A093005 -id:A093005 - cp's OEIS frontend

A159915 a(n) = floor((n+1)/4)*floor(n/2).

0, 0, 0, 1, 2, 2, 3, 6, 8, 8, 10, 15, 18, 18, 21, 28, 32, 32, 36, 45, 50, 50, 55, 66, 72, 72, 78, 91, 98, 98, 105, 120, 128, 128, 136, 153, 162, 162, 171, 190, 200, 200, 210, 231, 242, 242, 253, 276, 288, 288, 300, 325, 338, 338, 351, 378, 392, 392, 406, 435, 450, 450

Offset: 0

M. F. Hasler, May 01 2009, May 03 2009

Half the number of (n-2)-element subsets of {1,...,n} with odd sum of the elements.
This is half the antepenultimate column of A159916, cf. formula.
The number of subsets of {1,...,n} with n-2 elements, adding up to an odd integer, is always even (cf. examples), so we divide it by 2.
We prefer to include a(0)=a(1)=a(2)=0, even if it might seem more natural to start only at n=2 or n=3.
From the rational g.f. it can be seen that the sequence is a linear recurrence with constant coefficients (3,-5,7,-7,5,-3,1) of order 7.
A quasipolynomial of order 4 and degree 2. - Charles R Greathouse IV, Sep 18 2024

			a(0)=a(1)=0 since there are no subsets with -2 or -1 elements.
a(2)=0 since the sum of the elements of a 0-element subset is zero.
a(3)=1 since for n=3 we have two singleton subsets of {1,2,3}, {1} and {3}, with odd sum of elements.
a(4)=2 since for n=4 we have four 2-element subsets of {1,2,3,4} with odd sum: {1,2}, {2,3}, {1,4}, {3,4}.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A093005, A093353, A159916.

A159915:= func< n | Floor((n+1)/4)*Floor(n/2) >;
[A159915(n): n in [0..70]]; // G. C. Greubel, Sep 18 2024

Table[Floor[(n+1)/4]*Floor[n/2], {n,0,70}] (* G. C. Greubel, Sep 18 2024 *)

A159915(n)= polcoeff( (1-x+x^2)/((1-x)^3*(1+x^2)^2) + O(x^(n-2)), n-3);
a(n,t=[0,0,0,1,2,2,3],c=[1,-3,5,-7,7,-5,3]~)=while(n-->5,t=concat(vecextract(t,"^1"),t*c));t[n+2] /* Note: a(n+1,[0,0,0,0,1,2,2]) gives the same result as a(n) */

A159915(n)=(n+1)\4*(n\2) \\ M. F. Hasler, May 03 2009

def A159915(n): return ((n+1)//4)*(n//2)
[A159915(n) for n in range(71)] # G. C. Greubel, Sep 18 2024

G.f.: x^3*(1 - x + x^2)/(1 - 3*x + 5*x^2 - 7*x^3 + 7*x^4 - 5*x^5 + 3*x^6 - x^7) = x^3*(1-x+x^2)/((1-x)^3*(1+x^2)^2).
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n > 7.
For n > 2, a(n) = A159916(n*(n-1)/2 + n - 2)/2 = T(n,n-2)/2 as defined there.
From M. F. Hasler, May 03 2009: (Start)
a(n) = floor((n+1)/4)*floor(n/2).
a(2n+1) = A093005(n).
a(2n) = A093353(n-1) = floor(n/2)*n. (End)
a(n) ~ n^2/8. - Charles R Greathouse IV, Sep 18 2024

A182455 a(0)=1, a(n) = (a(n-1) mod (n+2))*(n+2).

Original entry on oeis.org

1, 3, 12, 10, 24, 21, 40, 36, 60, 55, 84, 78, 112, 105, 144, 136, 180, 171, 220, 210, 264, 253, 312, 300, 364, 351, 420, 406, 480, 465, 544, 528, 612, 595, 684, 666, 760, 741, 840, 820, 924, 903, 1012, 990, 1104, 1081, 1200, 1176, 1300, 1275, 1404

Offset: 0

Alex Ratushnyak, Apr 30 2012

			a(5) = (a(4) mod 7)*7 = (24 mod 7)*7 = 3*7 = 21.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A093005 (a(n)=(a(n-1) mod (n+1))*(n+1)).

a182455 n = a182455_list !! n
a182455_list = 1 : zipWith (*) (zipWith mod a182455_list [3..]) [3..]
-- Reinhard Zumkeller, May 01 2012

a=1
for n in range(1,55):
    print(a, end=', ')
    a = (a%(n+2)) * (n+2)

a(0)=1, a(n) = (a(n-1) mod (n+2))*(n+2).

