A000914 -id:A000914 - cp's OEIS frontend

A112461 Absolute value of coefficient of term [x^(n-5)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 5. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

5, 59, 374, 1694, 6149, 19019, 52052, 129272, 296582, 636922, 1293292, 2502604, 4644094, 8306914, 14382544, 24188824, 39633715, 63428365, 99360690, 152642490, 230345115, 341940885, 499969860, 720854160, 1025884860, 1442409540, 2005251864, 2758398104

Offset: 5

Paul Max Payton, Sep 23 2005

T. D. Noe and Bruno Berselli, Table of n, a(n) for n = 5..1004
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000217, A000914, A001844, A112459, A112460, A112462, A112463, A112464.

Table[(9n+5)/10! Product[n+i,{i,-4,4}],{n,5,40}] (* Harvey P. Dale, Apr 26 2019 *)

a(n) = ((9n+5)/10!) * Product_{i=-4..4} (n+i).

G.f.: x^5*(5+4*x)/(1-x)^11. - Colin Barker, Mar 28 2012

Offset changed from 1 to 5, formulas and b-file adapted by Bruno Berselli, Mar 29 2012

A112462 Absolute value of coefficient of term [x^(n-6)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 6. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

Original entry on oeis.org

6, 83, 611, 3185, 13195, 46228, 142324, 395148, 1007760, 2393430, 5349526, 11345698, 22985326, 44722580, 83947700, 152591660, 269449830, 463484385, 778439025, 1279189275, 2060359665, 3257868120, 5064210840, 7748481000, 11682325200, 17373286476, 25507265868

Offset: 6

Paul Max Payton, Sep 23 2005

T. D. Noe and Bruno Berselli, Table of n, a(n) for n = 6..1005

Cf. A000217, A000914, A001844, A112459, A112460, A112461, A112463, A112464.

a(n) = ((11n+6)/12!) * Product_{i=-5..5} (n+i).

G.f.: x^6*(6+5*x)/(1-x)^13. - Colin Barker, Mar 28 2012

Offset changed from 1 to 6, formulas and b-file adapted by Bruno Berselli, Mar 29 2012

A112464 Absolute value of coefficient of term [x^(n-8)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 8. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

Original entry on oeis.org

8, 143, 1343, 8823, 45543, 196707, 739347, 2483547, 7599867, 21492097, 56794705, 141485305, 334639305, 755863605, 1638428805, 3422280285, 6912424485, 13541987610, 25799313210, 47907161610, 86882479530, 154161302130, 268050218130, 457369908930, 766795640130

Offset: 8

Paul Max Payton, Sep 23 2005

Vincenzo Librandi and Bruno Berselli, Table of n, a(n) for n = 8..10008

Cf. A000217, A000914, A001844, A112459, A112460, A112461, A112462, A112463.

Table[(Times@@(n+Range[0,14])(15n+113))/16!,{n,30}] (* or *) CoefficientList[ Series[ (-8-7 x)/(-1+x)^17,{x,0,30}],x] (* Harvey P. Dale, Jul 24 2011 *)

a(n) = ((15n+8)/16!) * Product_{i=-7..7} (n+i).

G.f.: x^8*(8+7*x)/(1-x)^17. - Harvey P. Dale, Jul 24 2011

More terms from Harvey P. Dale, Jul 24 2011

Offset changed from 0 to 8, formulas and b-file adapted by Bruno Berselli, Mar 29 2012

A259749 Numbers that are congruent to {1,2,5,7,10,11,13,17,19,23} mod 24.

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 13, 17, 19, 23, 25, 26, 29, 31, 34, 35, 37, 41, 43, 47, 49, 50, 53, 55, 58, 59, 61, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 89, 91, 95, 97, 98, 101, 103, 106, 107, 109, 113, 115, 119, 121, 122, 125, 127, 130, 131, 133, 137, 139, 143, 145

Offset: 1

José María Grau Ribas, Jul 04 2015

Original name: Numbers n such that A259748(n) = 0.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-1).

