A083074 -id:A083074 - cp's OEIS frontend

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 43, 41, 1, 1, 6, 29, 94, 171, 122, 1, 1, 7, 41, 173, 469, 683, 365, 1, 1, 8, 55, 286, 1037, 2344, 2731, 1094, 1, 1, 9, 71, 439, 2001, 6221, 11719, 10923, 3281, 1, 1, 10, 89, 638, 3511, 14006, 37325, 58594, 43691, 9842, 1

Offset: 0

Paul Barry, Apr 21 2003

			Rows begin:
1  1   1    1     1      1       1        1         1 ...
1  2   5   14    41    122     365     1094      3281 ...  A007051
1  3  11   43   171    683    2731    10923     43691 ...  A007583
1  4  19   94   469   2344   11719    58594    292969 ...  A083065
1  5  29  173  1037   6221   37325   223949   1343693 ...  A083066
1  6  41  286  2001  14006   98041   686286   4804001 ...  A083067
1  7  55  439  3511  28087  224695  1797559  14380471 ...  A083068
1  8  71  638  5741  51668  465011  4185098  37665881 ...  A187709
1  9  89  889  8889  88889  888889  8888889  88888889 ...  A059482
1 10 109 1198 13177 144946 1594405 17538454 192922993 ...  A199760, etc.
Column 2: A000027;
column 3: A028387;
column 4: A083074;
column 5: A125082;
column 6: A125083.
Diagonals:
1,  2,  11,   94,  1037,  14006, ... A083069;
1,  3,  19,  173,  2001,  28087, ... A083071;
1,  4,  29,  286,  3511,  51668, ... A083072;
1,  5,  41,  439,  5741,  88889, ... A083073;
1,  5,  43,  469,  6221,  98041, ... A083070;
1, 14, 171, 2344, 37325, 686286, ... A191690.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 5, 1;
1, 4, 11, 14, 1;
1, 5, 19, 43, 41, 1;
1, 6, 29, 94, 171, 122, 1; etc.

Cf. rows: A007051, A007583, A059482, A083065 - A083068, A187709, A199760; columns: A000027, A028387, A083074, A125082, A125083; diagonals: A083069 - A083073, A191690.

Edited by Bruno Berselli, Jun 21 2013

A152618 a(n) = (n-1)^2*(n+1).

Original entry on oeis.org

1, 0, 3, 16, 45, 96, 175, 288, 441, 640, 891, 1200, 1573, 2016, 2535, 3136, 3825, 4608, 5491, 6480, 7581, 8800, 10143, 11616, 13225, 14976, 16875, 18928, 21141, 23520, 26071, 28800, 31713, 34816, 38115, 41616, 45325, 49248, 53391, 57760, 62361

Offset: 0

Philippe Deléham, Dec 09 2008

For n>0 this is the same under substitution of variables as d(d-2)^2, the number of connected components in Bertrand et al.: "We construct a polynomial of degree d in two variables whose Hessian curve has (d-2)^2 connected components using Viro patchworking. In particular, this implies the existence of a smooth real algebraic surface of degree d in RP^3 whose parabolic curve is smooth and has d(d-2)^2 connected components." - Jonathan Vos Post, Apr 30 2009

For n>0 a(n) is twice the area of the trapezoid created by plotting the four points (n-1,n), (n,n-1), (n*(n-1)/2,n*(n+1)/2), (n*(n+1)/2,(n-1)*n/2). - J. M. Bergot, Mar 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Benoît Bertand and Erwan Brugallé, On the number of connected components of the parabolic curve, Comptes Rendus Mathématique, Vol. 348, No. 5-6 (2010), pp. 287-289; arXiv preprint, arXiv:0904.4652 [math.AG], Apr 29 2009. - _Jonathan Vos Post_, Apr 30 2009
Jim Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 2024. See p. 9.

[(n-1)^2*(n+1): n in [0..50]]; // Vincenzo Librandi, Jun 25 2013

A152618:=n->(n-1)^2*(n+1); seq(A152618(k), k=0..100); # Wesley Ivan Hurt, Oct 06 2013

f[n_]:=(n-1)^2*(n+1);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
CoefficientList[Series[(9 x^2 - 4 x + 1)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2013 *)

a(n)=(n+1)*(n-1)^2 \\ Charles R Greathouse IV, Mar 21 2014

a(n) = n^3 - n^2 - n + 1 = A083074(n) + 2. - Jeremy Gardiner, Jun 23 2013
G.f.: (9*x^2 - 4*x + 1)/(1-x)^4. - Vincenzo Librandi, Jun 25 2013
a(n+1) = A005449(n) + A002414(n), n > 0. - Wesley Ivan Hurt, Oct 06 2013
Sum_{n>1} 1/a(n) = (1/24) * (2*Pi^2 - 9). - Enrique Pérez Herrero, May 31 2015
Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 3)/24. - Amiram Eldar, Jan 13 2021
E.g.f.: exp(x)*(x^3+2*x^2-x+1). - Nikolaos Pantelidis, Feb 06 2023

A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, -1, 0, 1, 0, 1, 0, 1, 1, 2, -1, 0, 1, 2, 3, -1, 1, 0, 1, 3, 4, 1, 1, -1, 0, 1, 4, 5, 5, 3, 1, 1, 0, 1, 5, 6, 11, 7, 5, 3, -1, 0, 1, 6, 7, 19, 13, 11, 7, -2, 1, 0, 1, 7, 8, 29, 21, 19, 13, 1, 0, -1, 0, 1, 8, 9, 41, 31, 29, 21, 14, 3, 0, 1, 0, 1, 9, 10, 55, 43, 41, 31, 43, 16, 5, 2, -1

Offset: 0

Alois P. Heinz, Jan 25 2023

The empty bit string is used as binary expansion of 0, so A(0,k) = 0.

