A011857 -id:A011857 - cp's OEIS frontend

A011848 a(n) = floor(binomial(n, 2)/2).

0, 0, 0, 1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 203, 217, 232, 248, 264, 280, 297, 315, 333, 351, 370, 390, 410, 430, 451, 473, 495, 517, 540, 564, 588, 612, 637, 663, 689, 715, 742, 770, 798

Offset: 0

N. J. A. Sloane, Dec 11 1996

Column sums of an array of the odd numbers repeatedly shifted 4 places to the right:
1 3 5 7  9 11 13 15 17...
         1  3  5  7  9...
                     1...
.........................
-------------------------
1 3 5 7 10 14 18 22 27...
Floor of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Beginning with a(4)=3, the sequence might be called the "off-axis" Ulam-Spiral numbers because they are the numbers in ascending order on the horizontal and vertical spokes (heading outward) starting with the first turning points on the spiral (i.e., 3, 5, 7 and 10). That is, starting with: 3 (upward); 5 (leftward); 7 (downward) and 10 (rightward). These are A033991 (starting at a(1)), A007742 (starting at a(1)), A033954 (starting at a(1)) and A001107 (starting at a(2)), respectively. These quadri-sections are summarized in the formulas of Sep 26 2015. - Bob Selcoe, Oct 05 2015
Conjecture: For n = 2, a(n) is the greatest k such that A123663(k) < A000217(n - 2). - Peter Kagey, Nov 18 2016
a(n) is also the matching number of the n-triangular graph, (n-1)-triangular honeycomb queen graph, (n-1)-triangular honeycomb bishop graphs, and (for n > 7) (n-1)-triangular honeycomb obtuse knight graphs. - Eric W. Weisstein, Jun 02 2017 and Apr 03 2018
After 0, 0, 0, add 1, then add 2 three times, then add 3, then add 4 three times, then add 5, etc.; i.e., first differences are A004524 = (0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, ...). - M. F. Hasler, May 09 2018
Let s(0) = s(1) = 1, s(-1) = s(2) = x, and s(n+2)*s(n-2) = s(n+1)*s(n-1) + s(n)^2 for all n in Z. Then s(n) = p(n) / x^e(n) is a Laurent polynomial in x with p(n) a polynomial with nonnegative integer coefficients of degree a(n) for all n in Z. If x = 1, then s(n) = p(n) = A006720(n+1). - Michael Somos, Mar 22 2023

			G.f. = x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 10*x^7 + 14*x^8 + 18*x^9 + 22*x^10 + ...
p(0) = p(1) = 1, p(2) = 1 + x, p(3) = 1 + x + x^3, p(4) = 1 + 2*x + 2*x^2 + x^3 + x^5. - _Michael Somos_, Mar 22 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Triangular Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

A column of triangle A011857.
First differences are in A004524.
Cf. A007318, A033991, A007742, A033954, A001107, A006720, A035608 (bisection), A156859 (bisection).

List([0..60],n->Int(Binomial(n,2)/2)); # Muniru A Asiru, Apr 05 2018

a011848 n = if n < 2 then 0 else flip div 2 $ a007318 n 2
-- Reinhard Zumkeller, Mar 04 2015

[ Floor(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

seq(floor(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Floor[n (n - 1)/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
CoefficientList[Series[x^3/((1 + x^2) (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 21 2013 *)
LinearRecurrence[{3, -4, 4, -4, 1}, {0, 0, 1, 3, 5}, {0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[Floor[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[1/4 (-1 + (-1 + n) n + Cos[n Pi/2] + Sin[n Pi/2]), {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Floor[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Apr 03 2018 *)

