A007877 -id:A007877 - cp's OEIS frontend

A251610 Determinants of the spiral knots S(4,k,(1,1,1)).

1, 4, 3, 0, 5, 12, 7, 0, 9, 20, 11, 0, 13, 28, 15, 0, 17, 36, 19, 0, 21, 44, 23, 0, 25, 52, 27, 0, 29, 60, 31, 0, 33, 68, 35, 0, 37, 76, 39, 0, 41, 84, 43, 0, 45, 92, 47, 0, 49, 100, 51, 0, 53, 108, 55, 0, 57, 116, 59, 0, 61, 124, 63, 0, 65, 132, 67, 0, 69, 140, 71, 0, 73, 148, 75, 0, 77, 156, 79

Offset: 1

Ryan Stees, Dec 05 2014

a(k) = det(S(4,k,(1,1,1))). These knots are also the torus knots T(4,k).

			For k=3, b(3)=sqrt(2)b(2)-b(1)=2-1=1, so det(S(4,3,(1,1,1)))=3*1^2=3.

Colin Barker, Table of n, a(n) for n = 1..1000
A. Breiland, L. Oesper, and L. Taalman, p-Coloring classes of torus knots, Online Missouri J. Math. Sci., 21 (2009), 120-126.
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Product of terms of A000027 and A007877.

B=vector(166); B[1]=1; B[2]=s;  \\ s := sqrt(2)
for(n=3,#B,B[n]=s*B[n-1]-B[n-2]);
B=substpol(B,s^2,2);
A=vector(#B,n,n*B[n]^2);
A=substpol(A,s^2,2)
\\ Joerg Arndt, Dec 06 2014

Vec(x*(x^4+2*x^3-2*x^2+2*x+1) / ((x-1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, Dec 07 2014

a(k) = det(S(4,k,(1,1,1))) = k*(b(k))^2, where b(1)=1, b(2)=sqrt(2), b(k)=sqrt(2)*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2).
From Colin Barker, Dec 06 2014: (Start)
b(k) = ((2-(-i)^k-i^k)*k)/2 where i=sqrt(-1).
b(k) = 2*b(k-1)-3*b(k-2)+4*b(k-3)-3*b(k-4)+2*b(k-5)-b(k-6).
G.f.: x*(x^4+2*x^3-2*x^2+2*x+1) / ((x-1)^2*(x^2+1)^2).
(End)

More terms from Joerg Arndt, Dec 06 2014

A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 2, 3, 2, 1, 0, 1, 2, 4, 4, 2, 2, 0, 1, 2, 5, 6, 4, 2, 1, 0, 1, 2, 6, 8, 7, 4, 2, 0, 0, 1, 2, 7, 10, 11, 8, 4, 2, 1, 0, 1, 2, 8, 12, 16, 14, 8, 4, 2, 2, 0, 1, 2, 9, 14, 22, 22, 15, 8, 4, 2, 1, 0, 1, 2, 10, 16, 29, 32, 26, 16, 8, 4, 2, 0

Offset: 0

Stefano Spezia, May 19 2024

A(n,k) is also the size of the Hamming ball in {0,1}^n of radius (k-1)/2 if k is odd and of the union of two Hamming balls in {0,1}^n of radius k/2-1 whose centers are of Hamming distance 1 if k is even.

			The array begins:
  1, 1, 2, 1,  0,  1,  2,  1, ...
  0, 1, 2, 2,  2,  2,  2,  2, ...
  0, 1, 2, 3,  4,  4,  4,  4, ...
  0, 1, 2, 4,  6,  7,  8,  8, ...
  0, 1, 2, 5,  8, 11, 14, 15, ...
  0, 1, 2, 6, 10, 16, 22, 26, ...
  0, 1, 2, 7, 12, 22, 32, 42, ...
  0, 1, 2, 8, 14, 29, 44, 64, ...
  ...

Noga Alon, Zhihan Jin, and Benny Sudakov, The Helly number of Hamming balls and related problems, arXiv:2405.10275 [math.CO], 2024. See p. 3.
S. L. Bezrukov, Specification of all maximal subsets of the unit cube with respect to given diameter. Problemy Peredachi Informatsii, pages 106-109, 1987. On ResearchGate.
G. Katona, Intersection theorems for systems of finite sets, Acta Mathematica Academiae Scientiarum Hungaricae 15, 329-337 (1964).
Daniel J. Kleitman, On a combinatorial conjecture of Erdös, Journal of Combinatorial Theory, Series A 1, 209-214, (1966).

Cf. A000007 (k=0), A000012 (k=1), A000124 (k=5), A000125 (k=7), A005843 (k=4), A006261 (k=11), A007395 (k=2), A008859 (k=13), A011782 (main diagonal), A014206, A046127 (k=8), A059173, A059174, A130130 (n=1), A158411 (n=2), A373006 (antidiagonal sums).

