A120070 -id:A120070 - cp's OEIS frontend

A204994 Least k such that n divides A120070(k+1), the k-th difference between distinct squares.

1, 2, 1, 2, 3, 5, 6, 2, 10, 14, 15, 5, 21, 27, 4, 9, 36, 31, 45, 14, 8, 65, 66, 7, 41, 90, 13, 27, 105, 23, 120, 12, 19, 152, 11, 31, 171, 189, 26, 18, 210, 40, 231, 65, 17, 275, 276, 16, 85, 96, 43, 90, 351, 61, 24, 33, 53, 434, 435, 23

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204994, A204892.

Mathematica
```
(See the program at A204994.)
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A142717 First (leftmost) odd term in the n-th row of triangle A120070.

Original entry on oeis.org

3, 5, 15, 21, 35, 45, 63, 77, 99, 117, 143, 165, 195, 221, 255, 285, 323, 357, 399, 437, 483, 525, 575, 621, 675, 725, 783, 837, 899, 957, 1023, 1085, 1155, 1221, 1295, 1365, 1443, 1517, 1599, 1677, 1763, 1845, 1935, 2021, 2115, 2205, 2303, 2397, 2499, 2597

Offset: 1

Paul Curtz, Sep 26 2008

nonn

Also: Records sequence of A100181.

The last (rightmost) term in the n-th row of triangle A120070 is A005408(n).

			The odd terms of A120070 build the irregular triangle
  3;
  5;
  15,7;
  21,9;
  35,27,11;
  45,33,13;
  63,55,39,15;
The leftmost column defines this sequence.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000466, A078371, A100181.

A142717[n_]:=(n+1)^2-If[OddQ[n],1,4];Array[A142717,100] (* or *)
LinearRecurrence[{2,0,-2,1},{3,5,15,21},100] (* Paolo Xausa, Dec 05 2023 *)

First differences: a(n+1)-a(n) = A142954(n).
From R. J. Mathar, Oct 24 2008: (Start)
a(n) = (n+1)^2-1 = A000466((n+1)/2) if n odd.
a(n) = (n+1)^2-4 = A078371(n/2-1) if n even.
a(n) = 2*a(n-1) -2*a(n-3) +a(n-4).
G.f.: x(3-x+5x^2-3x^3)/((1+x)(1-x)^3). (End)

Edited and extended by R. J. Mathar, Oct 24 2008

A143785 Antidiagonal sums of the triangle A120070.

Original entry on oeis.org

3, 8, 20, 36, 63, 96, 144, 200, 275, 360, 468, 588, 735, 896, 1088, 1296, 1539, 1800, 2100, 2420, 2783, 3168, 3600, 4056, 4563, 5096, 5684, 6300, 6975, 7680, 8448, 9248, 10115, 11016, 11988, 12996, 14079, 15200, 16400, 17640, 18963, 20328, 21780, 23276

Offset: 1

Paul Curtz, Sep 01 2008

Let b(n) be the sequence (0,0,0,3,8,20,36,...), with offset 0. Then b(n) is the number of triples (w,x,y) having all terms in {0,...,n} and w < range{w,x,y}. - Clark Kimberling, Jun 11 2012

Consider a(n) with two 0's prepended and offset 1. Call the new sequence b(n) and consider the partitions of n into two parts (p,q). Then b(n) represents the sum of all the products (p + q) * (q - p) where p <= q. - Wesley Ivan Hurt, Apr 12 2018

			First diagonal 3 = 3.
Second diagonal 8 = 8.
Third diagonal 5+15 = 20.
Fourth diagonal 24+12 = 36.

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A035006, A099721 (bisections).

Cf. A120070, A002620.

[(n+2)*(2*n^2+4*n-(-1)^n+1)/8: n in [1..50]]; // Vincenzo Librandi, Jan 22 2018

Rest@ CoefficientList[Series[x (3 + 2 x + x^2)/((1 + x)^2*(x - 1)^4), {x, 0, 44}], x] (* Michael De Vlieger, Dec 22 2017 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {3, 8, 20, 36, 63, 96}, 60] (* Vincenzo Librandi, Jan 22 2018 *)

Vec(x*(3+2*x+x^2)/((1+x)^2*(x-1)^4) + O(x^50)) \\ Colin Barker, May 07 2016

a(n+1) - a(n) = A032438(n+2).
a(n) = A006918(n-2) + 2*A006918(n-1) + 3*A006918(n). - R. J. Mathar, Jul 01 2011
G.f.: x*(3+2*x+x^2) / ( (1+x)^2*(x-1)^4 ). - R. J. Mathar, Jul 01 2011
a(n) = (n+2)*(2*n^2 + 4*n - (-1)^n + 1)/8. - Ilya Gutkovskiy, May 07 2016
From Colin Barker, May 07 2016: (Start)
a(n) = (n^3 + 4*n^2 + 4*n)/4 for n even.
a(n) = (n^3 + 4*n^2 + 5*n + 2)/4 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 6. (End)
a(n) = Sum_{k=1..n+1} floor((n+1)*k/2). - Wesley Ivan Hurt, Apr 01 2017
a(n) = (n+2)*floor((n+1)^2/4) ( = (n+2)*A002620(n+1) ) for n > 0. - Heinrich Ludwig, Dec 22 2017
E.g.f.: e^(-x) * (-2 + x + e^(2*x)*(2 + 19*x + 14*x^2 + 2*x^3))/8. - Iain Fox, Dec 29 2017

A141620 First differences of A120070.

