A053698 - OEIS

%I #81 May 11 2024 16:27:18

%S 1,4,15,40,85,156,259,400,585,820,1111,1464,1885,2380,2955,3616,4369,

%T 5220,6175,7240,8421,9724,11155,12720,14425,16276,18279,20440,22765,

%U 25260,27931,30784,33825,37060,40495,44136,47989,52060,56355,60880

%N a(n) = n^3 + n^2 + n + 1.

%C a(n) = 1111 in base n.

%C n^3 + n^2 + n + 1 = (n^2 + 1)*(n + 1), therefore a(n) is never prime. - _Alonso del Arte_, Apr 22 2014

%H Vincenzo Librandi, <a href="/A053698/b053698.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F For n >= 2, a(n) = (n^4-1)/(n-1) = A024002(n)/A024000(n) = A002522(n)*(n+1) = A002061(n+1) + A000578(n).

%F G.f.: (1+5*x^2) / (1-x)^4. - _Colin Barker_, Jan 06 2012

%F a(n) = -A062158(-n). - _Bruno Berselli_, Jan 26 2016

%F a(n) = Sum_{i=0..n} 2*n*(n-i)+1. - _Bruno Berselli_, Jan 02 2017

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - _Colin Barker_, Jan 02 2017

%F a(n) = A104878(n+3,n) = A055129(4,n) for n > 0. - _Mathew Englander_, Jan 06 2021

%F E.g.f.: exp(x)*(x^3+4*x^2+3*x+1). - _Nikolaos Pantelidis_, Feb 06 2023

%e a(2) = 15 because 2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15.

%e a(3) = 40 because 3^3 + 3^2 + 3 + 1 = 27 + 9 + 3 + 1 = 40.

%e a(4) = 85 because 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.

%e From _Bruno Berselli_, Jan 02 2017: (Start)

%e The terms of the sequence are provided by the row sums of the following triangle (see the seventh formula in the previous section):

%e .   1;

%e .   3,   1;

%e .   9,   5,   1;

%e .  19,  13,   7,   1;

%e .  33,  25,  17,   9,   1;

%e .  51,  41,  31,  21,  11,   1;

%e .  73,  61,  49,  37,  25,  13,  1;

%e .  99,  85,  71,  57,  43,  29, 15,  1;

%e . 129, 113,  97,  81,  65,  49, 33, 17,  1;

%e . 163, 145, 127, 109,  91,  73, 55, 37, 19,  1;

%e . 201, 181, 161, 141, 121, 101, 81, 61, 41, 21, 1;

%e ...

%e Columns from the first to the fifth, respectively: A058331, A001844, A056220 (after -1), A059993, A161532. Also, eighth column is A161549.

%e (End)

%p A053698:=n->n^3 + n^2 + n + 1; seq(A053698(n), n=0..50); # _Wesley Ivan Hurt_, Apr 22 2014

%t Table[n^3 + n^2 + n + 1, {n, 0, 39}] (* _Alonso del Arte_, Apr 22 2014 *)

%t FromDigits["1111", Range[0, 50]] (* _Paolo Xausa_, May 11 2024 *)

%o (Magma) [n^3+n^2+n+1: n in [0..50]]; // _Vincenzo Librandi_, May 01, 2011

%o (PARI) Vec((1 + 5*x^2) / (1 - x)^4 + O(x^50)) \\ _Colin Barker_, Jan 02 2017

%o (Python) def a(n): return (n**3+n**2+n+1) # _Torlach Rush_, May 08 2024

%Y Cf. A237627 (subset of semiprimes).

%Y Cf. A056106 (first differences).

%Y Cf. A062158, A027444, A104878, A055129.

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, Mar 23 2000