A011863 -id:A011863 - cp's OEIS frontend

A060494 a(n) = floor(n^4/64).

0, 0, 0, 1, 4, 9, 20, 37, 64, 102, 156, 228, 324, 446, 600, 791, 1024, 1305, 1640, 2036, 2500, 3038, 3660, 4372, 5184, 6103, 7140, 8303, 9604, 11051, 12656, 14430, 16384, 18530, 20880, 23447, 26244, 29283, 32580, 36147, 40000, 44152, 48620, 53418, 58564, 64072

Offset: 0

Henry Bottomley, Mar 21 2001

			a(9) = floor(9^4/64) = floor(6561/64) = floor(102.51562...) = 102.

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

Floor[Range[0,50]^4/64] (* or *) LinearRecurrence[ {4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1},{0,0,0,1,4,9,20,37,64,102,156,228,324,446,600,791,1024,1305,1640,2036},50] (* Harvey P. Dale, May 30 2014 *)

a(n) = { n^4\64 } \\ Harry J. Smith, Jul 06 2009

a(n) = floor(A000583(n)/64) = floor(A011863(n-1)/4). a(2n) = A059403(2n); a(2n-1) = A059403(2n-1) + A011861(n).
From R. J. Mathar, Mar 24 2011: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-16) - 4*a(n-17) + 6*a(n-18) - 4*a(n-19) + a(n-20).
G.f.: -x^3 *(1 - x^2 + 4*x^3 - 4*x^4 - 3*x^6 + 4*x^7 - 3*x^8 + 4*x^9 - 4*x^10 + 4*x^11 - x^12 + x^14 + 4*x^5) / ( (1+x) *(x^2+1) *(x^4+1) *(x^8+1) *(x-1)^5 ). (End)

A081276 Floor(n^3/8).

Original entry on oeis.org

0, 0, 1, 3, 8, 15, 27, 42, 64, 91, 125, 166, 216, 274, 343, 421, 512, 614, 729, 857, 1000, 1157, 1331, 1520, 1728, 1953, 2197, 2460, 2744, 3048, 3375, 3723, 4096, 4492, 4913, 5359, 5832, 6331, 6859, 7414, 8000, 8615, 9261, 9938, 10648, 11390, 12167, 12977

Offset: 0

Paul Barry, Mar 15 2003

a(2n) = n^3.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).

Cf. A002620, A011863, A038503.

[Floor(n^3/8): n in [0..50]]; // Vincenzo Librandi, Aug 07 2013

Floor[Range[0,50]^3/8] (* or *) LinearRecurrence[ {3,-3,1,0,0,0,0,1,-3,3,-1},{0,0,1,3,8,15,27,42,64,91,125},50] (* Harvey P. Dale, Jan 27 2012 *)

a(n) = floor(n^3/8).
G.f.: x^2*(-x^3-2*x^5+3*x^4+1+4*x^6+2*x^2-2*x^7+x^8)/((-1+x)^4*(1+x)*(1+x^2)*(x^4+1)). - R. J. Mathar, Jun 26 2009
a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=8, a(5)=15, a(6)=27, a(7)=42, a(8)=64, a(9)=91, a(10)=125, a(n)=3*a(n-1)-3*a(n-2)+a(n-3)+a(n-8)- 3*a(n-9)+ 3*a(n-10)-a (n-11). - Harvey P. Dale, Jan 27 2012

A175707 Number of ways to put n copies of 1,2,3,4 into sets.

Original entry on oeis.org

1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756, 351989816, 575711716, 921889652, 1447822620, 2233501928, 3389114724, 5064582169, 7461570579, 10848490675, 15579077786, 22115241763, 31054971635, 43166197978, 59427633555, 81077755892, 109673237289, 147158299390, 195946638641

Offset: 0

Thotsaporn Thanatipanonda, Dec 04 2010

nonn

Related to generalized Bell Numbers.
The n copies of each digit must be in different sets, and the sets must be nonempty.
Other definition: Number of ways to distribute n copies of 1,2,3,4 into an arbitrary number of (nonempty) sets. Due to the nature of sets, the same digit may not be several times in the same set.

			For n=1, the solution is the fourth term of Bell numbers A000110.
For n=2, one way to partition 2 copies of 1, 2 copies of 2, 2 copies of 3 and 2 copies of 4 is {1}{2}{34}{12}{34}. On the other hand {112}{34}{23}{4} is not allowed since the same numbers are in the same set {112}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls? (2009).
Doron Zeilberger, In How many ways can you reassemble several russian dolls?, arXiv:0909.3453 [math.CO], 2009.
Doron Zeilberger, BABUSHKAS; Local copy
Index entries for linear recurrences with constant coefficients, signature (7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1).

Cf. A000110, A011863, A020554, A165434.

a:= n-> (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) /(2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3)+ cos(n*Pi/3));
seq(a(n), n=0..40);
seq(SeqBrnDJ(n,4)[5], n=1..6); # using the Maple package BABUSHKAS (see links)

LinearRecurrence[{7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1}, {1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756}, 36] (* Jean-François Alcover, Nov 13 2018 *)

a(n) = (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) / (2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3) +cos(n*Pi/3)).
Recurrence: a(n) -7*a(n-1) +17*a(n-2) -8*a(n-3) -36*a(n-4) +60*a(n-5) -4*a(n-6) -56*a(n-7) +22*a(n-8) +22*a(n-9) +22*a(n-10) -56*a(n-11) -4*a(n-12) +60*a(n-13) -36*a(n-14) -8*a(n-15) +17*a(n-16) -7*a(n-17) +a(n-18) = 0.
G.f.: (x^10 +8*x^9 +51*x^8 +136*x^7 +252*x^6 +300*x^5 +252*x^4 +136*x^3 +51*x^2 +8*x+1) / ((x^2+x+1)*(x+1)^4*(x-1)^12).

A061003 Nearest integer to n^5/25.

Original entry on oeis.org

0, 1, 10, 41, 125, 311, 672, 1311, 2362, 4000, 6442, 9953, 14852, 21513, 30375, 41943, 56794, 75583, 99044, 128000, 163364, 206145, 257454, 318505, 390625, 475255, 573956, 688415, 820446, 972000, 1145166, 1342177, 1565416, 1817417

Offset: 1

N. J. A. Sloane, May 15 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A011863, A061004.

seq(round(n^5/25),n=1..35); # Muniru A Asiru, Jul 02 2018

Round[Range[40]^5/25] (* Harvey P. Dale, May 12 2024 *)

a(n) = { round(n^5/25) } \\ Harry J. Smith, Jul 16 2009

G.f.: x*(x^7+5x^6+x^5+10x^4+x^3+5x^2+x)/((1-x^5)*(1-x)^5).

cp's OEIS Frontend

A060494 a(n) = floor(n^4/64).

Views

Author

Keywords

Examples

Links

Programs

Mathematica

PARI

Formula

A081276 Floor(n^3/8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A175707 Number of ways to put n copies of 1,2,3,4 into sets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A061003 Nearest integer to n^5/25.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula