A272132 -id:A272132 - cp's OEIS frontend

A272129 a(n) = 32n^2 - 56n + 25.

25, 1, 41, 145, 313, 545, 841, 1201, 1625, 2113, 2665, 3281, 3961, 4705, 5513, 6385, 7321, 8321, 9385, 10513, 11705, 12961, 14281, 15665, 17113, 18625, 20201, 21841, 23545, 25313, 27145, 29041, 31001, 33025, 35113, 37265, 39481, 41761, 44105, 46513, 48985

Offset: 0

Vincenzo Librandi, Apr 26 2016

Subsequence of A001844.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001844, A272126, A272127, A272128, A272131, A272132, A272133, A245244.

Magma
```
[32*n^2 - 56*n + 25: n in [0..50]];
```

[32*n^2-56*n+25$n=0..40]; # Muniru A Asiru, Jan 28 2019

Table[32 n^2 - 56 n + 25, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{25,1,41},50] (* Harvey P. Dale, Jul 03 2018 *)

lista(nn) = for(n=0, nn, print1(32*n^2-56*n+25, ", ")); \\ Altug Alkan, Apr 26 2016

O.g.f.: (25 - 74*x + 113*x^2)/(1-x)^3.
E.g.f.: (25 - 24*x + 32*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
n*a(n) = 1 + 3^5*(n-1)/(n+1) + 5^5*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019

A272131 a(n) = 384n^3 - 1184n^2 + 1228*n - 427.

Original entry on oeis.org

-427, 1, 365, 2969, 10117, 24113, 47261, 81865, 130229, 194657, 277453, 380921, 507365, 659089, 838397, 1047593, 1288981, 1564865, 1877549, 2229337, 2622533, 3059441, 3542365, 4073609, 4655477, 5290273, 5980301, 6727865, 7535269, 8404817, 9338813, 10339561

Offset: 0

Vincenzo Librandi, Apr 26 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A272129, A272132, A245244.

[384*n^3 - 1184*n^2 + 1228*n - 427: n in [0..50]];

[384*n^3-1184*n^2+1228*n-427$n=0..35]; # Muniru A Asiru, Jan 28 2019

Table[384 n^3 - 1184 n^2 + 1228 n - 427, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{-427,1,365,2969},40] (* Harvey P. Dale, Aug 24 2024 *)

lista(nn) = for(n=0, nn, print1(384*n^3-1184*n^2+1228*n-427, ", ")); \\ Altug Alkan, Apr 26 2016

O.g.f.: (-427 + 1709*x - 2201*x^2 + 3223*x^3)/(1-x)^4.
E.g.f.: (-427 + 428*x - 32*x^2 + 384*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
See page 7 in Brent's paper:
a(n) = (2*n-1)^2*A272129(n) - 4*(n-1)^2*A272129(n-1).
A272132(n) =  (2*n-1)^2*a(n) - 4*(n-1)^2*a(n-1).
n*a(n) = 1 + 3^7*(n-1)/(n+1) + 5^7*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019

A272133 a(n) = 122880n^5 - 829440n^4 + 2258688n^3 - 3076288n^2 + 2079892*n - 555731.

Original entry on oeis.org

-555731, 1, 29525, 1657129, 16591741, 80872529, 269614501, 711754105, 1604794829, 3229552801, 5964902389, 10302521801, 16861638685, 26403775729, 39847496261, 58283149849, 82987617901, 115439059265, 157331655829, 210590358121, 277385630909, 360148198801

Offset: 0

Vincenzo Librandi, Apr 26 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16)
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A272132, A245244.

[122880*n^5 - 829440*n^4 + 2258688*n^3 -3076288*n^2 + 2079892*n - 555731: n in [0..30]];

[122880*n^5-829440*n^4+2258688*n^3-3076288*n^2+2079892*n-555731$n=0..30]; # Muniru A Asiru, Jan 28 2019

Table[122880 n^5 - 829440 n^4 + 2258688 n^3 - 3076288 n^2 + 2079892 n - 555731, {n, 0, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{-555731,1,29525,1657129,16591741,80872529},30] (* Harvey P. Dale, Feb 10 2021 *)

lista(nn) = for(n=0, nn, print1(122880*n^5 - 829440*n^4 + 2258688*n^3 - 3076288*n^2 + 2079892*n - 555731, ", ")); \\ Altug Alkan, Apr 26 2016

O.g.f.: (-555731 + 3334387*x - 8306446*x^2 + 12594614*x^3 - 1244143*x^4 + 8922919*x^5)/(1-x)^6.
E.g.f.: (-555731 + 555732*x - 263104*x^2 + 354048*x^3 + 399360*x^4 + 122880*x^5)*exp(x).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (2*n-1)^2*A272132(n) - 4*(n-1)^2*A272132(n-1), see page 7 in Brent's paper.
n*a(n) = 1 + 3^11*(n-1)/(n+1) + 5^11*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019

cp's OEIS Frontend

A272129 a(n) = 32n^2 - 56n + 25.

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A272131 a(n) = 384n^3 - 1184n^2 + 1228*n - 427.

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Magma

Maple

Mathematica

PARI

Formula

A272133 a(n) = 122880n^5 - 829440n^4 + 2258688n^3 - 3076288n^2 + 2079892*n - 555731.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

cp's OEIS Frontend

A272129 a(n) = 32*n^2 - 56*n + 25.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272131 a(n) = 384*n^3 - 1184*n^2 + 1228*n - 427.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272133 a(n) = 122880*n^5 - 829440*n^4 + 2258688*n^3 - 3076288*n^2 + 2079892*n - 555731.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272129 a(n) = 32n^2 - 56n + 25.

A272131 a(n) = 384n^3 - 1184n^2 + 1228*n - 427.

A272133 a(n) = 122880n^5 - 829440n^4 + 2258688n^3 - 3076288n^2 + 2079892*n - 555731.