A007590 -id:A007590 - cp's OEIS frontend

A171974 Integer part of the height of a regular tetrahedron with edge length n.

0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 53, 54, 55, 56, 57, 57, 58, 59

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

-3 <= 4*A171975(n) - 3*a(n) < 3;
a(n)*A171975(n) <= A007590(n);
floor(a(n)*A171971(n)/3) <= A171973(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron
Index entries for sequences related to Beatty sequences

Cf. A171972, A022840. Beatty sequence of A157697.

Cf. A007590, A171971, A171973, A171975.

a171974 = floor . (/ 3) . (* sqrt 6) . fromInteger
-- Reinhard Zumkeller, Dec 15 2012

a(n) = floor(n*sqrt(6)/3).

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10416 (the first 31 rows of the triangle)

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];

Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

T(n,k) = A010766(n^2,k).

A007009 Number of 3-voter voting schemes with n linearly ranked choices.

Original entry on oeis.org

1, 4, 12, 27, 54, 96, 160, 250, 375, 540, 756, 1029, 1372, 1792, 2304, 2916, 3645, 4500, 5500, 6655, 7986, 9504, 11232, 13182, 15379, 17836, 20580, 23625, 27000, 30720, 34816, 39304, 44217, 49572, 55404, 61731, 68590, 76000, 84000, 92610, 101871, 111804

Offset: 1

Daniel E. Loeb

With a(0) = 0 nontrivial integer solutions of (x + y)^3 = (x - y)^4. If x = a(n) then y = a(n + (-1)^n). - Thomas Scheuerle, Mar 22 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A034828 (first differences).

I:=[1,4,12,27,54,96,160]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)- 5*Self(n-3)+5*Self(n-4)+Self(n-5)-3*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Sep 21 2015

a:= n-> (Matrix([[0$4, 1, 4, 12, 27]]). Matrix(8, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [4, -4, -4, 10, -4, -4, 4, -1][i], 0)))^n)[1, 1]:
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 13 2008

LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {1, 4, 12, 27, 54, 96, 160}, 50] (* Vincenzo Librandi, Sep 21 2015 *)

Vec(x*(1-x^3)/((1-x)^4*(1-x^2)^2) + O(x^100)) \\ Colin Barker, Jan 07 2016

G.f.: x*(1-x^3)/((1-x)^4*(1-x^2)^2) = x*(1+x+x^2)/((1-x)^5*(1+x)^2).
a(n) = (1/2)*Sum_{k=1..n+1} k*floor(k/2)*ceiling(k/2). - Vladeta Jovovic, Apr 29 2006
a(n) = A006009(n)/4.
a(n) = A007590(n+2)*A007590(n+1)/8. - Richard R. Forberg, Dec 03 2013
For n > 1, a(n) = A000332(n+3) - A002624(n-2). - Antal Pinter, Sep 20 2015
a(n) = (n^4 + 6*n^3 + 12*n^2 + 8*n)/32 for n even; a(n) = (n^4 + 6*n^3 + 12*n^2 + 10*n + 3)/32 for n odd. - Colin Barker, Jan 07 2016

More terms from James Sellers, Sep 08 2000

A094953 Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 2, 8, 9, 4, 3, 12, 21, 16, 5, 3, 18, 39, 44, 25, 6, 4, 24, 66, 96, 80, 36, 7, 4, 32, 102, 184, 200, 132, 49, 8, 5, 40, 150, 320, 430, 372, 203, 64, 9, 5, 50, 210, 520, 830, 888, 637, 296, 81, 10, 6, 60, 285, 800, 1480, 1884, 1673, 1024, 414, 100, 11, 6

Offset: 2

Ralf Stephan, May 26 2004

			1
1 2
2 4 3
2 8 9 4
3 12 21 16 5
3 18 39 44 25 6
4 24 66 96 80 36 7

S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.

Columns 2-4 (+-offset) are A004526, A007590, A007518.
Row sums are A045883, diagonals include n, n^2, (n-1)(n^2-n+2)/2, (n-1)^2(n^+n+6), etc.
Cf. A045927.

T[n_, m_] := SeriesCoefficient[(m-1)x^(m+1)/(1+x)/(1-x)^m, {x, 0, n+1}];
Table[T[n, m], {n, 2, 13}, {m, 2, n}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

T(n,m)=polcoeff((m-1)*x^(m+1)/(1+x)/(1-x)^m,n)

G.f. of m-th column: [(m-1)x^(m+1)]/[(1+x)(1-x)^m].

A060155 Table T(n,k) by antidiagonals of floor(n^k/k) [n,k >= 1].

