A245288 -id:A245288 - cp's OEIS frontend

A260810 a(n) = n^2(3n^2 - 1)/2.

0, 1, 22, 117, 376, 925, 1926, 3577, 6112, 9801, 14950, 21901, 31032, 42757, 57526, 75825, 98176, 125137, 157302, 195301, 239800, 291501, 351142, 419497, 497376, 585625, 685126, 796797, 921592, 1060501, 1214550, 1384801, 1572352, 1778337, 2003926, 2250325, 2518776

Offset: 0

Bruno Berselli, Jul 31 2015

Pentagonal numbers with square indices.

After 0, a(k) is a square if k is in A072256.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Subsequence of A001318 and A245288 (see Formula field).
Cf. A000326, A193218 (first differences).
Cf. A000583 (squares with square indices), A002593 (hexagonal numbers with square indices).
Cf. A232713 (pentagonal numbers with pentagonal indices), A236770 (pentagonal numbers with triangular indices).

Magma
```
[n^2*(3*n^2-1)/2: n in [0..40]];
```

I:=[0,1,22,117,376]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Aug 23 2015

A260810:=n->n^2*(3*n^2 - 1)/2: seq(A260810(n), n=0..50); # Wesley Ivan Hurt, Apr 25 2017

Table[n^2 (3 n^2 - 1)/2, {n, 0, 40}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 22, 117, 376}, 40] (* Vincenzo Librandi, Aug 23 2015 *)

PARI
```
vector(40, n, n--; n^2*(3*n^2-1)/2)
```
Sage
```
[n^2*(3*n^2-1)/2 for n in (0..40)]
```

G.f.: x*(1 + x)*(1 + 16*x + x^2)/(1 - x)^5.
a(n) = a(-n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A245288(2*n^2).
a(n) = A001318(2*n^2-1) with A001318(-1) = 0.
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi^2/3 - sqrt(3)*Pi*cot(Pi/sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi*cosec(Pi/sqrt(3)) - Pi^2/6 - 3. (End)

A245467 a(n) = ( 4n^2 - 2n + 1 - (2n^2 - 6n + 1) * (-1)^n )/16.

Original entry on oeis.org

0, 0, 1, 2, 3, 7, 6, 15, 10, 26, 15, 40, 21, 57, 28, 77, 36, 100, 45, 126, 55, 155, 66, 187, 78, 222, 91, 260, 105, 301, 120, 345, 136, 392, 153, 442, 171, 495, 190, 551, 210, 610, 231, 672, 253, 737, 276, 805, 300, 876, 325, 950, 351, 1027, 378, 1107, 406

Offset: 0

Wesley Ivan Hurt, Jul 23 2014

For even n, the sequence gives the sum of the smallest parts of the partitions of n into two parts. For odd n, the sequence gives the sum of the largest parts of the partitions of n into two parts (see example).

Union of triangular numbers (A000217) and second pentagonal numbers (A005449). - Wesley Ivan Hurt, Oct 31 2015

			a(6) = 6; the partitions of 6 into two parts are: (5,1), (4,2), (3,3). Since 6 is even, we add the smallest parts in these partitions to get 6.
a(7) = 15; the partitions of 7 into two parts are: (6,1), (5,2), (4,3). Since 7 is odd, we add the largest parts in the partitions to get 15.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000217 (triangular numbers), A005449 (second pentagonal numbers).

Cf. A245288.

[( 4*n^2-2*n+1-(2*n^2-6*n+1)*(-1)^n )/16 : n in [0..50]];

A245467:=n->( 4*n^2-2*n+1-(2*n^2-6*n+1)*(-1)^n )/16: seq(A245467(n), n=0..50);

Table[(4n^2 - 2n + 1 - (2n^2 - 6n + 1) (-1)^n)/16, {n, 0, 50}]
CoefficientList[Series[- x^2 (x^3 + 2 x + 1)/((x - 1)^3 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,0,1,2,3,7},60] (* Harvey P. Dale, May 11 2019 *)

concat([0,0], Vec(-x^2*(x^3+2*x+1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ Colin Barker, Jul 23 2014

vector(100, n, n--; if(n%2==0, t=n/2; t*(t+1)/2, t*(3*t + 1)/2)) \\ Altug Alkan, Nov 01 2015

a(n) = floor(n/2) * (3*floor(n/2)+1) * (n mod 2)/2 + floor(n/2) * (floor(n/2)+1) * ((n+1) mod 2)/2.
From Colin Barker, Jul 23 2014: (Start)
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
G.f.: -x^2*(x^3+2*x+1) / ((x-1)^3*(x+1)^3). (End)
a(2n) = A000217(n), a(2n+1) = A005449(n). - Wesley Ivan Hurt, Oct 31 2015
Sum_{n>=2} 1/a(n) = 8 - Pi/sqrt(3) - 3*log(3). - Amiram Eldar, Aug 25 2022

cp's OEIS Frontend

A260810 a(n) = n^2(3n^2 - 1)/2.

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Formula

A245467 a(n) = ( 4n^2 - 2n + 1 - (2n^2 - 6n + 1) * (-1)^n )/16.

Views

Author

Keywords

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Magma

Maple

Mathematica

PARI

PARI

Formula

cp's OEIS Frontend

A260810 a(n) = n^2*(3*n^2 - 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

A245467 a(n) = ( 4*n^2 - 2*n + 1 - (2*n^2 - 6*n + 1) * (-1)^n )/16.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A260810 a(n) = n^2(3n^2 - 1)/2.

A245467 a(n) = ( 4n^2 - 2n + 1 - (2n^2 - 6n + 1) * (-1)^n )/16.