A111314 a(n) = a(n-1) + a(n-2) + 2 where a(0) = a(1) = 1.

1, 1, 4, 7, 13, 22, 37, 61, 100, 163, 265, 430, 697, 1129, 1828, 2959, 4789, 7750, 12541, 20293, 32836, 53131, 85969, 139102, 225073, 364177, 589252, 953431, 1542685, 2496118, 4038805, 6534925, 10573732, 17108659, 27682393, 44791054, 72473449, 117264505, 189737956

Offset: 0

Author

Parthasarathy Nambi, Nov 03 2005

Keywords

• easy
• nonn

Comments

This is the sequence A(1,1;1,1;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 17 2010

Links

• T. D. Noe, Table of n, a(n) for n=0..500
• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 13.
• Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
• Elif Tan and Ho-Hon Leung, On Leonardo p-Numbers, Integers (2023) Vol. 23, #A7.
• Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000045, A000071.

Programs

Maple

with(combinat): seq(fibonacci(n-1)+fibonacci(n+3)-2, n=0..35); # Zerinvary Lajos, Jan 31 2008

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + 2; Table[ a[n], {n, 0, 36}] (* Robert G. Wilson v *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+a[n-2]+2},a,{n,40}] (* Harvey P. Dale, Mar 27 2022 *)

Sage

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,1,1,1, lambda n: 2); [next(it) for i in range(1,38)] # Zerinvary Lajos, Jul 09 2008

Formula

a(n) = 2*F(n+1)-F(n+2)+F(n+3)-2, where F(n) is the n-th Fibonacci number. - Robert G. Wilson v, Nov 10 2005

G.f.: (2*x^2-x+1)/((x-1)*(x^2+x-1)). - T. D. Noe, Oct 19 2007

a(n) = F(n-1)+F(n+3)-2. - Zerinvary Lajos, Jan 31 2008

a(n) = 3*F(n+1)-2. - Olivier Pirson, Jun 30 2015

E.g.f.: 3*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2*exp(x). - Stefano Spezia, Jul 21 2024

Extensions

More terms from Robert G. Wilson v, Nov 07 2005