PARI
```
a(n) = binomial(n, 2)\2;
```

vector(100, n, n--; floor(n*(n-1)/4)) \\ Altug Alkan, Sep 30 2015

def a(n): return n*(n-1)//4 # Christoph B. Kassir, Oct 07 2022

[floor(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

G.f.: x^3*(1-x^2)/((1-x)^3*(1-x^4)).
G.f.: x^3/((1+x^2)*(1-x)^3). - Jon Perry, Mar 31 2004
a(n) = +3*a(n-1) -4*a(n-2) +4*a(n-3) -3*a(n-4) +a(n-5). - R. J. Mathar, Apr 15 2010
a(n) = floor((n/(1+e^(1/n)))^2). - Richard R. Forberg, Jun 19 2013
a(n) = floor(n*(n-1)/4). - T. D. Noe, Jun 20 2013
a(n) = (1/4) * ( n^2 - n - 1 + (-1)^floor(n/2) ). - Ralf Stephan, Aug 11 2013
a(n) = A054925(n) - A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
a(4*n) = A033991(n). a(4*n+1) = A007742(n). a(4*n+2) = A033954(n). a(4*n+3) = A001107(n+1). - Bob Selcoe, Sep 26 2015
E.g.f.: (sin(x) + cos(x) + (x^2 - 1)*exp(x))/4. - Ilya Gutkovskiy, Nov 18 2016
A054925(n) = a(-n). A035608(n) = a(2*n+1). Wesley Ivan Hurt, Jun 09 2014
A156859(n) = a(2*n+2). - Michael Somos, Nov 18 2016
Euler transform of length 4 sequence [ 3, -1, 0, 1]. - Michael Somos, Nov 18 2016
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=3} 1/a(n) = 40/9 - 2*Pi/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = 32/9 - 4*log(2). (End)
0 = a(n+2)*(a(n)*(a(n) -6*a(n+1) +4*a(n+2)) +a(n+1)*(8*a(n+1) -10*a(n+2)) + 3*a(n+2)^2) +a(n+3)*(a(n)*(+a(n) -2*a(n+1)) +a(n+2)*(2*a(n+1) -a(n+2))) for all n in Z. - Michael Somos, Mar 22 2023
2*a(n) + 2*a(n-2) = (n-1)*(n-2). - R. J. Mathar, Feb 12 2024

A011849 a(n) = floor(binomial(n,3)/3).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 11, 18, 28, 40, 55, 73, 95, 121, 151, 186, 226, 272, 323, 380, 443, 513, 590, 674, 766, 866, 975, 1092, 1218, 1353, 1498, 1653, 1818, 1994, 2181, 2380, 2590, 2812, 3046, 3293, 3553, 3826

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, 3, 3, -6, 3, 3, -6, 4, -1).

A column of triangle A011857.

[Floor(Binomial(n,3)/3): n in [0..50]]; // Vincenzo Librandi, Jun 19 2012

seq(floor(binomial(n,3)/3), n=0..42); # Zerinvary Lajos, Jan 12 2009

CoefficientList[Series[x^4*(x^2-x+1)*(x^3-x^2+1)/((-1+x)^4*(x^6+x^3+1)),{x,0,50}],x] (* Vincenzo Librandi, Jun 19 2012 *)

G.f.: x^4*(1-x+x^2)*(1-x^2+x^3)/((1-x)^3*(1-x^9)). - Ralf Stephan, Mar 05 2004
From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 3*a(n-3) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) + 3*a(n-7) - 6*a(n-8) + 4*a(n-9) - a(n-10).
G.f.: x^4*(x^2-x+1)*(x^3-x^2+1) / ( (-1+x)^4*(x^6+x^3+1) ). (End)
a(n) = (1/54) * ( n^3 - 3*n - 6 + [6,8,4,-12,8,4,-30,8,4](mod 9) ). - Ralf Stephan, Aug 11 2013

A011851 a(n) = floor(binomial(n,5)/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 11, 25, 50, 92, 158, 257, 400, 600, 873, 1237, 1713, 2325, 3100, 4069, 5266, 6729, 8500, 10626, 13156, 16146, 19656, 23751, 28501, 33982, 40275, 47467, 55651, 64926, 75398, 87179, 100388, 115151, 131601, 149879, 170133, 192519, 217201, 244351, 274150

Offset: 0

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 5, 5, -15, 20, -15, 5, 5, -15, 20, -15, 5, 5, -15, 20, -15, 5, 5, -15, 20, -15, 6, -1).