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 4, 9, 8, 5, 0, 6, 20, 21, 12, 6, 0, 10, 48, 64, 41, 18, 7, 0, 18, 121, 204, 156, 72, 24, 8, 0, 32, 312, 682, 625, 324, 114, 32, 9, 0, 56, 820, 2340, 2604, 1555, 600, 170, 40, 10, 0, 102, 2187, 8192, 11160, 7776, 3361, 1024, 243, 50, 11

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3)=[5^3/3]=[125/3]=41.
Rows start:
  1,  0,  0,   0,   0, ...
  2,  2,  2,   4,   6, ...
  3,  4,  9,  20,  48, ...
  4,  8, 21,  64, 204, ...
  5, 12, 41, 156, 625, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A000007, A000799, A092763, A129794, A129795, A129796, A129797, A129798, A129799, A060156.
Columns include A000027, A007590.
Diagonals include A000169.

T(n, k) = (A051129(n, k)-A060154(n, k))/k.

A080476 Floor( geometric mean of next n numbers ).

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, 72, 84, 98, 112, 128, 144, 162, 180, 200, 220, 242, 264, 288, 312, 338, 364, 392, 420, 450, 480, 512, 544, 578, 612, 648, 684, 722, 760, 800, 840, 882, 924, 968, 1012, 1058, 1104, 1152, 1200, 1250, 1300, 1352, 1404, 1458

Offset: 1

Amarnath Murthy, Mar 11 2003

Essentially the same as A007590: a(n) = A007590(n) for n>=2.
Also, floor( harmonic mean of next n numbers ).
Also, floor(sqrt(A131479(n)+1)). - Richard R. Forberg, Aug 04 2013

			a(4) = floor( (7*8*9*10)^(1/4) ) = 8.
a(4) = floor( 1/( (1/7 + 1/8 + 1/9 + 1/10 )*(1/4)) ) = 8.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000982, A007590, A131479.

PARI
```
a(n)=if(n<2,n>0,n^2\2);
```

a(n+3) = 2*a(n+2) - a(n+1) if n even, a(n+3) = 2*a(n+2) - a(n+1) + 2 if n odd, with a(1) = 1, a(2) = 2, a(3) = 4. - Yosu Yurramendi, Sep 12 2008
From Colin Barker, Aug 08 2013: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 5.
G.f.: x*(x^4 - 2*x^3 - 1)/((x - 1)^3*(x + 1)). (End)
E.g.f.: (2*x + x*(x + 1)*cosh(x) + (x^2 + x - 1)*sinh(x))/2. - Stefano Spezia, Feb 18 2023

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A171975 Integer part of the circumsphere radius of a regular tetrahedron with edge length n.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 45

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

-3 <= 4*a(n) - 3*A171974(n) < 3;

a(n)*A171974(n) <= A007590(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron

Cf. A171973, A171972, A022840. Beatty sequence of A187110.

a171975 = floor . (/ 4) . (* sqrt 6) . fromInteger
-- Reinhard Zumkeller, Dec 15 2012

a(n) = floor(n*sqrt(6)/4).

A241683 Total number of 2 X 2 squares appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 0, 0, 2, 12, 50, 220, 882, 3612, 14450, 58140, 232562, 931612, 3726450, 14911260, 59645042, 238602012, 954408050, 3817719580, 15270878322, 61083862812, 244335451250, 977343203100, 3909372812402

Offset: 0

Kival Ngaokrajang, Apr 27 2014

nonn

a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 2 columns) that appear as 2 X 2 squares in the Thue-Morse logical matrices after n stages. See links for more details.

Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4, 5, -20, -4, 16).

Cf. A010060.

{a0=0;print1(a0,", "); for (n=0,50, b=ceil(2*(2^n-1)/3); a=floor(b^2/2); print1(a,", "))}

a(n) = A007590(A000975(n - 1)).
Empirical g.f.: 2*x^3*(4*x^2-2*x-1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014
a(n) = (1/18) * (4^n - 3*2^n - 6*(-1)^n + (-2)^n - 2), n>0 (from g.f.). - Ralf Stephan, Apr 27 2014

A241684 The total number of rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 0, 4, 8, 32, 120, 464, 1848, 7312, 29240, 116624, 466488, 1864592, 7458360, 29827984, 119311928, 477225872, 1908903480, 7635526544, 30542106168, 122168075152, 488672300600, 1954687804304, 7818751217208, 31274999276432, 125099997105720, 500399966053264, 2001599864213048

Offset: 0

Kival Ngaokrajang, Apr 27 2014

a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 1 column or 1 row, 2 columns) that appear as rectangles in the Thue-Morse logical matrices after n stages. See links for more details.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4,5,-20,-4,16)

Cf. A010060.