A[n_,k_]:=If[OddQ[k],Sum[Binomial[n,i],{i,0,(k-1)/2}], Binomial[n-1,k/2-1]+Sum[Binomial[n,i],{i,0,k/2-1}]]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(n,k) = Sum_{i=0..(k-1)/2} binomial(n,i) if k is odd;
A(n,k) = binomial(n-1,k/2-1) + Sum_{i=0..k/2-1} binomial(n,i) if k is even.
A(n,3) = n+1.
A(n,6) = A014206(n-1).
A(n,9) = A000127(n+1).
A(n,10) = A059173(n) for n > 0.
A(n,12) = A059174(n) for n > 0.
A(0,k) = A007877(k) for k > 0.

A204040 Triangle T(n,k), read by rows, given by (0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, -4, 4, 6, 1, 0, -4, -8, 12, 8, 1, 0, 4, -24, -4, 24, 10, 1, 0, 12, -8, -60, 16, 40, 12, 1, 0, 4, 56, -84, -96, 60, 60, 14, 1, 0, -20, 88, 84, -272, -100, 136, 84, 16, 1, 0, -28, -40

Offset: 0

Philippe Deléham, Jan 27 2012

Antidiagonal sums : periodic sequence 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, ... (see A007877 or A098178).Riordan array (1, x*(1+x)/(1-x+2*x^2)) .

			Triangle begins :
1
0, 1
0, 2, 1
0, 0, 4, 1
0, -4, 4, 6, 1
0, -4, -8, 12, 8, 1
0, 4, -24, -4, 24, 10, 1
0, 12, -8, -60, 16, 40, 12, 1
0, 4, 56, -84, -96, 60, 60, 14, 1
0, -20, 88, 84, -272, -100, 136, 84, 16, 1

Cf. A005408.

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2, k-1) - 2*T(n-2,k).
G.f.: (1-x+2*x^2)/(1-(1+y)*x + (2-y)*x^2).
T(n,n) = n = A000012(n), T(n+1,n) = 2n = A005843(n), T(n+2,n) = A046092(n-1) for n>0, T(n+1,1) = A078050(n)*(-1)^n.
Sum_{k, 0<=k<=n} T(n,k) = A060747(n) = A005408(n-1).

A257931 Period 24 sequence [0, 1, 1, 0, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, -1, 0, 1, 2, 0, -1, -1].

Original entry on oeis.org

0, 1, 1, 0, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, -1, 0, 1, 2, 0, -1, -1, 0, 1, 1, 0, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, -1, 0, 1, 2, 0, -1, -1, 0, 1, 1, 0, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, -1, 0, 1, 2, 0, -1, -1, 0, 1, 1, 0, -2, -1, 0

Offset: 0

Michael Somos, May 13 2015

			G.f. = x + x^2 - 2*x^4 - x^5 + x^7 - x^10 - x^11 + x^13 + x^14 - x^17 + ...

Michael Somos, Rational function multiplicative coefficients.
Index entries for linear recurrences with constant coefficients, signature (1,-1,0,1,-1,1,0,-1,1,-1).

Cf. A049347, A057078.

a[ n_] := (-1)^Quotient[ n, 3] Sign[Mod[n, 3]] - If[ Mod[n, 4] > 0, 0, (-1)^Quotient[ n, 12] Sign[Mod[n, 12]]];

{a(n) = (-1)^(n\3) * (n%3>0) - if( n%4, 0, (-1)^(n\12) * (n%12>0))};

{a(n) = my(m = abs(n)); sign(n) * polcoeff( x * (1 - x^3) * (1 - x^5) / ((1 - x + x^2) * (1 - x^4 + x^8)) + x * O(x^m), m)};

Euler transform of length 24 sequence [1, -1, -2, 1, -1, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1].
a(n) is multiplicative with a(2) = 1, a(4) = -2, a(2^e) = 0 if e > 2, a(3^e) = 0^e, a(p^e) = 1, if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).
a(n) = -a(-n) = a(n+24), a(n) + a(n+12) = 2 * A128834(n) for all n in Z.
a(2*n + 1) = a(4*n + 2) = a(8*n + 1) = a(8*n + 7) = A057078(n). a(3*n) = a(8*n) = 0. a(4*n + 1) = a(8*n + 2) = -a(8*n + 5) = A049347(n-1). a(6*n + 1) = -a(6*n + 5) = 1. a(6*n + 2) = A007877(n-1).
G.f.: f(x) - f(x^4) where f(x) := x / (1 - x + x^2).
G.f.: x * (1 - x^3) * (1 - x^5) / ((1 - x + x^2) * (1 - x^4 + x^8)).

cp's OEIS Frontend

A251610 Determinants of the spiral knots S(4,k,(1,1,1)).

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A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

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A204040 Triangle T(n,k), read by rows, given by (0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A257931 Period 24 sequence [0, 1, 1, 0, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, -1, 0, 1, 2, 0, -1, -1].

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