A column of triangle A011857.

seq(floor(binomial(n,5)/5), n=0..37); # Zerinvary Lajos, Jan 12 2009

Floor[Binomial[Range[0,40],5]/5] (* Harvey P. Dale, Jan 21 2013 *)
CoefficientList[Series[x^6 (x^19 - 4 x^18 + 8 x^17 - 8 x^16 + 4 x^15 - 3 x^13 + 6 x^12 - 6 x^11 + 3 x^10 - 2 x^8 + 4 x^7 - 4 x^6 + 2 x^5 - x^3 + 2 x^2 - 2 x + 1)/((x - 1)^6 (x^20 + x^15 + x^10 + x^5 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)

G.f.: x^6*(x^19 -4*x^18 +8*x^17 -8*x^16 +4*x^15 -3*x^13 +6*x^12 -6*x^11 +3*x^10 -2*x^8 +4*x^7 -4*x^6 +2*x^5 -x^3 +2*x^2 -2*x +1) / ((x -1)^6*(x^20 +x^15 +x^10 +x^5 +1)). [Colin Barker, Jan 23 2013]

More terms from Colin Barker, Jan 23 2013

A011853 a(n) = floor( binomial(n,7)/7 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 17, 47, 113, 245, 490, 919, 1634, 2778, 4546, 7198, 11074, 16611, 24363, 35022, 49443, 68671, 93971, 126861, 169148, 222968, 290828, 375653, 480836, 610292, 768516, 960645

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,7,7,-28,56,-70,56,-28,7,7,-28, 56,-70,56,-28,7,7, -28,56,-70,56,-28,7,7,-28,56, -70,56,-28,7,7,-28,56,-70,56,-28,7,7,-28,56, -70,56,-28,8,-1).

Partial sums of A011797.

A column of triangle A011857.

Floor[Binomial[Range[0,40],7]/7] (* Harvey P. Dale, Oct 08 2012 *)

a(n)=binomial(n,7)\7 \\ Charles R Greathouse IV, Sep 28 2015

[floor(binomial(n,7)/7) for n in range(0,36)] # Zerinvary Lajos, Dec 01 2009

A011856 a(n) = floor(binomial(n,10)/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 28, 100, 300, 800, 1944, 4375, 9237, 18475, 35271, 64664, 114406, 196125, 326876, 531173, 843628, 1312311, 2003001, 3004501, 4435216, 6451224, 9256104, 13112814, 18357939

Offset: 0

N. J. A. Sloane

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, order 241.

A column of triangle A011857.

Array[Floor[Binomial[#, 10]/10] &, 36, 0] (* Michael De Vlieger, Jul 06 2018 *)

[floor(binomial(n,10)/10) for n in range(0,36)] # Zerinvary Lajos, Dec 01 2009

A101687 a(n) = Sum_{k=1..n} floor(binomial(n,k)/k).

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 23, 41, 73, 131, 232, 418, 758, 1388, 2558, 4742, 8837, 16547, 31111, 58705, 111133, 210996, 401645, 766367, 1465416, 2807595, 5388705, 10359733, 19946714, 38459504, 74250895, 143524561, 277742291, 538043338, 1043333607, 2025040417, 3933914770

Offset: 0

Ralf Stephan, Dec 15 2004

nonn

Row sums of triangle A011857.

Cf. A095718.

Table[Sum[Floor[Binomial[n,k]/k],{k,n}],{n,40}] (* Harvey P. Dale, Nov 14 2021 *)

Missing a(0)=0 inserted by Alois P. Heinz, Nov 14 2021

A011850 a(n) = floor(binomial(n,4)/4).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 8, 17, 31, 52, 82, 123, 178, 250, 341, 455, 595, 765, 969, 1211, 1496, 1828, 2213, 2656, 3162, 3737, 4387, 5118, 5937, 6851, 7866, 8990, 10230, 11594, 13090, 14726, 16511, 18453, 20562

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-6,6,-10,10,-6,6,-10,10,-6,6,-10,10,-5,1).

