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 A279101 a(n) = Sum_{k=0..n} ceiling((1 + sqrt(2))^k). 1

%I

%S 1,4,10,25,59,142,340,819,1973,4760,11486,27725,66927,161570,390056,

%T 941671,2273385,5488428,13250226,31988865,77227939,186444726,

%U 450117372,1086679451,2623476253,6333631936,15290740102,36915112117,89120964311,215157040714,519435045712,1254027132111,3027489309905,7309005751892

%N a(n) = Sum_{k=0..n} ceiling((1 + sqrt(2))^k).

%H G. C. Greubel, <a href="/A279101/b279101.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SilverRatio.html">Silver Ratio</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4,1,1).

%F G.f.: (1 + x - 2*x^2 - x^3 - x^4)/((1 - x)^2*(1 - x - 3*x^2 - x^3)).

%F a(n) = 3*a(n-1) - 4*a(n-3) + a(n-4) + a(n-5).

%F a(n) = (4*(1 + sqrt(2))^n + 2*sqrt(2)*(1 + sqrt(2))^n - 2*(-2 + sqrt(2))*(1 - sqrt(2))^n + 2*n - (-1)^n - 3)/4.

%F a(n) ~ s^(n+1)/(s-1), where s is the silver ratio (A014176).

%p Digits:=100: a:=n->add(ceil((1+sqrt(2))^k),k=0..n); seq(a(n),n=0..35); # _Muniru A Asiru_, Oct 11 2018

%t Accumulate[Table[Ceiling[(1 + Sqrt[2])^n], {n, 0, 33}]]

%t LinearRecurrence[{3, 0, -4, 1, 1}, {1, 4, 10, 25, 59}, 34]

%t CoefficientList[Series[(1 + x - 2*x^2 - x^3 - x^4)/((1 - x)^2*(1 - x - 3*x^2 - x^3)), {x, 0, 50}], x] (* or *)

%t a[n_]:=(4*(1 + Sqrt[2])^n + 2*Sqrt[2]*(1 + Sqrt[2] )^n - 2*(-2 + Sqrt[2] )*(1 - Sqrt[2] )^n + 2*n - (-1)^n - 3)/4; Simplify[Array[a, 50, 0]] (* _Stefano Spezia_, Oct 11 2018 *)

%o (PARI) x='x+O('x^40); Vec((1+x-2*x^2-x^3-x^4)/((1-x)^2*(1-x-3*x^2-x^3))) \\ _G. C. Greubel_, Oct 10 2018

%o (MAGMA) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-2*x^2-x^3-x^4)/((1-x)^2*(1-x-3*x^2-x^3)))); // _G. C. Greubel_, Oct 10 2018

%Y Cf. A014176, A020962.

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Dec 06 2016

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Last modified September 30 11:55 EDT 2020. Contains 337439 sequences. (Running on oeis4.)