			Square array A(n,k) begins:
   0,  0, 0,  0,  0,   0,   0,   0,   0,   0,   0, ...
   1,  1, 1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  -1,  0, 1,  2,  3,   4,   5,   6,   7,   8,   9, ...
   1,  2, 3,  4,  5,   6,   7,   8,   9,  10,  11, ...
  -1, -1, 1,  5, 11,  19,  29,  41,  55,  71,  89, ...
   1,  1, 3,  7, 13,  21,  31,  43,  57,  73,  91, ...
  -1,  1, 5, 11, 19,  29,  41,  55,  71,  89, 109, ...
   1,  3, 7, 13, 21,  31,  43,  57,  73,  91, 111, ...
  -1, -2, 1, 14, 43,  94, 173, 286, 439, 638, 889, ...
   1,  0, 3, 16, 45,  96, 175, 288, 441, 640, 891, ...
  -1,  0, 5, 20, 51, 104, 185, 300, 455, 656, 909, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-6, 10 give: A062157, A145037, A006257, A147991, A147992, A153777, A147993, A359925.
Rows n=0-10 give: A000004, A000012, A023443, A000027(k+1), A165900, A002061, A165900(k+1), A002061(k+1), A083074, A152618, A062158.
Main diagonal gives A360096.
Cf. A030300, A057427.

A:= proc(n, k) option remember; local m;
     `if`(n=0, 0, k*A(iquo(n, 2, 'm'), k)+2*m-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> (l-> add((2*l[i]-1)*k^(i-1), i=1..nops(l)))(Bits[Split](n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

G.f. for column k satisfies g_k(x) = k*(x+1)*g_k(x^2) + x/(1+x).
A(n,k) = k*A(floor(n/2),k)+2*(n mod 2)-1 for n>0, A(0,k)=0.
A(n,k) mod 2 = A057427(n) if k is even.
A(n,k) mod 2 = A030300(n) if k is odd and n>=1.
A(2^(n+1),1) + n = 0.

A125082 a(n) = n^4 - n^3 - n^2 - n - 1.

Original entry on oeis.org

-1, -3, 1, 41, 171, 469, 1037, 2001, 3511, 5741, 8889, 13177, 18851, 26181, 35461, 47009, 61167, 78301, 98801, 123081, 151579, 184757, 223101, 267121, 317351, 374349, 438697, 511001, 591891, 682021, 782069, 892737, 1014751, 1148861, 1295841, 1456489, 1631627, 1822101, 2028781, 2252561, 2494359

Offset: 0

Artur Jasinski, Nov 19 2006

For odd n > 1, a(n) * (n-1) / 2 represents the first integer in a sum of (2 * n^4) consecutive integers that equals n^9. - Patrick J. McNab, Dec 25 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..660
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A083074.

[n^4-n^3-n^2-n-1: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Table[n^4 - n^3 - n^2 - n - 1, {n, 0, 41}]
LinearRecurrence[{5,-10,10,-5,1},{-1,-3,1,41,171},40] (* Harvey P. Dale, Dec 30 2011 *)

a(n)=n^4-n^3-n^2-n-1 \\ Charles R Greathouse IV, Oct 07 2015

O.g.f.: -60/(-1+x)^3-66/(-1+x)^4-1/(-1+x)-24/(-1+x)^5-20/(-1+x)^2. - R. J. Mathar, Feb 26 2008

a(0)=-1, a(1)=-3, a(2)=1, a(3)=41, a(4)=171, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Dec 30 2011

A125083 a(n) = n^5-n^4-n^3-n^2-n-1.

Original entry on oeis.org

-1, -4, 1, 122, 683, 2344, 6221, 14006, 28087, 51668, 88889, 144946, 226211, 340352, 496453, 705134, 978671, 1331116, 1778417, 2338538, 3031579, 3879896, 4908221, 6143782, 7616423, 9358724, 11406121, 13797026, 16572947, 19778608, 23462069, 27674846, 32472031, 37912412, 44058593

Offset: 0

Artur Jasinski, Nov 19 2006

More generally, the ordinary generating function for the values of quintic polynomial b*n^5 + p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (b + p + q + k + m - 5*r)*x + (13*b + 5*p + q - k - 2*m + 5*r)*2*x^2 + (33*b - 3*q + 3*m - 5*r)*2*x^3 + (26*b - 10*p + 2*q + 2*k - 4*m + 5*r)*x^4 + (b - p + q - k + m - r)*x^5)/(1 - x)^6. - Ilya Gutkovskiy, Mar 31 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..580
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A125082, A083074.

[n^5-n^4-n^3-n^2-n-1: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Table[n^5 - n^4 - n^3 - n^2 - n - 1, {n, 0, 41}]

a(n) = n^5-n^4-n^3-n^2-n-1; \\ Michel Marcus, Mar 31 2016

G.f.: (-1 + 2*x + 10*x^2 + 76*x^3 + 31*x^4 + 2*x^5)/(1 - x)^6. - Ilya Gutkovskiy, Mar 31 2016

cp's OEIS Frontend

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

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A152618 a(n) = (n-1)^2*(n+1).

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A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Keywords

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Examples

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Crossrefs

Programs

Maple

Formula

A125082 a(n) = n^4 - n^3 - n^2 - n - 1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A125083 a(n) = n^5-n^4-n^3-n^2-n-1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula