Simone Severini - cp's OEIS frontend

A130198 Single paradiddle. In percussion, the paradiddle is a four-note drum sticking pattern consisting of two alternating notes followed by two notes on the same hand.

0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1

Offset: 0

Simone Severini, May 16 2007

Also the binary expansion of the constant 5/17 = 2^(-2) + 2^(-5) + 2^(-7) + ... - R. J. Mathar, Mar 27 2009

Period 8: repeat [0, 1, 0, 0, 1, 0, 1, 1]. - Wesley Ivan Hurt, Aug 23 2015

Wikipedia, Paradiddle
Index entries for sequences related to music
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A121262, A131078. - Jaume Oliver Lafont, Mar 19 2009

Cf. A165211.

[(1-(-1)^((n+5)*(n+6)*(n^2+11*n+32) div 8))/2 : n in [0..100]]; // Wesley Ivan Hurt, Aug 23 2015

A130198:= n -> [0, 1, 0, 0, 1, 0, 1, 1][(n mod 8)+1]: seq(A130198(n), n=0..100); # Wesley Ivan Hurt, Aug 23 2015

CoefficientList[Series[x*(1 - x + x^3)/((1 - x)*(1 + x^4)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 23 2015 *)

a(n)=((n%8>3)+(n%4==1))%2 \\ Jaume Oliver Lafont, Mar 19 2009

a(n)=210\2^(n%8)%2; \\ Jaume Oliver Lafont, Mar 24 2009

apply( A130198(n)=bittest(210,n%8), [0..99]) \\ M. F. Hasler, May 24 2019

def A130198(n): return n&1^bool(n+1&4) # Chai Wah Wu, Aug 30 2024

From R. J. Mathar, Mar 27 2009: (Start)
a(n) = a(n-8) = a(n-1) - a(n-4) + a(n-5).
G.f.: -x*(1+x^3-x)/((x-1)*(1+x^4)). (End)
a(n) = (1-(-1)^((n+5)*(n+6)*(n^2+11*n+32)/8))/2. - Wesley Ivan Hurt, Aug 23 2015
a(n) = A165211(n+5). - Wesley Ivan Hurt, Aug 23 2015

A126857 Maximal number of vertices in an integral circulant graph of degree n or n+1.

Original entry on oeis.org

6, 12, 30, 42, 120

Offset: 2

Simone Severini, Mar 11 2007

nonn

Nitin Saxena, Simone Severini and Igor Shparlinski, Parameters of Integral Circulant Graphs, Preprint, 2007.

A127546 a(n) = F(n)^2 + F(n+1)^2 + F(n+2)^2, where F(n) denotes the n-th Fibonacci number.

Original entry on oeis.org

2, 6, 14, 38, 98, 258, 674, 1766, 4622, 12102, 31682, 82946, 217154, 568518, 1488398, 3896678, 10201634, 26708226, 69923042, 183060902, 479259662, 1254718086, 3284894594, 8599965698, 22515002498, 58945041798, 154320122894, 404015326886, 1057725857762

Offset: 0

Simone Severini, Apr 01 2007

nonn

The following conjecture, if not already well-known, is probably easy to prove: a(n) = 3a(n-1)-a(n-2)-2(-1)^n, for n=4,5,6,... . (This has been verified up to n=1000.)

			a(2)=14 because F(2)^2+F(3)^2+F(4)^2=1+4+9=14.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad and Doron Zeilberger, Automatic Counting of Tilings of Skinny Plane Regions, arXiv preprint arXiv:1206.4864 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A061646.

with(combinat): a:=n->fibonacci(n)^2+fibonacci(n+1)^2+fibonacci(n+2)^2: seq(a(n),n=0..32); # Emeric Deutsch, Apr 04 2007
A000045 := proc(n) combinat[fibonacci](n) ; end: A127546 := proc(n) add( A000045(i+1)^2,i=n..n+2) ; end: for n from 1 to 33 do printf("%d, ",A127546(n)) ; od ; # R. J. Mathar, Apr 03 2007
with(combinat): seq(4*fibonacci(n+1)^2-2*(-1)^n, n=0..29)

Total/@(Partition[Fibonacci[Range[0,30]],3,1]^2) (* Harvey P. Dale, Oct 20 2011 *)

for(n=0,10,print1(4*fibonacci(n+1)^2-2*(-1)^n,", "))

a(n) = 2*A061646(n+1) = 4*F(n+1)^2-2*(-1)^(n+1). - Emeric Deutsch, Apr 04 2007; Gary Detlefs, Nov 27 2010
a(n) = 2*(F(n)^2+F(n+1)^2+F(n)*F(n+1)). - Emeric Deutsch, Apr 04 2007
G.f.: 2(1+x-x^2)/((1+x)(1-3x+x^2)). - R. J. Mathar, Nov 25 2008

Edited and extended by R. J. Mathar, Emeric Deutsch and John W. Layman, Apr 09 2007

A120339 a(n) = 3abc, where (a,b,c) is a Markoff triple. The first Markoff triple considered is (1,2,5) and the ordering is increasing.

