A095364 -id:A095364 - cp's OEIS frontend

A095369 Number of walks of length n between two nodes at distance 4 in the cycle graph C_9.

1, 1, 6, 7, 28, 36, 120, 165, 495, 716, 2003, 3018, 8024, 12512, 31977, 51357, 127110, 209475, 504736, 850840, 2003784, 3445885, 7956715, 13926276, 31609071, 56191734, 125640180, 226444616, 499685777, 911607609, 1988440598

Offset: 4

Herbert Kociemba, Jul 03 2004

In general, (2^n/m)*Sum_{r=0..m-1} cos(2*Pi*k*r/m)*cos(2*Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=4.

Michael De Vlieger, Table of n, a(n) for n = 4..3325
Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Cf. A095364, A095367, A095368.

Drop[CoefficientList[Series[-x^4/((1 + x) (-1 + 2 x) (1 - 3 x^2 + x^3)), {x, 0, 34}], x], 4] (* Michael De Vlieger, Jan 23 2022 *)

a(n) = (2^n/9)*Sum_{r=0..8} cos(8*Pi*r/9)*cos(2*Pi*r/9)^n.
G.f.: x^4/((1+x)(-1+2x)(1-3x^2+x^3)).
a(n) = a(n-1) + 5*a(n-2) - 4*a(n-3) - 5*a(n-4) + 2*a(n-5).

A095367 Number of walks of length n between two nodes at distance 2 in the cycle graph C_9.

Original entry on oeis.org

1, 0, 4, 0, 15, 1, 56, 9, 210, 56, 792, 299, 3003, 1470, 11441, 6868, 43776, 31008, 168151, 136629, 648208, 591261, 2507046, 2523676, 9726080, 10656387, 37839375, 44612702, 147600981, 185477216, 577147212, 766744608, 2261792303

Offset: 2

Herbert Kociemba, Jul 03 2004

In general (2^n/m)*Sum_{r=0..m-1} cos(2*Pi*k*r/m)*cos(2*Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=2.

Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Cf. A095364, A095368, A095369.

a(n) = (2^n/9)*Sum_{r=0..8} cos(4*Pi*r/9)*cos(2*Pi*r/9)^n.
G.f.: x^2(-1+x+x^2)/((1+x)*(-1+2x)*(1-3x^2+x^3));
a(n) = a(n-1) + 5*a(n-2) - 4*a(n-3) - 5*a(n-4) + 2*a(n-5).

A095368 Number of walks of length n between two nodes at distance 3 in the cycle graph C_9.

Original entry on oeis.org

1, 0, 5, 1, 21, 8, 84, 45, 330, 221, 1287, 1015, 5006, 4488, 19465, 19380, 75753, 82365, 295261, 346104, 1152944, 1442101, 4510830, 5969561, 17682795, 24582663, 69448446, 100804436, 273241161, 411921832, 1076832989

Offset: 3

Herbert Kociemba, Jul 03 2004

In general (2^n/m)*Sum_{r=0..m-1} cos(2Pi*k*r/m)*cos(2Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=3.

Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Cf. A095364, A095367, A095369.

LinearRecurrence[{1,5,-4,-5,2},{1,0,5,1,21},40] (* Harvey P. Dale, Dec 21 2024 *)

a(n) = (2^n/9)*Sum_{r=0..8} cos(2Pi*r/3)*cos(2Pi*r/9)^n.
G.f.: (-1+x)x^3/((1+x)(-1+2x)(1-3x^2+x^3)).
a(n) = a(n-1)+5a(n-2)-4a(n-3)-5a(n-4)+2a(n-5).

A378031 Cogrowth sequence for the 18-element group C6 X C3 = .

Original entry on oeis.org

1, 1, 2, 85, 926, 5461, 37130, 349525, 2973350, 22369621, 174174002, 1431655765, 11582386286, 91625968981, 729520967450, 5864062014805, 47006639297270, 375299968947541, 2999857885752002, 24019198012642645, 192222214478506046, 1537228672809129301

Offset: 0

Sean A. Irvine, Nov 14 2024

Sequence gives terms for n = 0 (mod 3), all other terms are 0.

Cf. A095364 (D9), A377627 (C6 X C2), A007613 (C3 X C3), A378109 (S3 X C3), A378110 (S3:C3).

G.f.: (36*x^4+99*x^3-14*x^2+6*x-1) / ((8*x-1) * (x+1) * (27*x^2+1)).

A378109 Cogrowth sequence of the 18-element group S3 X C3 = .

Original entry on oeis.org

1, 0, 1, 2, 3, 15, 32, 126, 351, 1094, 3321, 9801, 29768, 88452, 266085, 797162, 2390391, 7175547, 21516800, 64573362, 193700403, 581130734, 1743421725, 5230147077, 15690706952, 47071500840, 141215033961, 423644304722, 1270932117003, 3812799539655

Offset: 0

Sean A. Irvine, Nov 16 2024

Also called: D3 X C3. Gap identifier 18, 3.

Cf. A095364 (D9), A378031 (C6 X C3), A378110 (S3:C3), A001045 (S3).

G.f.: (3*x^8-3*x^6-16*x^5+6*x^4+7*x^3-2*x^2-3*x+1) / ((x-1) * (3*x-1) * (x^2+x+1) * (3*x^2+3*x+1) * (3*x^2-3*x+1)).

A378110 Cogrowth sequence of the 18-element group S3:C3 = .

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 53, 98, 397, 1058, 3341, 9658, 30053, 87386, 267877, 793682, 2397437, 7162066, 21552125, 64506026, 193839445, 580889738, 1743836117, 5229312706, 15692323949, 47067796610, 141222563821, 423628907162, 1270962692165, 3812740639930

Offset: 0

Sean A. Irvine, Nov 16 2024

Also called: C3^2:C2. Gap identifier 18, 4.

Cf. A095364 (D9), A378031 (C6 X C3), A378109 (S3 X C3), A001045 (S3).

G.f.: (6*x^5-4*x^4-8*x^3-5*x^2+1) / ((x-1) * (3*x-1) * (2*x+1) * (3*x^2+2*x+1)).

cp's OEIS Frontend

A095369 Number of walks of length n between two nodes at distance 4 in the cycle graph C_9.

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A095367 Number of walks of length n between two nodes at distance 2 in the cycle graph C_9.

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A095368 Number of walks of length n between two nodes at distance 3 in the cycle graph C_9.

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A378031 Cogrowth sequence for the 18-element group C6 X C3 = .

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A378109 Cogrowth sequence of the 18-element group S3 X C3 = .

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A378110 Cogrowth sequence of the 18-element group S3:C3 = .

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