A215005 a(n) = a(n-2) + a(n-1) + floor(n/2) + 1 for n > 1 and a(0)=0, a(1)=1.

0, 1, 3, 6, 12, 21, 37, 62, 104, 171, 281, 458, 746, 1211, 1965, 3184, 5158, 8351, 13519, 21880, 35410, 57301, 92723, 150036, 242772, 392821, 635607, 1028442, 1664064, 2692521, 4356601, 7049138, 11405756, 18454911, 29860685, 48315614, 78176318, 126491951, 204668289

Offset: 0

Author

Alex Ratushnyak, Jul 31 2012

Keywords

• nonn
• easy

Comments

If the seed is {1,1}: 1, 1, 4, 7, 14, 24, 42, 70, 117, 192, 315, 513, 835, 1355, 2198, 3561, 5768, 9338, 15116, 24464, 39591, 64066, 103669, 167747, ...

If the seed is {1,2}: A129696.

Same seed, but -1 in the formula instead of +1: b(n)=a(n-2)+1 for n>=2, i.e. 0, 1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 172, 282, 459, 747, 1212, 1966, 3185, 5159, 8352, 13520, 21881, 35411, 57302, 92724, 150037, 242773, 392822, ...

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Crossrefs

Cf. A129696 (same formula, seed {1,2}).

Cf. A000071 (a(n+1) = a(n-1) + a(n) + 1).

Cf. A000045.

Programs

Magma

[2*Fibonacci(n+2) -(2*n+9-(-1)^n)/4: n in [0..50]]; // G. C. Greubel, Apr 05 2024

Mathematica

LinearRecurrence[{2,1,-3,0,1}, {0,1,3,6,12}, 39] (* Jean-François Alcover, Oct 05 2017 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+Floor[(n+1)/2]+1}; NestList[nxt,{1,0,1},40][[;;,2]] (* Harvey P. Dale, Feb 20 2025 *)

PARI

concat(0, Vec(x*(1+x-x^2)/((1-x)^2*(1+x)*(1-x-x^2)) + O(x^100))) \\ Colin Barker, Sep 16 2015

Python

prpr = 0
prev = 1
for n in range(2,100):
    print(prpr, end=', ')
    curr = prpr+prev + 1 + n//2
    prpr = prev
    prev = curr

SageMath

[2*fibonacci(n+2) -(n+4+(n%2))//2 for n in range(51)] # G. C. Greubel, Apr 05 2024

Formula

a(n) = 2*Fibonacci(n+2) - (2*n + 9 - (-1)^n)/4. - Vaclav Kotesovec, Aug 11 2012

From Colin Barker, Sep 16 2015: (Start)

a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5) for n>4.

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

E.g.f.: 2*exp(x/2)*(cosh(sqrt(5)*x/2) + (3/sqrt(5))*sinh(sqrt(5)*x/2)) - (1/2)*((x+4)*cosh(x) + (x+5)*sinh(x)). - G. C. Greubel, Apr 05 2024