For k>0, a(2*k)=(k+1)*(2*k+4), a(2*k+1)=(k+1)*(2*k+3).

A024930 a(n) = sum of remainders of n mod 1,3,5,...,2k-1, where k = [ (n+1)/2 ].

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 3, 6, 6, 5, 9, 11, 16, 15, 13, 20, 27, 23, 31, 35, 34, 33, 43, 51, 57, 56, 56, 62, 75, 66, 80, 95, 96, 95, 99, 104, 121, 120, 122, 136, 155, 144, 164, 174, 163, 162, 184, 204, 220, 214, 218, 230, 255, 242, 252, 272, 277, 276, 304, 310, 339, 338, 328, 359, 372, 357

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A078471, A093005.

A093005:= [seq(n*ceil(n/2),n=1..100)]:
A078471:= ListTools:-PartialSums([seq(numtheory:-sigma(n/2^padic:-ordp(n,2)),n=1..100)]):
A093005-A078471; # Robert Israel, May 13 2019

a(n) = A093005(n) - A078471(n). - Robert Israel, May 13 2019

A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles.

Original entry on oeis.org

660, 1092, 1140, 1155, 1260, 1320, 1365, 1380, 1428, 1540, 1560, 1740, 1785, 1820, 1860, 1980, 1995, 2184, 2220, 2340, 2380, 2415, 2436, 2460, 2508, 2580, 2604, 2660, 2805, 2820, 2856, 2860, 2940, 3003, 3036, 3060, 3108, 3120, 3135, 3180, 3192, 3220, 3300

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 25 2008

nonn

			Examples of triples: 660.779.1021, 660.989.1189, 660.2989.3061, 660.4331.4381, 660.12091.12109, 660.27221.27229, 660.108899.108901
1092.1325.1717, 1092.1595.1933, 1092.6035.6133, 1092.8245.8317, 1092.33115.33133, 1092.74525.74533, 1092.298115.298117

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A056866 Orders of non-solvable groups. A093006 Referring to the triangle in A093005, sequence contains the least term with maximal number of divisors. A138605 Short legs of more than 3 primitive Pythagorean triangles. A033993 Numbers that are divisible by exactly four different primes.

PythagoreanAs[a_]:=(q={};k=0;Do[y=(a^2+b^2)^0.5;c=IntegerPart[y];If[c==y,p=0;If[GCD[a,b,c]==1,AppendTo[q,a.b.c];k++ ]],{b,a+1,a^2}];PrependTo[q,k];q);lst={};Do[If[PythagoreanAs[n][[1]]==7,Print[n];AppendTo[lst,n]],{n,6*10^2,2*10^3}];lst

a^2+b^2=c^2

More terms from Ray Chandler, Jan 22 2020

A199855 Inverse permutation to A210521.

Original entry on oeis.org

1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79

Offset: 1

Boris Putievskiy, Feb 04 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
  T(1,1)=1;
  T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
  ...
  T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
  T(2,n),   T(4,n-2), T(6,n-4), ..., T(n,2),
  T(1,n),   T(3,n-2), T(5,n-4), ..., T(n-1,2),
  T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
  ...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(2,2*m-1) to T(2*m,1)   length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m)   to T(2*m,2)   length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(1,2*m)   to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
  row  1 is alternation of elements A001844 and A084849,
  row  2 is alternation of elements A130883 and A058331,
  row  3 is alternation of elements A051890 and A096376,
  row  4 is alternation of elements A033816 and A005893,
  row  6 is alternation of elements A100037 and A093328;
  row  5  accommodates elements A097080 in odd places,
  row  7  accommodates elements A137882 in odd places,
  row 10  accommodates elements A100038 in odd places,
  row 14  accommodates elements A100039 in odd places;
  column 1 is A093005 and alternation of elements A000384 and A001105,
  column 2 is alternation of elements A046092 and A014105,
  column 3 is A105638 and alternation of elements A014106 and A056220,
  column 4 is alternation of elements A142463 and A014107,
  column 5 is alternation of elements A091823 and A054000,
  column 6 is alternation of elements A090288 and |A168244|,
  column 8 is alternation of elements A059993 and A033537;
  column  7 accommodates elements  A071355  in odd  places,
  column  9 accommodates elements |A147973| in even places,
  column 10 accommodates elements  A139570  in odd  places,
  column 13 accommodates elements  A130861  in odd  places.

