A020962 -id:A020962 - cp's OEIS frontend

A277789 a(n) = Sum_{k=0..n} (-1)^k*floor((1 + sqrt(2))^k).

1, -1, 4, -10, 23, -59, 138, -340, 813, -1973, 4752, -11486, 27715, -66927, 161558, -390056, 941657, -2273385, 5488412, -13250226, 31988847, -77227939, 186444706, -450117372, 1086679429, -2623476253, 6333631912, -15290740102, 36915112091, -89120964311, 215157040686, -519435045712, 1254027132081

Offset: 0

Ilya Gutkovskiy, Oct 31 2016

Alternating sum of A080039.

Robert Israel, Table of n, a(n) for n = 0..2610
Eric Weisstein's World of Mathematics, Silver Ratio
Index entries for linear recurrences with constant coefficients, signature (-1,4,0,-3,1).

Cf. A000129, A014176, A020962, A080039.

I:=[1,-1,4,-10,23]; [n le 5 select I[n] else -Self(n-1)+4*Self(n-2)-3*Self(n-4)+Self(n-5): n in [1..35]]; // Vincenzo Librandi, Nov 01 2016

f:= gfun:-rectoproc({a(n) = -a(n-1) + 4*a(n-2) - 3*a(n-4) + a(n-5),seq(a(i)=[ 1, -1, 4, -10, 23][i+1],i=0..4)},a(n),remember):
map(f, [$0..40]); # Robert Israel, Oct 31 2016

Accumulate[Table[(-1)^n Floor[(1 + Sqrt[2])^n], {n, 0, 32}]]
LinearRecurrence[{-1, 4, 0, -3, 1}, {1, -1, 4, -10, 23}, 33]

x='x+O('x^30); Vec((1-x^2-2*x^3)/((1-x)^2*(1+x)*(1+2*x-x^2))) \\ G. C. Greubel, Sep 30 2018

O.g.f.: (1 - x^2 - 2*x^3)/((1 - x)^2*(1 + x)*(1 + 2*x - x^2)).
E.g.f.: ((-4*sqrt(2)*sinh(sqrt(2)*x) - 1)*exp(-x) + (5 - 2*x)*exp(x))/4.
a(n) = -a(n-1) + 4*a(n-2) - 3*a(n-4) + a(n-5).
a(n) = (2*sqrt(2)*(-1 - sqrt(2))^n - 2*sqrt(2)*(sqrt(2) - 1)^n - (-1)^n - 2*n + 5)/4.
a(n) ~ (-1)^n*s^(n+1)/(s + 1), where s is the silver ratio (A014176).

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

Original entry on oeis.org

1, 4, 10, 25, 59, 142, 340, 819, 1973, 4760, 11486, 27725, 66927, 161570, 390056, 941671, 2273385, 5488428, 13250226, 31988865, 77227939, 186444726, 450117372, 1086679451, 2623476253, 6333631936, 15290740102, 36915112117, 89120964311, 215157040714, 519435045712, 1254027132111, 3027489309905, 7309005751892

Offset: 0

Ilya Gutkovskiy, Dec 06 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Silver Ratio
Index entries for linear recurrences with constant coefficients, signature (3,0,-4,1,1).

Cf. A014176, A020962.

m:=40; R:=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

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

Accumulate[Table[Ceiling[(1 + Sqrt[2])^n], {n, 0, 33}]]
LinearRecurrence[{3, 0, -4, 1, 1}, {1, 4, 10, 25, 59}, 34]
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 *)
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 *)

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

G.f.: (1 + x - 2*x^2 - x^3 - x^4)/((1 - x)^2*(1 - x - 3*x^2 - x^3)).
a(n) = 3*a(n-1) - 4*a(n-3) + a(n-4) + a(n-5).
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.
a(n) ~ s^(n+1)/(s-1), where s is the silver ratio (A014176).

cp's OEIS Frontend

A277789 a(n) = Sum_{k=0..n} (-1)^k*floor((1 + sqrt(2))^k).

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