A011915 -id:A011915 - cp's OEIS frontend

A011795 a(n) = floor(C(n,4)/5).

0, 0, 0, 0, 0, 1, 3, 7, 14, 25, 42, 66, 99, 143, 200, 273, 364, 476, 612, 775, 969, 1197, 1463, 1771, 2125, 2530, 2990, 3510, 4095, 4750, 5481, 6293, 7192, 8184, 9275, 10472, 11781, 13209, 14763, 16450, 18278, 20254, 22386, 24682, 27150, 29799, 32637, 35673, 38916, 42375, 46060, 49980, 54145, 58565, 63250, 68211, 73458, 79002, 84854, 91025, 97527, 104371

Offset: 0

N. J. A. Sloane, David Broadhurst

a(n-1) = number of aperiodic necklaces (Lyndon words) with 5 black beads and n-5 white beads.

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 11, 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A000031, A001037, A011915, A051168.
Same as A051170(n+1).
A column of triangle A011847.

[Floor(Binomial(n+1,5)/(n+1)): n in [0..70]]; // Vincenzo Librandi Jun 19 2012

seq(floor(binomial(n,4)/5), n=0.. 70); # Zerinvary Lajos, Jan 12 2009

CoefficientList[Series[x^5(1+x^3)/((1-x)^3(1-x^2)(1-x^5)),{x,0,70}],x] (* Vincenzo Librandi, Jun 19 2012 *)
CoefficientList[Series[x^4/5 (1/(1-x)^5-1/(1- x^5)),{x,0,70}],x] (* Herbert Kociemba, Oct 16 2016 *)

a(n)=binomial(n,4)\5 \\ Charles R Greathouse IV, Oct 07 2015

[binomial(n,4)//5 for n in range(71)] # G. C. Greubel, Oct 20 2024

G.f.: x^5*(1+x^3)/((1-x)^3*(1-x^2)*(1-x^5)) = x^5*(1-x+x^2)/((1-x)^5*(1+x+x^2+x^3+x^4)).

a(n) = floor(binomial(n+1,5)/(n+1)). - Gary Detlefs, Nov 23 2011

A011932 a(n) = floor( n(n-1)(n-2)*(n-3)/22 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 16, 38, 76, 137, 229, 360, 540, 780, 1092, 1489, 1985, 2596, 3338, 4228, 5285, 6529, 7980, 9660, 11592, 13800, 16309, 19145, 22336, 25910, 29896, 34325, 39229, 44640, 50592, 57120, 64260, 72049, 80525, 89728, 99698, 110476, 122105, 134629, 148092, 162540, 178020, 194580, 212269, 231137

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(12*Binomial(n,4)/11): n in [0..80]]; // G. C. Greubel, Nov 03 2024

Table[Floor[(n(n-1)(n-2)(n-3))/22],{n,0,60}] (* Harvey P. Dale, Nov 25 2017 *)

a(n)=n*(n-1)*(n-2)*(n-3)\22 \\ Charles R Greathouse IV, Oct 18 2022

[12*binomial(n,4)//11 for n in range(81)] # G. C. Greubel, Nov 03 2024

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-11) - 4*a(n-12) + 6*a(n-13) - 4*a(n-14) + a(n-15) for n > 14.
G.f.: x^4*(1 + x + 2*x^2 + x^4 + 2*x^5 + x^6 + 2*x^8 + x^9 + x^10)/((1-x)^4*(1-x^11)). (End)

More terms added by G. C. Greubel, Nov 03 2024

A011933 a(n) = floor( n(n-1)(n-2)*(n-3)/23 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 36, 73, 131, 219, 344, 516, 746, 1044, 1424, 1899, 2483, 3193, 4044, 5055, 6245, 7633, 9240, 11088, 13200, 15600, 18313, 21365, 24783, 28596, 32833, 37523, 42699, 48392, 54636, 61466, 68916, 77024, 85827, 95363, 105673, 116796, 128775, 141653, 155473, 170280, 186120, 203040, 221088

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A000332, A011915, A052762.