Cf. A000914.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259750, A259751, A259752, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1, n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]  == 0 & ]
Rest@ CoefficientList[Series[x (1 + x^2) (1 + 2 x^2 - x^3 + 2 x^4 - 2 x^5 + 3 x^6 + x^7)/((1 - x)^2*(1 - x + x^2 - x^3 + x^4) (1 + x + x^2 + x^3 + x^4)), {x, 0, 61}], x] (* Michael De Vlieger, Aug 25 2016 *)
Select[Range[150],MemberQ[{1,2,5,7,10,11,13,17,19,23},Mod[#,24]]&] (* or *) LinearRecurrence[{2,-2,2,-2,2,-2,2,-2,2,-1},{1,2,5,7,10,11,13,17,19,23},70] (* Harvey P. Dale, Jan 15 2022 *)

Vec(x*(1+x^2)*(1+2*x^2-x^3+2*x^4-2*x^5+3*x^6+x^7)/((1-x)^2*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)) + O(x^100)) \\ Colin Barker, Aug 25 2016

A259748(a(n)) = Sum_{x*y: x,y in Z/a(n)Z, x<>y} = 0.

G.f.: x*(1+x^2)*(1+2*x^2-x^3+2*x^4-2*x^5+3*x^6+x^7) / ((1-x)^2*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)). - Colin Barker, Aug 25 2016

Better name from Danny Rorabaugh, Oct 22 2015

A259751 Numbers that are congruent to {8, 16} mod 24.

Original entry on oeis.org

8, 16, 32, 40, 56, 64, 80, 88, 104, 112, 128, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 256, 272, 280, 296, 304, 320, 328, 344, 352, 368, 376, 392, 400, 416, 424, 440, 448, 464, 472, 488, 496, 512, 520, 536, 544, 560, 568, 584, 592, 608, 616, 632

Offset: 1

José María Grau Ribas, Jul 06 2015

Original name: Numbers n such that n/A259748(n) = 4.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000914, A001651.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259752, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1,  n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]/# == 1/4 & ]

Vec(8*x*(1+x+x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ Colin Barker, Aug 26 2016

A259748(a(n))/a(n) = 1/4.
a(n) = 8*A001651. - Danny Rorabaugh, Oct 22 2015
From Colin Barker, Aug 26 2016: (Start)
a(n) = 12*n-2*(-1)^n-6.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: 8*x*(1+x+x^2) / ((1-x)^2*(1+x)).
(End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/72. - Amiram Eldar, Dec 31 2021
E.g.f.: 2*(4 + (6*x - 3)*exp(x) - exp(-x)). - David Lovler, Sep 05 2022

Better name from Danny Rorabaugh, Oct 22 2015

A259752 a(n) = 24*n - 18.

Original entry on oeis.org

6, 30, 54, 78, 102, 126, 150, 174, 198, 222, 246, 270, 294, 318, 342, 366, 390, 414, 438, 462, 486, 510, 534, 558, 582, 606, 630, 654, 678, 702, 726, 750, 774, 798, 822, 846, 870, 894, 918, 942, 966, 990, 1014, 1038, 1062, 1086, 1110, 1134, 1158, 1182, 1206

Offset: 1

José María Grau Ribas, Jul 12 2015

Original name: Numbers n such that n/A259748(n) = 6.

Partial sums give A152746. - Leo Tavares, Jul 29 2023

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000914, A016813, A063128, A063148, A152746.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259751, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1,  n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]/# == 1/6 & ]

A259748(a(n))/a(n) = 1/6.
a(n) = 6*A016813(n-1). - Michel Marcus, Jul 18 2015
G.f.: 6*x*(3*x+1)/(x-1)^2. - Alois P. Heinz, Jul 29 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 6*(exp(x)*(4*x - 3) + 3).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Better name from Danny Rorabaugh, Oct 22 2015

A259754 Numbers that are congruent to {3,9,15,18,21} mod 24.

Original entry on oeis.org

3, 9, 15, 18, 21, 27, 33, 39, 42, 45, 51, 57, 63, 66, 69, 75, 81, 87, 90, 93, 99, 105, 111, 114, 117, 123, 129, 135, 138, 141, 147, 153, 159, 162, 165, 171, 177, 183, 186, 189, 195, 201, 207, 210, 213, 219, 225, 231, 234, 237, 243, 249, 255, 258, 261, 267

Offset: 1

José María Grau Ribas, Jul 12 2015

Original name: Numbers n such that n/A259748(n) = 3/2.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A000914.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259751, A259752, A259755.

