A123203 - OEIS

%I #46 Sep 15 2024 02:46:07

%S 1,2,7,20,49,110,235,488,997,2018,4063,8156,16345,32726,65491,131024,

%T 262093,524234,1048519,2097092,4194241,8388542,16777147,33554360,

%U 67108789,134217650,268435375,536870828,1073741737,2147483558

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

%C An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - _Johannes W. Meijer_, Aug 15 2010

%H Harvey P. Dale, <a href="/A123203/b123203.txt">Table of n, a(n) for n = 1..1000</a>

%H Tamas Lengyel, <a href="https://doi.org/10.1016/j.jnt.2014.09.015">On p-adic properties of the Stirling numbers of the first kind</a>, Journal of Number Theory, 148 (2015) 73-94.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).

%F Binomial transform of [1, 1, 4, 4, 4, ...].

%F Equals row sums of triangle A131061.

%F From _Johannes W. Meijer_, Aug 15 2010; corrected by _Colin Barker_, Jul 28 2012: (Start)

%F a(n) = 2^(1+n) - 3*n.

%F a(n) = 3*A000295(n-1) + A000079(n-1).

%F (End)

%F G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - _Colin Barker_, Jul 28 2012

%F a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - _Colin Barker_, Jul 29 2012

%F E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - _G. C. Greubel_, Sep 14 2024

%e a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).

%e a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).

%t Table[2^(n+1) - 3*n, {n,40}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 15 2008 *)

%t LinearRecurrence[{4,-5,2},{1,2,7},40] (* _Harvey P. Dale_, Mar 30 2024 *)

%o (Magma) [2^(n+1) -3*n: n in [1..40]]; // _G. C. Greubel_, Sep 14 2024

%o (SageMath)

%o def A123203(n): return 2^(n+1) -3*n

%o [A123203(n) for n in range(1,41)] # _G. C. Greubel_, Sep 14 2024

%Y Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Jun 13 2007

%E More terms from _Vladimir Joseph Stephan Orlovsky_, Nov 15 2008

%E Title changed by _G. C. Greubel_, Sep 14 2024