, 22369620, -22369620, 89478484, -89478484, 357913940, -357913940, 1431655764, -1431655764, 5726623060

Offset: 0

Paul Barry, Aug 26 2003

Binomial transform is A046717 (with extra leading 1).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,4,4).

CoefficientList[Series[(1 + x - 4 x^2)/((1 + x) (1 - 4 x^2)), {x, 0, 33}], x] (* or *) {1, 0}~Join~LinearRecurrence[{-1, 4, 4}, {0, 4, -4}, 32] (* Michael De Vlieger, Mar 17 2017 *)

Vec((1 + x - 4*x^2) / ((1 + x)*(1 - 2*x)*(1 + 2*x)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*n+1) = A002450(n) = A001045(2*n).
a(2*n+2) = -A002450(n) = -A001045(2*n).
a(n) = 2^n/6 - (-2)^n/2 + 4*(-1)^n/3.
a(n) = -a(n-1)+4*a(n-2)+4*a(n-3) for n>2.

A202389 Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -2, 2, 3, 1, -2, -5, 3, 5, 1, 3, -5, -10, 5, 8, -1, 3, 9, -10, -20, 8, 13, -1, -4, 9, 22, -20, -38, 13, 21, 1, -4, -14, 22, 51, -38, -71, 21, 34, 1, 5, -14, -40, 51, 111, -71, -130, 34, 55

Offset: 0

Philippe Deléham, Dec 18 2011

T(n,n) = A000045(n+1) = Fibonacci(n+1).

			Triangle begins :
1
1, 1
-1, 1, 2
-1, -2, 2, 3
1, -2, -5, 3, 5
1, 3, -5, -10, 5, 8
-1, 3, 9, -10, -20, 8, 13

Cf. A000007, A000045, A000244, A040000, A046717, A057077, A084938, A124137, A123585.

With[{m = 9}, CoefficientList[CoefficientList[Series[(1+x)/(1-y*x+(1-y^2)*x
^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k) = if (k<0, 0, if (nMichel Marcus, Feb 17 2020

T(n,k) = T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n

G.f.: (1+x)/(1-y*x+(1-y^2)*x^2).

Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A046717(n), A000007(n), A057077(n), A040000(n), A000244(n) for x = -2, -1, 0, 1, 2 respectively.

a(52) corrected by Georg Fischer, Feb 17 2020

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cp's OEIS Frontend

A127620 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 6 with the steps (1,0), (0, 1), (2,0) and (0,2).

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A164907 a(n) = (3*3^n-(-1)^n)/2.

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A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).

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A165553 a(n) = (3/2)*(1+(-3)^(n-1)).

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A190943 a(n) = 8*a(n-1) + 27*a(n-2), with a(0)=0, a(1)=1.

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Magma

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Maxima

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A326347 Number of unordered pairs of 4-colorings of an n-cycle that differ in the coloring of exactly one vertex.

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PARI

Formula

A375248 Expansion of (1 - x)/(1 - 2*x - 3*x^2)^(7/2).

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Mathematica

PARI

Formula

A378796 Number of minimal edge cuts in the n-sun graph.

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A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

A190943 a(n) = 8a(n-1) + 27a(n-2), with a(0)=0, a(1)=1.

A375248 Expansion of (1 - x)/(1 - 2x - 3x^2)^(7/2).

A087213 Expansion of (1+x-4x^2) / ((1+x)(1-4*x^2)).