Original entry on oeis.org

30, 195, 870, 1326, 9078, 29406, 37830, 62211, 188355, 426390, 998790, 1756950, 2922510, 8430270, 20031171, 33929310, 41996670, 57206526, 82401006, 137295678, 216148803, 941038566, 1152597606, 1418915886, 1879906755, 2668548675, 3870146310, 6449974275, 9365063430, 18263765550, 39154389150, 44208781350

Offset: 1

Simone Severini, Jun 22 2006

nonn

Tom Ace, Markoff numbers.
Simone Severini, Homepage.

Cf. A002559.

Corrected and extended by T. D. Noe, Jan 28 2011

A117393 Number of different classes of entangling power of permutation matrices of dimension d^2, acting on a d X d (tensor product) Hilbert space.

Original entry on oeis.org

1, 2, 15, 65, 190, 447

Offset: 1

Lieven Clarisse and Simone Severini, Apr 25 2006

a(1), a(2) and a(3) were obtained via an exhaustive search; Lower bounds on a(4) through a(10) are 65, 190, 447, 890, 1603, 2675, 4200. - Lieven Clarisse, Jan 29 2017

I moved a(4)-a(6) back to the data field, even though they have not been proved. If they are deleted the sequence melts away. - N. J. A. Sloane, Jan 31 2017

Lieven Clarisse, Sibasish Ghosh, Simone Severini and Anthony Sudbery, Entangling Power of Permutations, Phys. Rev. A 72, 012314 (2005).

Moved terms a(4) and above to comments since those were not obtained via an exhaustive search (but believed to be correct). - Lieven Clarisse, Jan 29 2017. I restored a(4)-a(6), see note above. - N. J. A. Sloane, Jan 31 2017

A114533 Permanent of the n X n matrix with numbers prime(1),prime(2),...,prime(n^2) in order across rows.

Original entry on oeis.org

1, 2, 29, 3746, 1919534, 2514903732, 6571874957648, 30662862975835376, 228731722381012564816, 2641049525155781555257440, 43818773386947889568479502592, 1014966115357067575070490776083200, 31412851866841234377483875199638978304

Offset: 0

Simone Severini, Feb 15 2006

nonn

Previous name was : "a(n) = permanent of the n X n matrix M defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i is the i-th prime number".

Vaclav Kotesovec, Table of n, a(n) for n = 0..36 (terms 0..20 from Alois P. Heinz)

Cf. A067276, A232773.

with(LinearAlgebra):
a:= n->`if`(n=0, 1, Permanent(Matrix(n, (i, j)->ithprime((i-1)*n+j)))):
seq(a(n), n=0..12);  # Alois P. Heinz, Dec 23 2013

a[n_] := Permanent[Table[Prime[(i-1)*n+j], {i, 1, n}, {j, 1, n}]]; a[0]=1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Jan 07 2016, adapted from Maple *)
Join[{1},Table[Permanent[Partition[Prime[Range[n^2]],n]],{n,15}]] (* Harvey P. Dale, Aug 03 2019 *)

permRWN(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); n1=n-1; sg=1; m=1; nc=0; in=vector(n); x=in; for(i=1,n,x[i]=a[i,n]-sum(j=1,n,a[i,j])/2); p=prod(i=1,n,x[i]); while(m,sg=-sg; j=1; if((nc%2)!=0,j++; while(in[j-1]==0,j++)); in[j]=1-in[j]; z=2*in[j]-1; nc+=z; m=nc!=in[n1]; for(i=1,n,x[i]+=z*a[i,j]); p+=sg*prod(i=1,n,x[i])); return(2*(2*(n%2)-1)*p)
for(n=1,19,a=matrix(n,n,i,j,prime((i-1)*n+j)); print1(permRWN(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 11 2007

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p)
for(n=1,23,a=matrix(n,n,i,j,prime((i-1)*n+j));print1(permRWNb(a)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007

{a(n) = matpermanent(matrix(n, n, i, j, prime((i-1)*n+j)))}
for(n=0, 25, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 13 2021

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 11 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007
New name from Michel Marcus, Nov 30 2013
a(0) inserted and a(12) by Alois P. Heinz, Dec 23 2013

A114534 The n-th entry of the sequence is the value of the permanent of an n X n matrix M defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i is the i-th Fibonacci number.