			The start of the sequence as table:
   1,  4,  5,  11,  13,  22,  25,  37,  41,  56,  61, ...
   2,  3,  7,   9,  16,  19,  29,  33,  46,  51,  67, ...
   6, 12, 14,  23,  26,  38,  42,  57,  62,  80,  86, ...
   8, 10, 17,  20,  30,  34,  47,  52,  68,  74,  93, ...
  15, 24, 27,  39,  43,  58,  63,  81,  87, 108, 115, ...
  18, 21, 31,  35,  48,  53,  69,  75,  94, 101. 123, ...
  28, 40, 44,  59,  64,  82,  88, 109, 116, 140, 148, ...
  32, 36, 49,  54,  70,  76,  95, 102, 124, 132, 157, ...
  45, 60, 65,  83,  89, 110, 117, 141, 149, 176, 185, ...
  50, 55, 71,  77,  96, 103, 125, 133, 158, 167, 195, ...
  66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  6;
  11,  7, 12,  8;
  13,  9, 14, 10, 15;
  22, 16, 23, 17, 24, 18;
  25, 19, 26, 20, 27, 21, 28;
  37, 29, 38, 30, 39, 31, 40, 32;
  41, 33, 42, 34, 43, 35, 44, 36, 45;
  56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
  61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
   1;
   4, 2, 5, 3, 6;
  11, 7,12, 8,13, 9,14,10,15;
  22,16,23,17,24,18,25,19,26,20,27,21,28;
  37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
  56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
  ...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4

T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.

a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).

A329965 a(n) = ((1+n)floor(1+n/2))(n!/floor(1+n/2)!)^2.

Original entry on oeis.org

1, 2, 6, 72, 240, 7200, 25200, 1411200, 5080320, 457228800, 1676505600, 221298739200, 821966745600, 149597947699200, 560992303872000, 134638152929280000, 508633022177280000, 155641704786247680000, 591438478187741184000, 224746621711341649920000

Offset: 0

Peter Luschny, Dec 04 2019

nonn

Cf. A212303, A057977, A052510, A123072, A093005, A262033, A329964.

A329965 := n -> ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2:
seq(A329965(n), n=0..19);

ser := Series[(1 - Sqrt[1 - 4 x^2] - 4 x^2 (1 - x - Sqrt[1 - 4 x^2]))/(2 x^2 (1 - 4 x^2)^(3/2)), {x, 0, 22}]; Table[n! Coefficient[ser, x, n], {n, 0, 20}]
Table[(1+n)Floor[1+n/2](n!/Floor[1+n/2]!)^2,{n,0,30}] (* Harvey P. Dale, Oct 01 2023 *)

from fractions import Fraction
def A329965():
    x, n = 1, Fraction(1)
    while True:
        yield int(x)
        m = n if n % 2 else 4/(n+2)
        n += 1
        x *= m * n
a = A329965(); [next(a) for i in range(36)]

a(n) = n!*A212303(n+1).
a(n) = (n+1)!*A057977(n).
a(n) = A093005(n+1)*A262033(n)^2.
a(n) = A093005(n+1)*A329964(n).
a(2*n) = A052510(n) (n >= 0).
a(2*n+1) = A123072(n+1) (n >= 0).
a(n) = n! [x^n] (1 - sqrt(1 - 4*x^2) - 4*x^2*(1 - x - sqrt(1 - 4*x^2)))/(2*x^2*(1 - 4*x^2)^(3/2)).