A column of triangle A011857.

seq(floor(binomial(n,4)/4), n=0.. 39); # Zerinvary Lajos, Jan 12 2009

Floor[Binomial[Range[0,50],4]/4] (* or *) LinearRecurrence[ {5,-10,10,-6,6,-10,10,-6,6,-10,10,-6,6,-10,10,-5,1},{0,0,0,0,0,1,3,8,17,31,52,82,123,178,250,341,455},50] (* Harvey P. Dale, Mar 25 2013 *)

a(n) = +5*a(n-1) -10*a(n-2) +10*a(n-3) -6*a(n-4) +6*a(n-5) -10*a(n-6) +10*a(n-7) -6*a(n-8) +6*a(n-9) -10*a(n-10) +10*a(n-11) -6*a(n-12) +6*a(n-13) -10*a(n-14) +10*a(n-15) -5*a(n-16) +a(n-17). [R. J. Mathar, Apr 15 2010]

G.f.: x^5*(-3*x^7-3*x^3+3*x^8-2*x^9+x^10-x^5+2*x^6-2*x+3*x^2+2*x^4+1) / ( (1-x)^5*(x^4+1)*(x^8+1) ). [R. J. Mathar, Apr 15 2010]

A011852 a(n) = floor(binomial(n,6)/6).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 14, 35, 77, 154, 286, 500, 834, 1334, 2062, 3094, 4522, 6460, 9044, 12435, 16824, 22432, 29516, 38371, 49335, 62790, 79170, 98962, 122713, 151032, 184594, 224150, 270526, 324632

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (7, -22, 41, -50, 41, -22, 7, 0, -6, 15, -19, 9, 9, -19, 15, -6, 0, 6, -15, 19, -9, -9, 19, -16, 13, -22, 35, -35, 22, -13, 16, -19, 9, 9, -19, 15, -6, 0, 6, -15, 19, -9, -9, 19, -15, 6, 0, -7, 22, -41, 50, -41, 22, -7, 1).

A column of triangle A011857.

a(n) = binomial(n, 6)\6; \\ Michel Marcus, Jan 28 2018

[floor(binomial(n,6)/6) for n in range(0,37)] # Zerinvary Lajos, Dec 01 2009

A011854 a(n) = floor(binomial(n,8)/8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 20, 61, 160, 375, 804, 1608, 3038, 5469, 9447, 15746, 25436, 39971, 61289, 91933, 135196, 195284, 277509, 388513, 536518, 731615, 986090, 1314787, 1735519, 2269525, 2941977

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -10, 10, -36, 84, -126, 126, -84, 36, -9, 1).

A column of triangle A011857.

a(n) = binomial(n, 8)\8; \\ Michel Marcus, Jan 28 2018

[floor(binomial(n,8)/8) for n in range(0,36)] # Zerinvary Lajos, Dec 01 2009

A011855 a(n) = floor(binomial(n,9)/9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 24, 79, 222, 556, 1271, 2701, 5402, 10264, 18662, 32658, 55268, 90798, 145278, 226997, 347172, 520758, 767433, 1112778, 1589683, 2240008, 3116533, 4285233, 5827917, 7845273

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 9, 9, -45, 120, -210, 252, -210, 120, -45, 10, -1).

A column of triangle A011857.

Floor[Binomial[Range[0,80],9]/9] (* Harvey P. Dale, Aug 30 2016 *)

a(n) = binomial(n,9)\9; \\ Michel Marcus, Jan 29 2018

[floor(binomial(n,9)/9) for n in range(0,36)] # Zerinvary Lajos, Dec 01 2009

cp's OEIS Frontend

A011848 a(n) = floor(binomial(n, 2)/2).

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A011849 a(n) = floor(binomial(n,3)/3).

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A011851 a(n) = floor(binomial(n,5)/5).

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A011853 a(n) = floor( binomial(n,7)/7 ).

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A011856 a(n) = floor(binomial(n,10)/10).

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A101687 a(n) = Sum_{k=1..n} floor(binomial(n,k)/k).

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A011850 a(n) = floor(binomial(n,4)/4).

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A011852 a(n) = floor(binomial(n,6)/6).