A079908 -id:A079908 - cp's OEIS frontend

A079921 Solution to the Dancing School Problem with n girls and n+2 boys: f(n,2).

3, 7, 14, 26, 46, 79, 133, 221, 364, 596, 972, 1581, 2567, 4163, 6746, 10926, 17690, 28635, 46345, 75001, 121368, 196392, 317784, 514201, 832011, 1346239, 2178278, 3524546, 5702854, 9227431, 14930317, 24157781, 39088132, 63245948, 102334116, 165580101

Offset: 1

Jaap Spies, Jan 28 2003

nonn

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 123.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.
E. S. Egge and T. Mansour, 132-avoiding two-stack sortable permutations....
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A079908-A079928, A001924.

Cf. Essentially the same as A001924.

with(genfunc): Fz := 1/((-1+z)^2 * (1-z-z^2)); seq(rgf_term(Fz,z,n), n=1..30);

CoefficientList[Series[(-z^3 + z^2 + 2*z - 3)/((z - 1)^2 (z^2 + z - 1)), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
LinearRecurrence[{3,-2,-1,1},{3,7,14,26},40] (* Harvey P. Dale, Oct 17 2022 *)

a(n) = a(n-1)+a(n-2)+n+1, a(1)=3, a(2)=7.
G.f.: 1/((1-x)^2*(1-x-x^2)).
F(n+5) - n - 4, F(n) = A000045(n).
a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Wesley Ivan Hurt, Dec 03 2021

More terms from Jaap Spies, Dec 15 2006

A079922 Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3).

Original entry on oeis.org

4, 13, 36, 90, 212, 478, 1044, 2227, 4664, 9627, 19640, 39684, 79544, 158364, 313464, 617365, 1210588, 2364713, 4603388, 8934142, 17291756, 33385018, 64311660, 123634471, 237233712, 454429239, 869095472, 1659708488

Offset: 1

Jaap Spies, Jan 28 2003

nonn

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.

Cf. A079908-A079928.

Empirical g.f.: -x*(x^7-4*x^4+2*x^3+3*x-4) / ((x-1)^2*(x^3+x^2+x-1)^2). - Colin Barker, Jan 04 2015

More terms from Jaap Spies, Dec 15 2006

A079923 Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4).

Original entry on oeis.org

5, 21, 76, 246, 738, 2108, 5794, 15458, 40296, 103129, 260019, 647617, 1596800, 3904260, 9479292, 22879520, 54947836, 131406732, 313129592, 743878505, 1762572329, 4167010597, 9832766588, 23164353834

Offset: 1

Jaap Spies, Jan 28 2003

nonn

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.

Cf. A079908-A079928.

More terms from Jaap Spies, Dec 14 2006

A079924 Solution to the Dancing School Problem with n girls and n+5 boys: f(n,5).

Original entry on oeis.org

6, 31, 140, 566, 2104, 7364, 24720, 80196, 253072, 780902, 2365772, 7058469, 20789082, 60560175, 174763208, 500245052, 1421824896, 4016278792, 11283371280, 31547434008, 87827653936, 243578509132, 673221043496

Offset: 1

Jaap Spies, Jan 28 2003

nonn

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.

Cf. A079908-A079928.

More terms from Jaap Spies, Dec 14 2006

A079925 Solution to the Dancing School Problem with n girls and n+6 boys: f(n,6).

Original entry on oeis.org

7, 43, 234, 1146, 5150, 21652, 86608, 334072, 1249768, 4557284, 16266830, 57031078, 196933710, 671224467, 2262089361, 7548882573, 24975936372, 82012110724, 267505920876, 867390073384

Offset: 1

Jaap Spies, Jan 28 2003

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.

Cf. A079908-A079928.

Corrected by Jaap Spies, Feb 01 2004

More terms Dec 14 2006

A079927 Solution to the Dancing School Problem with n girls and n+8 boys: f(n,8).

Original entry on oeis.org

9, 73, 536, 3590, 22162, 127604, 693552, 3598120, 17990600, 87396728, 413977192, 1918222840, 8719846960, 38983643908, 171764779170, 747190081890, 3213760467348, 13684132415133, 57742830924831, 241687792906641

Offset: 1

Jaap Spies, Jan 28 2003

nonn

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.