Original entry on oeis.org

5, -3, 10, -3, -5, 17, -3, -5, -7, 26, -3, -5, -7, -9, 37, -3, -5, -7, -9, -11, 50, -3, -5, -7, -9, -11, -13, 65, -3, -5, -7, -9, -11, -13, -15, 82, -3, -5, -7, -9, -11, -13, -15, -17, 101, -3, -5, -7, -9, -11, -13, -15, -17, -19, 122, -3, -5, -7, -9, -11, -13, -15, -17, -19, -21

Offset: 0

Paul Curtz, Aug 23 2008

sign

Cf. A120070. Essentially the same as A133128.

a(n) = mix (from 2) n^2+1 (or A002522(n+2)), (ever from -3) successive -A005408.

a(n) = ((A^2+3*A-2*n)*(-A^2+A+2*(n+4))-(B^2+3*B-2*(n-1))*(-B^2+B+2*(n+3)))/4, where A = floor((sqrt(8*n+9)-1)/2) and B = floor((sqrt(8*(n-1)+9)-1)/2) = floor((sqrt(8*n+1)-1)/2). - Luce ETIENNE, May 31 2017

A133128 Triangle of first differences of A120070 with a leftmost column of A002522.

Original entry on oeis.org

2, 5, -3, 10, -3, -5, 17, -3, -5, -7, 26, -3, -5, -7, -9, 37, -3, -5, -7, -9, -11, 50, -3, -5, -7, -9, -11, -13, 65, -3, -5, -7, -9, -11, -13, -15, 82, -3, -5, -7, -9, -11, -13, -15, -17, 101, -3, -5, -7, -9, -11, -13, -15, -17, -19, 122, -3, -5, -7, -9, -11, -13, -15, -17, -19, -21, 145, -3, -5, -7, -9, -11, -13, -15, -17, -19, -21, -23, 170, -3, -5, -7, -9, -11, -13

Offset: 0

Paul Curtz, Aug 27 2008

The flattened triangle is essentially the same as A141620.

			The triangle starts
2;
5, -3;
10,-3,-5;
17,-3,-5,-7;
26,-3,-5,-7,-9;

A133128 := proc(n,m) if m>= 1 then -2*m-1 ; else (n+1)^2+1 ; fi; end: seq(seq(A133128(n,m),m=0..n),n=0..15) ; # R. J. Mathar, Nov 22 2009

T(n,0) = A002522(n+1).
T(n,m) = -2m-1, m>0 .
sum_{m=0..n} T(n,m) = 2 (row sums).
T(n,n) = -A005408(n), n>0.

Edited and extended by R. J. Mathar, Nov 22 2009

A141616 Even terms in A120070.

Original entry on oeis.org

8, 12, 24, 16, 32, 20, 48, 40, 24, 60, 48, 28, 80, 72, 56, 32, 96, 84, 64, 36, 120, 112, 96, 72, 40, 140, 128, 108, 80, 44, 168, 160, 144, 120, 88, 48, 192, 180, 160, 132, 96, 52, 224, 216, 200, 176, 144, 104, 56, 252, 240, 220, 192, 156, 112, 60, 288, 280, 264, 240, 208, 168, 120, 64, 320, 308, 288

Offset: 1

Paul Curtz, Aug 23 2008

All terms are multiples of 4.

Row length L = Ceiling(n/2 - 1), thus the smallest value of n in A120070 to produce even terms is n = 3. - Michael De Vlieger, Apr 14 2016

			Irregular triangle:
n    Even values of A120070(n)
3    8
4    12
5    24    16
6    32    20
7    48    40    24
8    60    48    28
9    80    72    56    32
10   96    84    64    36
11  120   112    96    72    40
12  140   128   108    80    44
...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Table[n^2 - k^2, {n, 3, 18}, {k, n}] /. m_ /; Or[OddQ@ m, m == 0] -> Nothing // Flatten (* Michael De Vlieger, Apr 14 2016, after Alonso del Arte at A120070 *)

More terms from Michael De Vlieger, Apr 14 2016

A100181 Odd terms in A120070.

Original entry on oeis.org

3, 5, 15, 7, 21, 9, 35, 27, 11, 45, 33, 13, 63, 55, 39, 15, 77, 65, 45, 17, 99, 91, 75, 51, 19

Offset: 1

Paul Curtz, Aug 29 2008

nonn

Edited by N. J. A. Sloane, Aug 30 2008

A141595 Binomial transform of A120070.

