A008611 -id:A008611 - cp's OEIS frontend

A106251 Expansion of (1-x+x^2+x^3+x^5)/(1-x-x^6+x^7).

1, 0, 1, 2, 2, 3, 4, 3, 4, 5, 5, 6, 7, 6, 7, 8, 8, 9, 10, 9, 10, 11, 11, 12, 13, 12, 13, 14, 14, 15, 16, 15, 16, 17, 17, 18, 19, 18, 19, 20, 20, 21, 22, 21, 22, 23, 23, 24, 25, 24, 25, 26, 26, 27, 28, 27, 28, 29, 29, 30, 31, 30, 31, 32, 32, 33, 34, 33, 34, 35, 35, 36, 37, 36, 37, 38

Offset: 0

Paul Barry, Apr 27 2005

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

Cf. A008611, A106249, A106250.

LinearRecurrence[{1,0,0,0,0,1,-1},{1,0,1,2,2,3,4},80] (* Harvey P. Dale, Oct 04 2021 *)

G.f.: (1+x^2+2x^3+2x^4+3x^5+2x^6+3x^7+2x^8+x^9+x^10)/(1-x^6)^2; a(n)=sum{k=0..n, -mu(k mod 6)}.

A283431 a(n) is the number of zeros of the Hermite H(n, x) polynomial in the open interval (-1, +1).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9

Offset: 0

Michel Lagneau, May 16 2017

nonn

The Hermite polynomials satisfy the following recurrence relation:
H(0,x) = 1,
H(1,x) = 2*x,
H(n,x) = 2*x*H(n-1,x) - 2*(n-1)*H(n-2,x).
The first few Hermite polynomials are:
H(0,x) = 1
H(1,x) = 2x
H(2,x) = 4x^2 - 2
H(3,x) = 8x^3 - 12x
H(4,x) = 16x^4 - 48x^2 + 12
H(5,x) = 32x^5 - 160x^3 + 120x

			a(5) = 3 because the zeros of H(5,x) = 32x^5 - 160x^3 + 120x are x1 = -2.0201828..., x2 = -.9585724..., x3 = 0., x4 = .9585724... and x5 = 2.020182... with three roots x2, x3 and x4 in the open interval (-1, +1).

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Index entries for sequences related to Hermite polynomials

Cf. A054373, A054374, A059343, A008611, A096713, A257564, A285872.

for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(HermiteH(n,x)),x,real)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od:

a[n_] := Length@ List@ ToRules@ Reduce[ HermiteH[n, x] == 0 && -1 < x < 1, x]; Array[a, 82, 0] (* Giovanni Resta, May 17 2017 *)

Conjecture: a(n) = A257564(n+2).

A355399 a(n) is the failed skew zero forcing number of C^2_n.

Original entry on oeis.org

0, 1, 2, 4, 3, 4, 6, 5, 6, 8, 6, 8, 10, 8, 10, 12, 10, 12, 14, 12, 14, 16, 14, 16, 18, 16, 18, 20, 18, 20, 22, 20, 22, 24, 22, 24, 26, 24, 26, 28, 26, 28, 30, 28, 30, 32, 30, 32, 34, 32, 34, 36, 34, 36, 38, 36, 38, 40, 38, 40, 42, 40, 42, 44, 42, 44, 46, 44, 46

Offset: 3

Darren Narayan, Andrew E. Vick, and Aidan Johnson, Jun 30 2022

Given a graph G where each vertex is initially considered filled or unfilled, we apply the skew color change rule, which states that a vertex v becomes filled if and only if it is the unique empty neighbor of some other vertex in the graph. The failed skew zero forcing number of G, is the maximum cardinality of any subset S of vertices on which repeated application of the color change rule will not result in all vertices being filled. Note that C^2_n = Ci_n(1,2) is the square of C_n.

Thomas Ansill, Bonnie Jacob, Jaime Penzellna, and Daniel Saavedra, Failed skew zero forcing on a graph, Linear Algebra and its Applications, vol. 509 (2016), 40-63.
Aidan Johnson, Andrew Vick, Rigoberto Flórez, and Darren A. Narayan, Failed Skew Zero Forcing Numbers of Path Powers and Circulant Graphs, AppliedMath (2025) Vol. 5, No. 2, 32.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A008611, A343648.

a(n) = 2*floor(n/3) + 2*(ceiling(n/(3*floor(n/3) + 1)) - floor(n/(3*floor(n/3) +1 )) - 1) for n >= 11.

a(n) = 2*A008611(n-3) for n >= 11.

More terms from Stefano Spezia, Jun 30 2022

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A106251 Expansion of (1-x+x^2+x^3+x^5)/(1-x-x^6+x^7).

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A283431 a(n) is the number of zeros of the Hermite H(n, x) polynomial in the open interval (-1, +1).

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A355399 a(n) is the failed skew zero forcing number of C^2_n.

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