A001638 -id:A001638 - cp's OEIS frontend

A111574 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,-1,2,3.

1, -1, 2, 3, 3, 4, 9, 15, 22, 35, 59, 96, 153, 247, 402, 651, 1051, 1700, 2753, 4455, 7206, 11659, 18867, 30528, 49393, 79919, 129314, 209235, 338547, 547780, 886329, 1434111, 2320438, 3754547, 6074987, 9829536, 15904521, 25734055, 41638578, 67372635

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

Floretion Algebra Multiplication Program, FAMP Code: -4baseiseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1).

Cf. A001638, A111569, A111570, A111571, A111572, A111573, A111575, A111576.

LinearRecurrence[{1,0,1,1},{1,-1,2,3},40] (* Harvey P. Dale, Jan 24 2017 *)

G.f.: (-1+2*x-3*x^2)/((x^2+x-1)*(x^2+1)).

Name clarified by Robert C. Lyons, Feb 06 2025

A362750 Number of total dominating sets in the n-double cone graph.

Original entry on oeis.org

4, 16, 79, 336, 1144, 4351, 17224, 67936, 267919, 1063216, 4233904, 16882191, 67380304, 269142736, 1075602319, 4299846976, 17192621224, 68752838911, 274965310744, 1099740514416, 4398645585679, 17593754283616, 70372850295904, 281485727082511, 1125928050595744

Offset: 1

Eric W. Weisstein, May 02 2023

The n-double cone graph is defined for n >= 3. The sequence has been extended to n=1 using the formula/recurrence. - Andrew Howroyd, May 03 2023

Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Double Cone Graph
Eric Weisstein's World of Mathematics, Total Dominating Set
Index entries for linear recurrences with constant coefficients, signature (7,-15,18,-24,-6,27,-15,13,-4).

Cf. A000032, A001638, A056594, A302603.

Table[1 + 4 (-1)^n + 4^n + LucasL[2 n] + 4 LucasL[n] Cos[n Pi/2], {n, 20}] (* Eric W. Weisstein, Sep 09 2023 *)
Table[(2 Cos[n Pi/2] + Fibonacci[n + 1] + Fibonacci[n - 1])^2 + 4^n - 1, {n, 20}] (* Eric W. Weisstein, Sep 09 2023 *)
LinearRecurrence[{7, -15, 18, -24, -6, 27, -15, 13, -4}, {4, 16, 79, 336, 1144, 4351, 17224, 67936, 267919}, 20] (* Eric W. Weisstein, Sep 09 2023 *)
CoefficientList[Series[4/(1 - 4 x) + 1/(1 - x) - 4/(1 + x) + (3 - 2 x)/(1 + (-3 + x) x) - 4 x (3 + 2 x^2)/(1 + 3 x^2 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 09 2023 *)

a(n) = {(fibonacci(n+1) + fibonacci(n-1) + I^n + (-I)^n)^2 + 4^n - 1} \\ Andrew Howroyd, May 03 2023

From Andrew Howroyd, May 03 2023: (Start)
a(n) = A001638(n)^2 + 4^n - 1.
a(n) = (A000032(n) + 2*A056594(n))^2 + 4^n - 1.
a(2*n-1) = A302603(4*n-1).
a(n) = 7*a(n-1) - 15*a(n-2) + 18*a(n-3) - 24*a(n-4) - 6*a(n-5) + 27*a(n-6) - 15*a(n-7) + 13*a(n-8) - 4*a(n-9) for n > 9.
G.f.: x*(4 - 12*x + 27*x^2 - 49*x^3 - 215*x^4 + 369*x^5 - 237*x^6 + 207*x^7 - 64*x^8)/((1 - x)*(1 + x)*(1 - 4*x)*(1 - 3*x + x^2)*(1 + 3*x^2 + x^4)).
(End)

a(1)-a(2) prepended and a(16) and beyond from Andrew Howroyd, May 03 2023

A232965 Number of circular n-bit strings that, when circularly shifted by 3 bits, do not have coincident 1's in any position.

Original entry on oeis.org

1, 3, 1, 7, 11, 27, 29, 47, 64, 123, 199, 343, 521, 843, 1331, 2207, 3571, 5832, 9349, 15127, 24389, 39603, 64079, 103823, 167761, 271443, 438976, 710647, 1149851, 1860867, 3010349, 4870847, 7880599, 12752043, 20633239, 33386248, 54018521

Offset: 1

Gideon J. Kuhn, Dec 02 2013

K[n;s] = L[n/gcd(n,s)]^gcd(n,s) counts circular n-bit strings that, when circularly shifted by s bits, do not have coincident 1's in any position. K[n,s] = #{x|((x<<

K[n;1] = L[n] is the Lucas sequence; K[n;2] is the Fielder sequence A001638; K[n;3] is this sequence.

			K[1;3] = L[1] = 1; K[2;3] = L[2] = 3; K[3;3] = L[1] = 1; K[4;3] = L[4] = 7;
K[5;3] = L[5] = 11; K[6;3] = L[2]^3 = 27; K[7;3] = L[7] = 29; K[8;3] = L[8] = 47.

Rick L. Shepherd, Table of n, a(n) for n = 1..4750

Cf. A000032 (Lucas sequence), A001638 (Fielder sequence).

int gcd(int n, int s)//Return the gcd of n and s
int raiseToPower(int n, int d)//Return n^d
#define N 40
#define S 3
int Lucas[N+1] = {2,1,3,4,7,1,18,....};
main()
{
int n;
for(n = 1; n < N; n++)
printf("%i: %i\n",n,raiseToPower(Lucas[n/gcd(n,S)],gcd(n,S)));
return;
}

A232965[n_] := LucasL[n/#]^# & [GCD[n, 3]]; Array[A232965, 50] (* Paolo Xausa, Feb 25 2025 *)

L(n) = fibonacci(n-1) + fibonacci(n+1);
a(n) = L(n/gcd(n,3))^gcd(n,3) \\ Rick L. Shepherd, Jan 23 2014

a(n) = A000032(n/gcd(n,3))^gcd(n,3).
K[n;3] satisfies the (empirical) linear recurrence a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) +a(n-5) + a(n-6) - a(n-7) - a(n-8), n > 8, derived from the denominator polynomial (1+phi^(-1)*x)*(1-phi*x)*(1-phi^(-1)*x^3)*(1+phi*x^3) of the generating function, where phi = (1+sqrt(5)/2), the golden ratio.
Empirical g.f.: -x*(x-1)*(8*x^6+15*x^5+9*x^4+4*x^3+3*x+1) / ((x^2+x-1)*(x^6-x^3-1)). - Colin Barker, Oct 10 2015

More terms from Rick L. Shepherd, Jan 23 2014

Previous Showing 11-13 of 13 results.

cp's OEIS Frontend

A111574 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,-1,2,3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A362750 Number of total dominating sets in the n-double cone graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A232965 Number of circular n-bit strings that, when circularly shifted by 3 bits, do not have coincident 1's in any position.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Mathematica

PARI

Formula

Extensions