A272131 -id:A272131

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page 1

A272129

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Subsequence of A001844.

LINKS

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).

FORMULA

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

MAPLE

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

MATHEMATICA

Table[32 n^2 - 56 n + 25, {n, 0, 40}]

LinearRecurrence[{3, -3, 1}, {25, 1, 41}, 50] (* Harvey P. Dale, Jul 03 2018 *)

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Apr 26 2016

STATUS

approved

A272132

a(n) = 6144*n^4 - 29184*n^3 + 52416*n^2 - 41840*n + 12465.

+10
4

12465, 1, 3281, 68385, 388849, 1305665, 3307281, 7029601, 13255985, 22917249, 37091665, 57004961, 84030321, 119688385, 165647249, 223722465, 295877041, 384221441, 491013585, 618658849, 769710065, 946867521, 1152978961, 1391039585, 1664192049, 1975726465

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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 (5,-10,10,-5,1).

FORMULA

O.g.f.: (12465 - 62324*x + 127926*x^2 - 72660*x^3 + 142049*x^4)/(1-x)^5.

E.g.f.: (12465 - 12464*x + 7872*x^2 + 7680*x^3 + 6144*x^4)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

See page 7 in Brent's paper:

a(n) = (2*n-1)^2*A272131(n) - 4*(n-1)^2*A272131(n-1).

A272133(n) = (2*n-1)^2*a(n) - 4*(n-1)^2*a(n-1).

n*a(n) = 1 + 3^9*(n-1)/(n+1) + 5^9*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019

MAPLE

[6144*n^4-29184*n^3+52416*n^2-41840*n+12465$n=0..30]; # Muniru A Asiru, Jan 28 2019

MATHEMATICA

Table[6144 n^4 - 29184 n^3 + 52416 n^2 - 41840 n + 12465, {n, 0, 40}]

LinearRecurrence[{5, -10, 10, -5, 1}, {12465, 1, 3281, 68385, 388849}, 30] (* Harvey P. Dale, Aug 06 2022 *)

PROG

(Magma) [6144*n^4 - 29184*n^3 + 52416*n^2 - 41840*n + 12465: n in [0..40]];

(PARI) lista(nn) = for(n=0, nn, print1(6144*n^4-29184*n^3+52416*n^2-41840*n+12465, ", ")); \\ Altug Alkan, Apr 26 2016