Cf. A079908-A079928.

Corrected by Jaap Spies, Feb 01 2004

More terms Dec 14 2006

A180276 Primes of the form n^3 + 3*n - 1.

Original entry on oeis.org

3, 13, 139, 233, 8059, 9323, 19763, 42979, 103963, 125149, 175783, 185363, 216179, 373463, 422099, 456763, 636313, 729269, 778963, 885023, 1061513, 1331329, 1367963, 1561243, 2000753, 2744419, 3724339, 4657963, 6435413, 6968443

Offset: 1

Graziano Aglietti (mg5055(AT)mclink.it), Aug 23 2010

nonn

Subsequence of primes in the sequence defined as b(n) = n^3 + 3*n - 1 = 3, 13, 35, 75, 139, 233, 363, 535, ... = A079908(n) - 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

[ a: n in [0..250] | IsPrime(a) where a is (n^3+3*n-1)]; // Vincenzo Librandi, Jan 30 2011

Select[Table[n^3+3n-1,{n,200}],PrimeQ] (* Harvey P. Dale, Sep 08 2024 *)

a(n)=n^3+3*n-1;
for(i=0,10^3,if(isprime(a(i)),print1(a(i), ", ")))

Offset set to 1 by R. J. Mathar, Aug 25 2010

A180358 n^8+8n.

Original entry on oeis.org

0, 9, 272, 6585, 65568, 390665, 1679664, 5764857, 16777280, 43046793, 100000080, 214358969, 429981792, 815730825, 1475789168, 2562890745, 4294967424, 6975757577, 11019960720, 16983563193, 25600000160, 37822859529, 54875873712

Offset: 0

Odimar Fabeny, Aug 30 2010

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf.: A132411, A079908, A180354, A180355, A180356, A180357.

Table[n^8+8n,{n,0,30}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{0,9,272,6585,65568,390665,1679664,5764857,16777280},30] (* Harvey P. Dale, Jan 03 2012 *)

a(n)= +9*a(n-1) -36*a(n-2) +84*a(n-3) -126*a(n-4) +126*a(n-5) -84*a(n-6) +36*a(n-7) -9*a(n-8) +a(n-9). G.f.: x*(-9-191*x-4461*x^2-15339*x^3-15899*x^4-4125*x^5-303*x^6+7*x^7) / (x-1)^9 . [From R. J. Mathar, Sep 19 2010]

a(0) corrected by R. J. Mathar, Sep 19 2010

A079916 Solution to the Dancing School Problem with 11 girls and n+11 boys: f(11,n).

Original entry on oeis.org

1, 12, 972, 19640, 260019, 2365772, 16266830, 89700624, 413977192, 1650607040, 5826331440, 18558391936, 54055214144, 145576033920, 365883104080, 865023114560, 1936764883296, 4130528893504, 8433028861040

Offset: 0

Jaap Spies, Jan 28 2003

nonn

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).

Cf. A079908-A079928.

Corrected by Jaap Spies, Feb 01 2004

A079917 Solution to the Dancing School Problem with 12 girls and n+12 boys: f(12,n).

Original entry on oeis.org

1, 13, 1581, 39684, 647617, 7058469, 57031078, 363862092, 1918222840, 8641355080, 34132685120, 120629547584, 387665694976, 1145875468544, 3145428883520

Offset: 0

Jaap Spies, Jan 28 2003

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.

Cf. A079908-A079928.

Corrected by Jaap Spies, Feb 01 2004

cp's OEIS Frontend

A079921 Solution to the Dancing School Problem with n girls and n+2 boys: f(n,2).

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A079922 Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3).

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A079923 Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4).

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A079924 Solution to the Dancing School Problem with n girls and n+5 boys: f(n,5).

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A079925 Solution to the Dancing School Problem with n girls and n+6 boys: f(n,6).

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A079927 Solution to the Dancing School Problem with n girls and n+8 boys: f(n,8).

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A180276 Primes of the form n^3 + 3*n - 1.

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A180358 n^8+8n.

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A079916 Solution to the Dancing School Problem with 11 girls and n+11 boys: f(11,n).

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