A[n_] := A[n] = Sum[a b, {a, 1, n}, {b, a + 1, n}]; Select[Range[200], Mod[A[#], #]/# == 2/3 &]
Rest@ CoefficientList[Series[3 x (1 + x) (1 + x + x^2 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)), {x, 0, 56}], x] (* Michael De Vlieger, Aug 25 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{3,9,15,18,21,27},60] (* Harvey P. Dale, Aug 30 2016 *)

Vec(3*x*(1+x)*(1+x+x^2+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)) + O(x^100)) \\ Colin Barker, Aug 25 2016

A259748(a(n))/a(n) = 2/3.
a(n) = 3*A047584(n). - Michel Marcus, Jul 18 2015
From Colin Barker, Aug 25 2016: (Start)
a(n) = a(n-1)+a(n-5)-a(n-6) for n>6.
G.f.: 3*x*(1+x)*(1+x+x^2+x^4) / ((1-x)^2*(1+x+x^2+x^3+x^4)).
(End)

Better name from Danny Rorabaugh, Oct 22 2015

A112459 Absolute value of coefficient of term [x^(n-3)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 3. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

Original entry on oeis.org

3, 23, 98, 308, 798, 1806, 3696, 6996, 12441, 21021, 34034, 53144, 80444, 118524, 170544, 240312, 332367, 452067

Offset: 3

Paul Max Payton, Sep 23 2005

Cf. A000217, A000914, A001844, A112460, A112461, A112462, A112463, A112464.

a(n) = n*(n^2-4)*(n^2-1)*(5*n+3)/6!.

G.f.: x^3*(3+2*x)/(1-x)^7. - Colin Barker, Mar 28 2012

Offset changed from 1 to 3 and formulas adapted by Bruno Berselli, Mar 29 2012

A112460 Absolute value of coefficient of term [x^(n-4)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 4. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

Original entry on oeis.org

4, 39, 207, 795, 2475, 6633, 15873, 34749, 70785, 135850, 247962, 433602, 730626, 1191870, 1889550, 2920566, 4412826, 6532713, 9493825, 13567125, 19092645, 26492895, 36288135, 49113675, 65739375, 87091524, 114277284, 148611892, 191648820, 245213100, 311438028

Offset: 4

Paul Max Payton, Sep 23 2005

Cf. A000217, A000914, A001844, A112459, A112461, A112462, A112463, A112464.

Drop[Table[(7n^8+4n^7-98n^6-56n^5+343n^4+196n^3-252n^2-144n)/40320,{n,40}],3] (* Harvey P. Dale, Dec 15 2013 *)

a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)*(n+2)*(n+3)*(7*n+4)/8!.

G.f.: x^4*(4+3*x)/(1-x)^9. - Colin Barker, Mar 28 2012

A112463 Absolute value of coefficient of term [x^(n-7)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 7. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

Original entry on oeis.org

7, 111, 930, 5480, 25500, 99756, 341088, 1046520, 2936070, 7638950, 18631932, 42969336, 94348300, 198354300, 401166000, 783610920, 1483311285, 2728813725, 4891144350, 8560278000, 14656684680, 24591569640, 40493836800, 65527390800, 104329399500, 163608855372

Offset: 7

Paul Max Payton, Sep 23 2005

Cf. A000217, A000914, A001844, A112459, A112460, A112461, A112462, A112464.

a(n) = ((13*n+7)/14!) * Product_{i=-6..6} (n+i).

G.f.: x^7*(7+6*x)/(1-x)^15. - Colin Barker, Mar 28 2012

Offset changed by Alois P. Heinz, Mar 28 2012

cp's OEIS Frontend

A112461 Absolute value of coefficient of term [x^(n-5)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 5. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

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A112462 Absolute value of coefficient of term [x^(n-6)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 6. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

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A112464 Absolute value of coefficient of term [x^(n-8)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 8. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

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A259749 Numbers that are congruent to {1,2,5,7,10,11,13,17,19,23} mod 24.

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A259751 Numbers that are congruent to {8, 16} mod 24.

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A259752 a(n) = 24*n - 18.

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A259754 Numbers that are congruent to {3,9,15,18,21} mod 24.

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A112459 Absolute value of coefficient of term [x^(n-3)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 3. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.

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