Ian Stewart - cp's OEIS frontend

A265645 a(n) = n^2 * floor(n/2).

0, 0, 4, 9, 32, 50, 108, 147, 256, 324, 500, 605, 864, 1014, 1372, 1575, 2048, 2312, 2916, 3249, 4000, 4410, 5324, 5819, 6912, 7500, 8788, 9477, 10976, 11774, 13500, 14415, 16384, 17424, 19652, 20825, 23328, 24642, 27436, 28899, 32000, 33620, 37044, 38829, 42592, 44550, 48668, 50807, 55296, 57624, 62500

Offset: 0

Ian Stewart, Apr 06 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A000290, A004526.

List([0..55], n -> n^2*Int(n/2)); # Muniru A Asiru, Apr 05 2018

seq' x = x^2 * (x `div` 2)
map seq' [0..50]

[n^2*Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Apr 04 2018

Table[n^2 Floor[n/2], {n, 0, 50}] (* Vincenzo Librandi, Apr 04 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,0,4,9,32,50,108},60] (* Harvey P. Dale, May 19 2019 *)

a(n) = n^2*(n\2); \\ Altug Alkan, Apr 14 2016

concat(vector(2), Vec(x^2*(4+5*x+11*x^2+3*x^3+x^4)/((1-x)^4*(1+x)^3) + O(x^50))) \\ Colin Barker, Apr 14 2016

a(n) = A000290(n)*A004526(n). - Michel Marcus, Apr 14 2016
G.f.: x^2*(4 + 5*x + 11*x^2 + 3*x^3 + x^4)/((1 - x)^4*(1 + x)^3). - Ilya Gutkovskiy, Apr 14 2016; corrected by Colin Barker, Apr 14 2016
From Colin Barker, Apr 14 2016: (Start)
a(n) = n^2*(2*n + (-1)^n - 1)/4.
a(n) = n^3/2 for n even.
a(n) = n^2*(n-1)/2 for n odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>6. (End)
Sum_{n>=2} 1/a(n) = zeta(3)/4 - Pi^2/4 - 2*log(2) + 4. - Amiram Eldar, Mar 15 2024

A213077 a(n) = round(n^2 - sqrt(n)).

Original entry on oeis.org

0, 0, 3, 7, 14, 23, 34, 46, 61, 78, 97, 118, 141, 165, 192, 221, 252, 285, 320, 357, 396, 436, 479, 524, 571, 620, 671, 724, 779, 836, 895, 955, 1018, 1083, 1150, 1219, 1290, 1363, 1438, 1515, 1594, 1675, 1758, 1842, 1929, 2018, 2109, 2202, 2297, 2394

Offset: 0

Ian Stewart, Jun 04 2012

nonn

			0^2 - sqrt(0) = 0;
1^2 - sqrt(1) = 0;
2^2 - sqrt(2) = 3,
3^2 - sqrt(3) = 7;
4^2 - sqrt(4) = 14.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A056847.

seq(round(n^2-sqrt(n)), n=0..100); # Robert Israel, Jul 29 2022

Table[Round[n^2 - Sqrt[n]], {n, 0, 100}] (* T. D. Noe, Jun 06 2012 *)

count = 0
while count < 50:
    ns = count * count
    ns = ns - math.sqrt(count)
    ns = round(ns)
    print(ns, end=',')
    count += 1

from math import isqrt
def A213077(n): return n**2-(m:=isqrt(n))-int((n-m*(m+1)<<2)>=1) # Chai Wah Wu, Jul 29 2022

A180724 a(n) = n^2 + largest prime < n^2.

Original entry on oeis.org

7, 16, 29, 48, 67, 96, 125, 160, 197, 234, 283, 336, 389, 448, 507, 572, 641, 720, 797, 880, 963, 1052, 1147, 1244, 1349, 1456, 1557, 1680, 1787, 1914, 2045, 2176, 2309, 2448, 2587, 2736, 2883, 3032, 3197, 3350, 3523, 3696, 3869, 4042, 4229, 4416, 4601

Offset: 2

Ian Stewart, Sep 18 2010

Harvey P. Dale, Table of n, a(n) for n = 2..1000

#+NextPrime[#,-1]&/@(Range[2,50]^2) (* Harvey P. Dale, Apr 08 2023 *)

a(n) = n^2 + precprime(n^2-1); \\ Michel Marcus, Aug 23 2013

4 + 3 = 7; 9 + 7 = 16; 16 + 13 = 29; 25 + 23 = 48;
a(n) = n^2+A053001(n). - R. J. Mathar, Sep 19 2010
2n^2 - O(n^1.05) < a(n) < 2n^2. (Probably a much tighter lower bound is true.) - Charles R Greathouse IV, Jan 31 2023

A180046 a(n+1) = a(n-3) + a(n-2) - a(n-1) + a(n) starting with 1, 2, 3, 4.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 8, 11, 12, 14, 21, 30, 35, 40, 56, 81, 100, 115, 152, 218, 281, 330, 419, 588, 780, 941, 1168, 1595, 2148, 2662, 3277, 4358, 5891, 7472, 9216, 11993, 16140, 20835, 25904, 33202, 44273, 57810, 72643, 92308, 121748, 159893, 203096, 257259

Offset: 1

Ian Stewart, Aug 08 2010

			1 2 3 4 (by definition); 1 + 2 - 3 + 4 = 4, 2 + 3 - 4 + 4 = 5, 3 + 4 - 4 + 5 = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,1).

Cf. A100329.

import Data.List (zipWith4)
a180046 n = a180046_list !! (n-1)
a180046_list = [1..4] ++ zipWith4 (((((+) .) . (+)) .) . (-))
                          (drop 3 a180046_list) (drop 2 a180046_list)
                            (tail a180046_list) a180046_list
-- Reinhard Zumkeller, Oct 08 2014

LinearRecurrence[{1,-1,1,1},{1,2,3,4},50] (* Harvey P. Dale, Mar 18 2015 *)

Vec(x*(1+x)*(2*x^2+1)/(1-x+x^2-x^3-x^4)+O(x^99)) \\ Charles R Greathouse IV, Jul 06 2011

G.f.: -x*(1+x)*(2*x^2+1)/(-1+x-x^2+x^3+x^4). - R. J. Mathar, Aug 14 2010

a(n) = (-1)^(n)*(A100329(n-1)-A100329(n)-2*A100329(n-2)+2*A100329(n-3)) with A100329(-1) = A100329(-2) = 0. - Johannes W. Meijer, Jul 06 2011

More terms from R. J. Mathar, Aug 14 2010

Name edited by Michel Marcus, Feb 20 2025

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User: Ian Stewart

A265645 a(n) = n^2 * floor(n/2).

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A213077 a(n) = round(n^2 - sqrt(n)).

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A180724 a(n) = n^2 + largest prime < n^2.

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A180046 a(n+1) = a(n-3) + a(n-2) - a(n-1) + a(n) starting with 1, 2, 3, 4.

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