A196289 -id:A196289 - cp's OEIS frontend

A147854 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where the pairs of integers (x,y) and (z,t) are not proportional.

520, 975, 2040, 2080, 3567, 3900, 4680, 7215, 7800, 8160, 8320, 8775, 9840, 13000, 13920, 14268, 15600, 18360, 18720, 19680, 24375, 25480, 28860, 30160, 31200, 32103, 32640, 33280, 35100, 39360, 40545, 42120, 47775, 51000, 52000, 53040

Offset: 1

Max Alekseyev, Nov 17 2008, Nov 19 2008

nonn

Positive integers n such that n^2 = s^4*A147858(m)*A147858(k) for positive integers s and kA147856.
Euler proved that if n^2 = (x^4 - y^4)*(z^4 - t^4) then a,b,c (if n is even) or 4a,4b,4c (if n is odd) form a triple of integers with all pairwise sums and differences being squares, where a=(x^4+y^4)*(z^4+t^4)/2, b=(n^2+(2xyzt)^2)/2 and c=(n^2-(2xyzt)^2)/2. Note that a,b,c are pairwise distinct if and only if (x,y) and (z,t) are not proportional.
4*A196289(n) = 4*(n^9 - n) belong to this sequence since (4*(n^9 - n))^2 = ((n^4+2*n^2-1)^4 - (n^4-2*n^2-1)^4) * (n^4 - 1).

Cf. A147856, A147857, A147858.

A147856 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where x>y and z>t are distinct pairs of integers with gcd(x,y)=gcd(z,t)=1.

Original entry on oeis.org

520, 975, 2040, 3567, 7215, 7800, 9840, 13920, 19680, 30160, 40545, 53040, 57720, 62985, 95120, 108225, 138040, 151320, 180960, 230880, 247520, 286200, 289952, 352495, 473280, 535353, 546975, 720945, 769600, 1048560, 1141920, 1210560

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

Positive integers n such that n^2 = A147858(m)*A147858(k) for positive integers kA147854: any element n of A147854 is of the form a(k)*s^2 for some positive integer s.

4*A196289(2*k) and A196289(2*k+1) belong to this sequence.

A196290 a(n) = n^9 + n.

Original entry on oeis.org

0, 2, 514, 19686, 262148, 1953130, 10077702, 40353614, 134217736, 387420498, 1000000010, 2357947702, 5159780364, 10604499386, 20661046798, 38443359390, 68719476752, 118587876514, 198359290386, 322687697798, 512000000020, 794280046602, 1207269217814, 1801152661486, 2641807540248

Offset: 0

Vincenzo Librandi, Oct 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A195859, A196288, A196289.

Magma
```
[n^9+n: n in [0..30]]
```

Table[n^9+n,{n,0,40}] (* Harvey P. Dale, Oct 14 2011 *)

a(n)=n^9+n \\ Charles R Greathouse IV, Nov 21 2011

G.f.: 2*x*(1+247*x+7318*x^2+44089*x^3+78130*x^4+44089*x^5+7318*x^6+247*x^7+x^8) / (x-1)^10 . a(n) = 2*A168116(n). - R. J. Mathar, Oct 13 2011

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Wesley Ivan Hurt, May 04 2023

A196291 a(n) = n^10 - n.

Original entry on oeis.org

0, 0, 1022, 59046, 1048572, 9765620, 60466170, 282475242, 1073741816, 3486784392, 9999999990, 25937424590, 61917364212, 137858491836, 289254654962, 576650390610, 1099511627760, 2015993900432, 3570467226606, 6131066257782, 10239999999980, 16679880978180, 26559922791402, 41426511213626

Offset: 0

Vincenzo Librandi, Oct 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A195859, A196288, A196289, A196290

Magma
```
[n^10-n: n in [0..30]];
```

Table[n^10 - n, {n, 0, 40}] (* and *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 1022, 59046, 1048572, 9765620, 60466170, 282475242, 1073741816, 3486784392, 9999999990}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=n^10-n \\ Charles R Greathouse IV, Nov 22 2011

A196292 a(n) = n^10 + n.

Original entry on oeis.org

0, 2, 1026, 59052, 1048580, 9765630, 60466182, 282475256, 1073741832, 3486784410, 10000000010, 25937424612, 61917364236, 137858491862, 289254654990, 576650390640, 1099511627792, 2015993900466, 3570467226642, 6131066257820, 10240000000020, 16679880978222, 26559922791446

Offset: 0

Vincenzo Librandi, Oct 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A168118, A195859, A196288, A196289, A196290, A196291.

Magma
```
[n^10+n: n in [0..30]];
```

a(n) = 2 * A168118(n). - Alois P. Heinz, Jul 28 2025

A288604 a(n) = (n^9 - n)/10.

Original entry on oeis.org

0, 51, 1968, 26214, 195312, 1007769, 4035360, 13421772, 38742048, 99999999, 235794768, 515978034, 1060449936, 2066104677, 3844335936, 6871947672, 11858787648, 19835929035, 32268769776, 51199999998, 79428004656, 120726921777, 180115266144, 264180754020

Offset: 1

Seiichi Manyama, Jun 11 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A164938, A196289.

Table[(n^9-n)/10,{n,30}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,51,1968,26214,195312,1007769,4035360,13421772,38742048,99999999},30] (* Harvey P. Dale, Jun 11 2019 *)

concat(0, Vec(3*x^2*(17 + 486*x + 2943*x^2 + 5204*x^3 + 2943*x^4 + 486*x^5 + 17*x^6) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Jun 11 2017

a(n)=(n^9-n)/10 \\ Charles R Greathouse IV, Jun 11 2017

a(n) = (n^9 - n)/10 = A196289(n)/10.

G.f.: 3*x^2*(17 + 486*x + 2943*x^2 + 5204*x^3 + 2943*x^4 + 486*x^5 + 17*x^6) / (1 - x)^10. - Colin Barker, Jun 11 2017

cp's OEIS Frontend

A147854 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where the pairs of integers (x,y) and (z,t) are not proportional.

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A147856 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where x>y and z>t are distinct pairs of integers with gcd(x,y)=gcd(z,t)=1.

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A196290 a(n) = n^9 + n.

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A196291 a(n) = n^10 - n.

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Magma

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A196292 a(n) = n^10 + n.

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Magma

Formula

A288604 a(n) = (n^9 - n)/10.

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Mathematica

PARI

PARI

Formula