[(8+3*2^n+2*4^n+(-1)^n*(24-2^n))/18: n in [0..30]]; // Vincenzo Librandi, Sep 29 2017

CoefficientList[Series[-4*x^2*(8*x^3 - 5*x^2 - 2*x + 1)/((x - 1)*(x + 1)*(2*x - 1)*(2*x + 1)*(4*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)

x='x+O('x^50); concat([0,0], Vec(-4*x^2*(8*x^3-5*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)))) \\ G. C. Greubel, Sep 28 2017

a(n) = A007590(A005578(n+1)) - (A139598(A000975(n-2)) + A007590(A000975(n-1))).
G.f.: -4*x^2*(8*x^3-5*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014
a(n) = (8 + 3*2^n + 2*4^n + (-1)^n*(24 - 2^n))/18, n>0. - R. J. Mathar, May 04 2014

Terms a(21) onward added by G. C. Greubel, Sep 28 2017

A334892 Number T(n,k) of k-element subsets of [n] avoiding 3-term arithmetic progressions and containing n if n>0; triangle T(n,k), n>=0, 0<=k<=A003002(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 4, 1, 0, 1, 5, 8, 3, 0, 1, 6, 12, 6, 0, 1, 7, 18, 15, 0, 1, 8, 24, 26, 4, 0, 1, 9, 32, 47, 20, 0, 1, 10, 40, 67, 40, 7, 0, 1, 11, 50, 102, 80, 18, 0, 1, 12, 60, 140, 140, 53, 6, 0, 1, 13, 72, 194, 236, 110, 16, 1

Offset: 0

Alois P. Heinz, May 14 2020

T(n,k) is defined for all n >= 0 and k >= 0. The triangle contains only elements with 0 <= k <= A003002(n). T(n,k) = 0 for k > A003002(n).

			  1;
  0, 1;
  0, 1,  1;
  0, 1,  2;
  0, 1,  3,  2;
  0, 1,  4,  4,   1;
  0, 1,  5,  8,   3;
  0, 1,  6, 12,   6;
  0, 1,  7, 18,  15;
  0, 1,  8, 24,  26,   4;
  0, 1,  9, 32,  47,  20;
  0, 1, 10, 40,  67,  40,   7;
  0, 1, 11, 50, 102,  80,  18;
  0, 1, 12, 60, 140, 140,  53,   6;
  0, 1, 13, 72, 194, 236, 110,  16,  1;
  0, 1, 14, 84, 248, 342, 198,  42,  3;
  0, 1, 15, 98, 326, 532, 377, 100, 10;
  ...

Fausto A. C. Cariboni, Rows n = 0..80, flattened (rows n = 0..40 from Alois P. Heinz)
Eric Weisstein's World of Mathematics, Nonaveraging Sequence
Wikipedia, Arithmetic progression
Wikipedia, Salem-Spencer set
Index entries related to non-averaging sequences

Columns k=0-3 give: A000007, A057427, A000027(n-1), A007590(n-2).
Row sums give A334893.
Last elements of rows give A334894.
Cf. A003002, A334187.

b:= proc(n, s) option remember; `if`(n=0, x, b(n-1, s)+ `if`(
      ormap(j-> 2*j-n in s, s), 0, expand(x*b(n-1, s union {n}))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
            `if`(n=0, 1, b(n-1, {n}))):
seq(T(n), n=0..16);

b[n_, s_] := b[n, s] = If[n == 0, x, b[n-1, s] + If[
     AnyTrue[s, MemberQ[s, 2#-n]&], 0, Expand[x*b[n-1, s ~Union~ {n}]]]];
T[n_] := If[n == 0, {1}, CoefficientList[b[n-1, {n}], x]];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)

T(0,k) = A334187(0,k), T(n,k) = A334187(n,k) - A334187(n-1,k) for n > 0.

cp's OEIS Frontend

A171974 Integer part of the height of a regular tetrahedron with edge length n.

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A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

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A007009 Number of 3-voter voting schemes with n linearly ranked choices.

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A094953 Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.

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A060155 Table T(n,k) by antidiagonals of floor(n^k/k) [n,k >= 1].

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A080476 Floor( geometric mean of next n numbers ).

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PARI

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A171975 Integer part of the circumsphere radius of a regular tetrahedron with edge length n.

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Haskell

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A241683 Total number of 2 X 2 squares appearing in the Thue-Morse sequence logical matrices after n stages.

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