A329970 a(n) = (-1)^(n + 1) * n * ceiling(n/2) + Sum_{k=1..n} (-1)^k * k^2 * floor(n/k).

Original entry on oeis.org

0, 0, -2, 3, 0, -3, -7, 16, 2, -15, -21, 31, 24, -15, -57, 34, 25, -17, -27, 77, 8, -99, -111, 155, 117, -36, -140, 40, 25, -80, -96, 259, 112, -157, -249, 202, 183, -156, -354, 224, 203, -40, -62, 342, -21, -524, -548, 562, 488, -34, -358, 194, 167, -262

Offset: 1

Ilya Gutkovskiy, Nov 26 2019

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004125, A051126, A093005, A154585, A309176.

Table[(-1)^(n + 1) n Ceiling[n/2] + Sum[(-1)^k k^2 Floor[n/k], {k, 1, n}], {n, 1, 54}]
nmax = 54; CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x)^2 (1 + x)^3) + 1/(1 - x) Sum[(-1)^k k^2 x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(k + 1) Mod[n, k] k, {k, 1, n}], {n, 1, 54}]

a(n) = (-1)^(n + 1)*n*ceil(n/2) + sum(k=1, n, (-1)^k * k^2 * (n\k)); \\ Michel Marcus, Sep 20 2021

G.f.: x * (1 - x + 2*x^2) / ((1 - x)^2 * (1 + x)^3) + (1/(1 - x)) * Sum_{k>=1} (-1)^k * k^2 * x^k / (1 - x^k).

a(n) = Sum_{k=1..n} (-1)^(k + 1) * (n mod k) * k.

A356288 Sum of numbers in n-th upward diagonal of triangle the sum of {1; 2,3; 4,5,6; 7,8,9,10; ...} and {1; 2,3; 3,4,5; 4,5,6,7; ...}.

Original entry on oeis.org

2, 4, 13, 20, 40, 55, 90, 116, 170, 210, 287, 344, 448, 525, 660, 760, 930, 1056, 1265, 1420, 1672, 1859, 2158, 2380, 2730, 2990, 3395, 3696, 4160, 4505, 5032, 5424, 6018, 6460, 7125, 7620, 8360, 8911, 9730, 10340, 11242, 11914, 12903, 13640, 14720, 15525, 16700

Offset: 1

Torlach Rush, Aug 02 2022

			   2 = A079824(1) + A093005(1) =  1 + 1.
   4 = A079824(2) + A093005(2) =  2 + 2.
  13 = A079824(3) + A093005(3) =  7 + 6.
  20 = A079824(4) + A093005(4) = 12 + 8.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A079824, A093005.

def a(n): return (n * ((n + n % 2) // 2)) + (15 + 25*n + 15*(n**2) + 14*(n**3) - 3*(((-1)**n))*(5 + n*(3 + n))) // 96

a(n) = (n * ceiling(n/2)) + ((15 + 25*n + 15*n^2 + 14*n^3 - 3*(((-1)^n))*(5 + n*(3 + n))) / 96).
a(n) = A079824(n) + A093005(n).
G.f.: x*(2 + 2*x + 3*x^2 + x^3 - x^4)/((1 - x)^4*(1 + x)^3). - Stefano Spezia, Aug 19 2022

cp's OEIS Frontend

A159915 a(n) = floor((n+1)/4)*floor(n/2).

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A182455 a(0)=1, a(n) = (a(n-1) mod (n+2))*(n+2).

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A024930 a(n) = sum of remainders of n mod 1,3,5,...,2k-1, where k = [ (n+1)/2 ].

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Maple

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A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles.

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Extensions

A199855 Inverse permutation to A210521.

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A329965 a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.

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A329970 a(n) = (-1)^(n + 1) * n * ceiling(n/2) + Sum_{k=1..n} (-1)^k * k^2 * floor(n/k).

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A356288 Sum of numbers in n-th upward diagonal of triangle the sum of {1; 2,3; 4,5,6; 7,8,9,10; ...} and {1; 2,3; 3,4,5; 4,5,6,7; ...}.

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A329965 a(n) = ((1+n)floor(1+n/2))(n!/floor(1+n/2)!)^2.