A027974 a(n) = Sum_{k=1..n+1} A027960(n+1, n+1+k).

1, 5, 14, 35, 81, 180, 389, 825, 1726, 3575, 7349, 15020, 30561, 61965, 125294, 252795, 509161, 1024100, 2057549, 4130225, 8284926, 16609455, 33282989, 66669660, 133507081, 267285605, 535010414, 1070731475, 2142612801, 4287086100, 8577182549, 17159235945

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

The former name, a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A027960, was in error with the data given. [This double summation gives A023537(n+1), or A027960(n+2, n+4) for n >= 0]. - G. C. Greubel, Jun 08 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Philipp Emanuel Weidmann, The Sequencer OEIS Survey
• Susanne Wienand, Suggestion for a proof of Philipp Emanuel Weidman's conjecture concerning A027983
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Crossrefs

Cf. A000032, A000045, A027960, A101220.

Programs

GAP

List([0..30], n-> 2^(n+3) - Lucas(1,-1,n+4)[2]); # G. C. Greubel, Sep 26 2019

Magma

[2^(n+3) - Lucas(n+4): n in [0..30]]; // G. C. Greubel, Sep 26 2019

Maple

with(combinat); f:=fibonacci; seq(2^(n+3) - f(n+5) - f(n+3), n=0..30); # G. C. Greubel, Sep 26 2019

Mathematica

Table[2^(n+3) - LucasL[n+4], {n,0,30}] (* G. C. Greubel, Sep 26 2019 *)

PARI

vector(31, n, f=fibonacci; 2^(n+2) - f(n+4) - f(n+2)) \\ G. C. Greubel, Sep 26 2019

SageMath

def A027974(n): return 2**(n+3) - lucas_number2(n+4,1,-1)
[A027974(n) for n in range(31)] # G. C. Greubel, Sep 26 2019; Jun 08 2025

Formula

a(n) = 2^(n+3) - Fibonacci(n+5) - Fibonacci(n+3).

a(n) = A101220(4, 2, n+1).

G.f.: (1+2*x)/((1-2*x)*(1-x-x^2)). - R. J. Mathar, Sep 22 2008

a(n) = 2*a(n-1) + A000032(n+1). - David A. Corneth, Apr 16 2015

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). - Colin Barker, Feb 17 2016

From G. C. Greubel, Jun 08 2025: (Start)

a(n) = 2^(n+3) - A000032(n+4).

E.g.f.: 8*exp(2*x) - exp(x/2)*( 7*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2) ). (End)