Original entry on oeis.org

1, 132, 1460808, 6357011889600, 44491520971919463292800, 2082476039060691409777705387034081280, 2712373659248840873249840585282508476815021942277876736, 410721884168854528740774423430429149549703377187068577398353854503625738636800

Offset: 2

Simone Severini, Feb 15 2006

nonn

Conjecture: The rank of the matrix M is 2 for every n.

			For n=2, M=[0,1;1,2];
For n=3, M=[0,1,1;2,3,5;8,13,21].

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p)
a(n) = permRWNb(matrix(n,n,i,j,fibonacci((i-1)*n+j-1))); \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007

a(n) = matpermanent(matrix(n,n,i,j,fibonacci((i-1)*n+j-1))); \\ Michel Marcus, Apr 08 2025

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007

A114530 a(n) = permanent of a bordered n X n (1,-1)-matrix with the following property: the elements on the border are 1; if we concatenate the rows of the matrix to form a vector v of length n^2, v_i = -1 if i is not a prime. The border of a matrix consists of the first and the last row and the first and the last column.

Original entry on oeis.org

6, 0, 8, 144, -80, -384, -2816, -15360, -125184, 5322240, -22966272, 36771840, -887224320, 1488936960, -217760382976, -1484266291200, -45948014198784, 65021593190400, -3267216288645120, -856122753024000, -3180322010587725824

Offset: 3

Simone Severini, Feb 15 2006

sign

Simone Severini, Home page

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p)
for(n=3,23,a=matrix(n,n,i,j,if(i==1||j==1||i==n||j==n,1,-1+2*isprime((i-1)*n+j)));print1(permRWNb(a)","))
\\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007

More terms (and corrected definition) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 15 2007

A110956 a(n) = permanent of the n X n matrix M of zeros and ones defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i = 0 if i is an entry from the sequence A045506 (inscribe 2 spheres of curvature 2 inside sphere of curvature -1, continue to inscribe spheres where possible; the sequence gives list of curvatures).

Original entry on oeis.org

2, 6, 20, 0, 368, 2016

Offset: 3

Simone Severini, Sep 26 2005

nonn

Cf. A045506.

A110947 a(n) = permanent of an n X n matrix M of zeros and ones defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i = 1 only if i = 1 or a multiple of 2.

Original entry on oeis.org

1, 1, 1, 0, 4, 0, 36, 0, 576, 0, 14400, 0, 518400, 0, 25401600, 0, 1625702400, 0, 131681894400, 0, 13168189440000, 0, 1593350922240000, 0, 229442532802560000, 0, 38775788043632640000, 0, 7600054456551997440000, 0

Offset: 1

Simone Severini, Sep 25 2005

nonn

Odd-indexed terms are the same as A001044.

a(n)={my(A=matrix(n,n,i,j,1-((i-1)*n+j)%2)); A[1,1]=1; matpermanent(A)} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007

a(n)=if(n==2,1,if(n%2,((n-1)/2)!^2)) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007

a(1)=a(2)=1 and for n>2: a(n)=0 if n=2*k, a(n)=k!^2 if n=2*k+1. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007

Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007

cp's OEIS Frontend

User: Simone Severini

A130198 Single paradiddle. In percussion, the paradiddle is a four-note drum sticking pattern consisting of two alternating notes followed by two notes on the same hand.

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A126857 Maximal number of vertices in an integral circulant graph of degree n or n+1.

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A127546 a(n) = F(n)^2 + F(n+1)^2 + F(n+2)^2, where F(n) denotes the n-th Fibonacci number.

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A120339 a(n) = 3abc, where (a,b,c) is a Markoff triple. The first Markoff triple considered is (1,2,5) and the ordering is increasing.

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A117393 Number of different classes of entangling power of permutation matrices of dimension d^2, acting on a d X d (tensor product) Hilbert space.

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A114533 Permanent of the n X n matrix with numbers prime(1),prime(2),...,prime(n^2) in order across rows.

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A114534 The n-th entry of the sequence is the value of the permanent of an n X n matrix M defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i is the i-th Fibonacci number.

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