[Floor(24*Binomial(n,4)/23): n in [0..80]]; // G. C. Greubel, Nov 03 2024

Table[Floor[(n(n-1)(n-2)(n-3))/23],{n,0,60}] (* Harvey P. Dale, Jun 22 2011 *)

a(n) = n*(n-1)*(n-2)*(n-3)\23; \\ Michel Marcus, Jun 14 2017

[24*binomial(n,4)//23 for n in range(81)] # G. C. Greubel, Nov 03 2024

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-23) - 4*a(n-24) + 6*a(n-25) - 4*a(n-26) + a(n-27) for n > 26.
G.f.: x^4*(1+x^2)*(1 + x + x^3 - x^5 + 4*x^6 - x^7 - x^8 + 2*x^9 + 2*x^11 - x^12 - x^13 + 4*x^14 - x^15 + x^17 + x^19 + x^20)/((1-x)^4*(1-x^23)). (End)

More terms added by G. C. Greubel, Nov 03 2024

A011935 a(n) = floor( n(n-1)(n-2)*(n-3)/25 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 14, 33, 67, 120, 201, 316, 475, 686, 960, 1310, 1747, 2284, 2937, 3720, 4651, 5745, 7022, 8500, 10200, 12144, 14352, 16848, 19656, 22800, 26308, 30206, 34521, 39283, 44520, 50265, 56548, 63403, 70862, 78960, 87734, 97219, 107452, 118473

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(24*Binomial(n,4)/25): n in [0..80]]; // G. C. Greubel, Nov 02 2024

Floor[24*Binomial[Range[0,80], 4]/25] (* G. C. Greubel, Nov 02 2024 *)
Table[Floor[Times@@(n-Range[0,3])/25],{n,0,40}]  (* or *) LinearRecurrence[{4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1},{0,0,0,0,0,4,14,33,67,120,201,316,475,686,960,1310,1747,2284,2937,3720,4651,5745,7022,8500,10200,12144,14352,16848,19656},40] (* Harvey P. Dale, Jan 18 2025 *)

[24*binomial(n,4)//25 for n in range(81)] # G. C. Greubel, Nov 02 2024

a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +a(n-25) -4*a(n-26) +6*a(n-27) -4*a(n-28) +a(n-29). - R. J. Mathar, Apr 15 2010

G.f.: x^5*(4 -2*x +x^2 +3*x^3 -2*x^4 +5*x^5 -3*x^6 +4*x^7 -2*x^8 +3*x^9 +2*x^10 -2*x^11 +2*x^12 +3*x^13 -2*x^14 +4*x^15 -3*x^16 +5*x^17 -2*x^18 +3*x^19 +x^20 -2*x^21 +4*x^22)/((1-x)^4*(1-x^25)). - G. C. Greubel, Nov 02 2024

A011936 a(n) = floor( n(n-1)(n-2)*(n-3)/26 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 13, 32, 64, 116, 193, 304, 456, 660, 924, 1260, 1680, 2196, 2824, 3577, 4472, 5524, 6752, 8173, 9808, 11676, 13800, 16200, 18900, 21924, 25296, 29044, 33193, 37772, 42808, 48332, 54373, 60964, 68136, 75924, 84360, 93480, 103320, 113916, 125308, 137533, 150632, 164644, 179612, 195577

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(12*Binomial(n,4)/13): n in [0..80]]; // G. C. Greubel, Oct 29 2024

A011936:=n->floor(n*(n-1)*(n-2)*(n-3)/26): seq(A011936(n), n=0..100); # Wesley Ivan Hurt, Feb 03 2017

Floor[12*Binomial[Range[0,80], 4]/13] (* G. C. Greubel, Oct 29 2024 *)

[12*binomial(n,4)//13 for n in range(81)] # G. C. Greubel, Oct 29 2024

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-13) - 4*a(n-14) + 6*a(n-15) - 4*a(n-16) + a(n-17) for n > 16.
G.f.: x^5*(4 -3*x +4*x^2 -2*x^3 +4*x^4 -2*x^5 +4*x^6 -2*x^7 +4*x^8 -3*x^9 +4*x^10)/((1-x)^4*(1-x^13)). (End)

More terms added by G. C. Greubel, Oct 29 2024

A011937 a(n) = floor( n(n-1)(n-2)*(n-3)/27 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 13, 31, 62, 112, 186, 293, 440, 635, 889, 1213, 1617, 2115, 2720, 3445, 4306, 5320, 6502, 7871, 9445, 11244, 13288, 15600, 18200, 21112, 24360, 27968, 31964, 36373, 41223, 46542, 52360, 58706, 65613, 73112, 81235, 90017, 99493, 109697, 120667, 132440, 145053, 158546, 172960, 188334

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(8*Binomial(n,4)/9): n in [0..80]]; // G. C. Greubel, Oct 29 2024

