A022403 a(0)=a(1)=3; thereafter a(n) = a(n-1) + a(n-2) + 1.

3, 3, 7, 11, 19, 31, 51, 83, 135, 219, 355, 575, 931, 1507, 2439, 3947, 6387, 10335, 16723, 27059, 43783, 70843, 114627, 185471, 300099, 485571, 785671, 1271243, 2056915, 3328159, 5385075, 8713235, 14098311, 22811547, 36909859, 59721407, 96631267, 156352675, 252983943, 409336619, 662320563

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1)

Crossrefs

See A022406 for a similar sequence.

Programs

Magma

[4*Fibonacci(n+1) - 1: n in [0..40]]; // G. C. Greubel, Mar 01 2018

Mathematica

Table[4*Fibonacci[n+1] -1,{n, 0, 31}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
RecurrenceTable[{a[0]==a[1]==3,a[n]==a[n-1]+a[n-2]+1},a,{n,40}] (* or *) LinearRecurrence[{2,0,-1},{3,3,7},50] (* Harvey P. Dale, Jan 10 2021 *)

PARI

for(n=0, 40, print1(4*fibonacci(n+1) -1, ", ")) \\ G. C. Greubel, Mar 01 2018

Formula

From R. J. Mathar, Mar 11 2011: (Start)

a(n+1) - a(n) = A022087(n).

G.f.: ( 3-3*x+x^2 ) / ( (x-1)*(x^2+x-1) ). (End)

a(n) = 4*Fibonacci(n+1) - 1. - G. C. Greubel, Mar 01 2018

a(n) = (-1 + (2^(1-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5)). - Colin Barker, Mar 02 2018

Extensions

Terms a(32) onward added by G. C. Greubel, Mar 01 2018