A165157 Zero followed by partial sums of A133622.

0, 1, 3, 4, 7, 8, 12, 13, 18, 19, 25, 26, 33, 34, 42, 43, 52, 53, 63, 64, 75, 76, 88, 89, 102, 103, 117, 118, 133, 134, 150, 151, 168, 169, 187, 188, 207, 208, 228, 229, 250, 251, 273, 274, 297, 298, 322, 323, 348, 349, 375, 376, 403, 404, 432, 433, 462, 463, 493, 494, 525

Offset

0,3

Comments

Interleaving of A055998 and A034856.

a(n+1) is the number of distinct end states in a one-interaction knowledge model (see Sada Link).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Ali Sada, End-state variations in a one-contact information model, 2025.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(0) = 0, a(2*n) = a(2*n-1) + n + 1, a(2*n+1) = a(2*n) + 1.

a(n) = (n^2+10*n)/8 if n is even, a(n) = (n^2+8*n-1)/8 if n is odd.

a(2*k) = A055998(k) = k*(k+5)/2; a(2*k+1) = A034856(k+1) = k*(k+5)/2+1.

From Klaus Brockhaus, Sep 06 2009: (Start)

a(n) = 2*a(n-2)-a(n-4)+1 for n > 3; a(0)=0, a(1)=1, a(2)=3, a(3)=4.

a(n) = (2*n*(n+9)-1+(2*n+1)*(-1)^n)/16.

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

a(n) = n+binomial(1+floor(n/2),2). - Mircea Merca, Feb 18 2012

From Stefano Spezia, Jul 10 2020: (Start)

E.g.f.: (x*(9 + x)*cosh(x) + (-1 + 11*x + x^2)*sinh(x))/8.

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

Sum_{n>=1} 1/a(n) = 181/75 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Dec 23 2025

Example

From Stefano Spezia, Jul 10 2020: (Start)
Illustration of the initial terms for n > 0:
o    o      o      o         o        o
     o o    o o    o o       o o      o o
            o      o         o        o
                   o o o     o o o    o o o
                             o        o
                                      o o o o
(1)  (3)   (4)    (7)       (8)      (12)
(End)

Mathematica

a[n_] := If[EvenQ[n], (n^2+10*n)/8, (n^2+8*n-1)/8]; Array[a, 65, 0] (* Amiram Eldar, Dec 23 2025 *)

Prog

(Magma) m:=60; T:=[ 1+(1+(-1)^n)*n/4: n in [1..m] ]; [0] cat [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..m] ]; // Klaus Brockhaus, Sep 06 2009
(Magma) [ n le 2 select n-1 else n le 4 select n else 2*Self(n-2)-Self(n-4)+1: n in [1..61] ]; // Klaus Brockhaus, Sep 06 2009
(Haskell)
a165157 n = a165157_list !! n
a165157_list = scanl (+) 0 a133622_list
-- Reinhard Zumkeller, Feb 20 2015
(PARI) a(n) = n + (n\=2) * (n+1)/2; \\ Ruud H.G. van Tol, Oct 31 2025

Crossrefs

Equals -1+A101881.

a(n) = A117142(n+2)-2 = A055802(n+6)-3 = A114220(n+5)-3 = A134519(n+3)-3.

Cf. A133622, A055998, A034856.

Sequence in context: A026449 A286904 A282166 * A182079 A129819 A025032

Adjacent sequences: A165154 A165155 A165156 * A165158 A165159 A165160

Keyword

nonn,easy

Author

Jaroslav Krizek, Sep 05 2009

Extensions

Edited and extended by Klaus Brockhaus, Sep 06 2009

Status

approved