Table[Floor[n (n-1)(n-2)(n-3)/27], {n, 0, 60}] (* Wesley Ivan Hurt, Jan 02 2024 *)

a(n)=n*(n-1)*(n-2)*(n-3)\27 \\ Charles R Greathouse IV, Oct 18 2022

[8*binomial(n,4)//9 for n in range(81)] # G. C. Greubel, Oct 29 2024

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-27) - 4*a(n-28) + 6*a(n-29) - 4*a(n-30) + a(n-31) for n > 30.
G.f.: x^5*(4 -3*x +3*x^2 +2*x^4 -x^5 +4*x^6 -2*x^7 +x^8 +3*x^9 -x^11 +4*x^12 -x^13 +3*x^15 +x^16 -2*x^17 +4*x^18 -x^19 +2*x^20 +3*x^22 -3*x^23 +4*x^24)/((1-x)^5*(1 +x +x^2)*(1 +x^3 +x^6)*(1 +x^9 +x^18)). (End)

More terms added by G. C. Greubel, Oct 29 2024

A011938 a(n) = floor( n(n-1)(n-2)*(n-3)/28 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 12, 30, 60, 108, 180, 282, 424, 612, 858, 1170, 1560, 2040, 2622, 3322, 4152, 5130, 6270, 7590, 9108, 10842, 12814, 15042, 17550, 20358, 23490, 26970, 30822, 35074, 39750, 44880, 50490, 56610, 63270, 70500, 78334, 86802, 95940, 105780

Offset: 0

N. J. A. Sloane

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(n*(n-1)*(n-2)*(n-3)/28): n in [0..50]]; // Vincenzo Librandi, May 27 2016

Table[Floor[n (n-1)(n-2)(n-3)/28], {n, 0, 50}] (* Vincenzo Librandi, May 27 2016 *)
LinearRecurrence[{4,-6,4,-1,0,0,1,-4,6,-4,1},{0,0,0,0,0,4,12,30,60,108,180}, 50] (* Harvey P. Dale, Aug 10 2024 *)

concat(vector(5), Vec(2*x^5*(2-2*x+3*x^2-2*x^3+2*x^4)/((1-x)^5*(1+x+x^2+x^3+x^4+x^5+x^6)) + O(x^50))) \\ Colin Barker, May 25 2016

[6*binomial(n,4)//7 for n in range(61)] # G. C. Greubel, Oct 27 2024

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-7) - 4*a(n-8) + 6*a(n-9) - 4*a(n-10) + a(n-11). - Chai Wah Wu, May 25 2016

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

A011939 a(n) = floor( n(n-1)(n-2)*(n-3)/29 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 12, 28, 57, 104, 173, 273, 409, 591, 828, 1129, 1506, 1969, 2532, 3207, 4009, 4953, 6053, 7328, 8793, 10468, 12372, 14524, 16944, 19656, 22680, 26040, 29760, 33864, 38380, 43332, 48748, 54657, 61088, 68069, 75633, 83809, 92631, 102132, 112345, 123306, 135049, 147612, 161031, 175345

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(24*Binomial(n,4)/29): n in [0..60]]; // G. C. Greubel, Oct 27 2024

Table[Floor[Times@@(n-Range[0,3])/29],{n,0,60}] (* or *) LinearRecurrence[{4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1},{0,0,0,0,0,4,12,28,57,104,173,273,409,591,828,1129,1506,1969,2532,3207,4009,4953,6053,7328,8793,10468,12372,14524,16944,19656,22680,26040,29760},60] (* Harvey P. Dale, Jun 09 2024 *)
Floor[24*Binomial[Range[0,60], 4]/29] (* G. C. Greubel, Oct 27 2024 *)

a(n)=n*(n-1)*(n-2)*(n-3)\29 \\ Charles R Greathouse IV, Oct 18 2022

[24*binomial(n,4)//29 for n in range(61)] # G. C. Greubel, Oct 27 2024

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-29) - 4*a(n-30) + 6*a(n-31) - 4*a(n-32) + a(n-33) for n > 32.
G.f.: x^5*(4 -4*x +4*x^2 +x^3 -x^5 +5*x^6 -4*x^7 +5*x^8 -x^9 +3*x^11 -2*x^12 +4*x^13 -2*x^14 +3*x^15 -x^17 +5*x^18 -4*x^19 +5*x^20 -x^21 +x^23 +4*x^24 -4*x^25 +4*x^26)/((1-x)^4*(1-x^29)). (End)

More terms added by G. C. Greubel, Oct 27 2024

A011940 a(n) = floor(n(n-1)(n-2)*(n-3)/30).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 12, 28, 56, 100, 168, 264, 396, 572, 800, 1092, 1456, 1904, 2448, 3100, 3876, 4788, 5852, 7084, 8500, 10120, 11960, 14040, 16380, 19000, 21924, 25172, 28768, 32736, 37100, 41888, 47124, 52836, 59052, 65800, 73112, 81016, 89544, 98728, 108600, 119196, 130548, 142692, 155664, 169500

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A011795, A011915.