Original entry on oeis.org

3, 11, 24, 57, 137, 310, 672, 1445, 3135, 6834, 14797, 31605, 66642, 139500, 291697, 611517, 1285388, 2702278, 5664348, 11813505, 24503911, 50606865, 104273395, 214794252, 442965900, 914940122, 1891691613, 3910617099, 8072908510, 16626013425, 34146007356, 69946108176

Offset: 0

Paul Curtz, Aug 21 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A120070.

A120070:= [n^2-k^2: k in [1..n-1], n in [2..100]];
A141595:= func< n | (&+[Binomial(n,k)*A120070[k+1]: k in [0..n]]) >;
[A141595(n): n in [0..40]]; // G. C. Greubel, Sep 15 2024

A120070= Table[n^2 - k^2, {n,2,100}, {k,n-1}]//Flatten;
A141595[n_]:= Sum[Binomial[n, k]*A120070[[k+1]], {k,0,n}];
Table[A141595[n], {n,0,40}] (* G. C. Greubel, Sep 15 2024 *)

A120070=flatten([[n^2 -k^2 for k in range(1,n)] for n in range(2,101)])
def A141595(n): return sum(binomial(n,k)*A120070[k] for k in range(n+1))
[A141595(n) for n in range(41)] # G. C. Greubel, Sep 15 2024

a(n) = Sum_{k=0..n} binomial(n,k)*A120070(k). - G. C. Greubel, Sep 15 2024

Terms a(8) onward added by G. C. Greubel, Sep 15 2024

A164561 Triangle with elements A120070(m,n)/A120072(m,n) read by rows, m>=2, 1<=n

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 16, 27, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 16, 1, 4, 1, 1, 1, 9, 1, 1, 9, 1, 1, 1, 16, 1, 4, 25, 16, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 27, 128, 1, 108, 1, 16, 9, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 4, 1, 16, 49, 4, 1, 16, 1, 4, 1, 1, 1, 27, 1

Offset: 2

Paul Curtz, Aug 16 2009

			The triangle starts in row m=2 as
1;
1, 1;
1, 4, 1;
1, 1, 1, 1;
1, 16, 27, 4, 1;
1, 1, 1, 1, 1, 1;
1, 4, 1, 16, 1, 4, 1;

A120070 := proc(m,n) m^2-n^2 ; end: A120072 := proc(m,n) numer(1/n^2-1/m^2) ; end:
A164561 := proc(m,n) A120070(m,n)/A120072(m,n) ; end: seq( seq(A164561(m,n),n=1..m-1), m=2..30) ; # R. J. Mathar, Aug 19 2009

Edited and extended by R. J. Mathar, Aug 19 2009

A166492 Table of numerators of A120070(n,m)/A002260(n,m), 1 <= m < n.

Original entry on oeis.org

3, 8, 5, 15, 6, 7, 24, 21, 16, 9, 35, 16, 9, 5, 11, 48, 45, 40, 33, 24, 13, 63, 30, 55, 12, 39, 14, 15, 80, 77, 24, 65, 56, 15, 32, 17, 99, 48, 91, 21, 15, 32, 51, 9, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21, 143, 70, 45, 32, 119, 18, 95, 10, 7, 22, 23, 168, 165, 160, 153, 144

Offset: 2

Paul Curtz, Oct 15 2009

Numerators of the fractions (n+m)*(n-m)/m.
The numerical values are between A120070(n,m) and A120072(n,m), see A164561.
If we "flatten" the table (enumerate the sequence starting at 1 instead of using the double index), the positions where a common factor is removed from the numerator A120070 and denominator A002260 are at 5, 12, 13, etc., as given by A076537.

Cf. A129326.

A120070 := proc(n,m) (n+m)*(n-m) ; end:
A002260 := proc(n,m) m ; end:
A166492 := proc(n,m) A120070(n,m)/A002260(n+1,m) ; numer(%) ; end: seq(seq(A166492(n,m),m=1..n-1),n=2..17) ; # R. J. Mathar, Oct 21 2009

Edited and extended by R. J. Mathar, Oct 21 2009

cp's OEIS Frontend

A204994 Least k such that n divides A120070(k+1), the k-th difference between distinct squares.

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A142717 First (leftmost) odd term in the n-th row of triangle A120070.

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A143785 Antidiagonal sums of the triangle A120070.

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A141620 First differences of A120070.

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A133128 Triangle of first differences of A120070 with a leftmost column of A002522.

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A141616 Even terms in A120070.

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A100181 Odd terms in A120070.

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A141595 Binomial transform of A120070.

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A164561 Triangle with elements A120070(m,n)/A120072(m,n) read by rows, m>=2, 1<=n

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