[Floor(n*(n-1)*(n-2)*(n-3)/30): n in [0..60]]; // Vincenzo Librandi, Jun 19 2012

CoefficientList[Series[4*x^5*(1-x+x^2)/((1-x)^4*(1-x^5)),{x,0,60}],x] (* Vincenzo Librandi, Jun 19 2012 *)
LinearRecurrence[{4,-6,4,-1,1,-4,6,-4,1},{0,0,0,0,0,4,12,28,56},60] (* Harvey P. Dale, Nov 13 2017 *)
Floor[4*Binomial[Range[0,60], 4]/5] (* G. C. Greubel, Oct 27 2024 *)

[4*binomial(n,4)//5 for n in range(61)] # G. C. Greubel, Oct 27 2024

a(n) = 4 * A011795(n).
From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-5) - 4*a(n-6) + 6*a(n-7) - 4*a(n-8) + a(n-9).
G.f.: 4*x^5*(1-x+x^2) / ((1-x)^5*(1+x+x^2+x^3+x^4) ). (End)

A011941 a(n) = floor(n(n-1)(n-2)*(n-3)/31).

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 11, 27, 54, 97, 162, 255, 383, 553, 774, 1056, 1409, 1842, 2369, 3000, 3750, 4633, 5663, 6855, 8226, 9793, 11574, 13587, 15851, 18387, 21216, 24360, 27840, 31680, 35904, 40536, 45603, 51131, 57147, 63678, 70753, 78402, 86655, 95543

Offset: 0

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A011915.

[Floor(24*Binomial(n,4)/31): n in [0..60]]; // G. C. Greubel, Oct 26 2024

f:= n -> floor(n*(n-1)*(n-2)*(n-3)/31):
map(f, [$0..100]); # Robert Israel, Feb 12 2017

Floor[24*Binomial[Range[0, 60], 4]/31] (* G. C. Greubel, Oct 26 2024 *)

a(n) = n*(n-1)*(n-2)*(n-3)\31; \\ Altug Alkan, Feb 12 2017

[24*binomial(n,4)//31 for n in range(61)] # G. C. Greubel, Oct 26 2024

G.f.: (3-x+x^2+2*x^4+x^5+x^7+2*x^9+x^10-x^12+5*x^13-4*x^14+5*x^15-x^16+x^18 +2*x^19+x^21+x^23+2*x^24+x^26-x^27+3*x^28)*x^5/((1-x)^4*(1-x^31)). - Robert Israel, Feb 12 2017

cp's OEIS Frontend

A011795 a(n) = floor(C(n,4)/5).

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A011932 a(n) = floor( n*(n-1)*(n-2)*(n-3)/22 ).

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A011933 a(n) = floor( n*(n-1)*(n-2)*(n-3)/23 ).

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A011935 a(n) = floor( n*(n-1)*(n-2)*(n-3)/25 ).

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A011936 a(n) = floor( n*(n-1)*(n-2)*(n-3)/26 ).

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Programs

Magma

Maple

Mathematica

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A011937 a(n) = floor( n*(n-1)*(n-2)*(n-3)/27 ).

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Programs

Magma

Mathematica

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SageMath

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Extensions

A011938 a(n) = floor( n*(n-1)*(n-2)*(n-3)/28 ).

A011932 a(n) = floor( n(n-1)(n-2)*(n-3)/22 ).

A011933 a(n) = floor( n(n-1)(n-2)*(n-3)/23 ).

A011935 a(n) = floor( n(n-1)(n-2)*(n-3)/25 ).

A011936 a(n) = floor( n(n-1)(n-2)*(n-3)/26 ).

A011937 a(n) = floor( n(n-1)(n-2)*(n-3)/27 ).

A011938 a(n) = floor( n(n-1)(n-2)*(n-3)/28 ).

A011939 a(n) = floor( n(n-1)(n-2)*(n-3)/29 ).

A011940 a(n) = floor(n(n-1)(n-2)*(n-3)/30).

A011941 a(n) = floor(n(n-1)(n-2)*(n-3)/31).