A073371 Convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n >= 0, with itself.

1, 2, 7, 16, 41, 94, 219, 492, 1101, 2426, 5311, 11528, 24881, 53398, 114083, 242724, 514581, 1087410, 2291335, 4815680, 10097401, 21126862, 44117867, 91963996, 191384541, 397682154, 825190479, 1710033272, 3539371201, 7317351686

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Comments

PSumSIGN transform of A045883(n-1). - Michael Somos, Jul 10 2003

Numbers of the form ((6*m+4)*2^m + (-1)^(m-1)*(3*m+4))/27. - Artur Jasinski, Feb 09 2007

With [0, 0, 0] prepended, this is an "autosequence" of the first kind, whose companion is [0, 0, 2, 3, 12, 25, 66, ...], that is A099429. - Jean-François Alcover, Jul 10 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wieb Bosma, Signed bits and fast exponentiation, J. Th. Nombres de Bordeaux, 13 no. 1 (2001), p. 27-41.
• OEIS Wiki, Autosequence

Crossrefs

Second (m=1) column of triangle A073370.

Cf. A001045, A062111, A099429, A127976.

Programs

Magma

[((5+3*n)*2^(n+2) + (-1)^n*(7+3*n))/27: n in [0..40]]; // G. C. Greubel, Sep 28 2022

Mathematica

Table[((6n+4)*2^n + (-1)^(n-1)(3n+4))/27, {n, 100}] (* Artur Jasinski, Feb 09 2007 *)

PARI

a(n) = if(n<-3, 0, ((5+3*n)*2^(n+2)+(7+3*n)*(-1)^n)/27)

SageMath

def A073371(n): return ((5+3*n)*2^(n+2) + (-1)^n*(7+3*n))/27
[A073371(n) for n in range(40)] # G. C. Greubel, Sep 28 2022

Formula

a(n) = Sum_{k=0..n} b(k) * b(n-k), where b(k) = A001045(k+1).

a(n) = Sum_{k=0..floor(n/2)} (n-k+1) * binomial(n-k, k) * 2^k.

a(n) = ((n+1)*U(n+1) + 4*(n+2)*U(n))/9 with U(n) = A001045(n+1), n>=0.

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

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

a(n) = ((5+3*n)*2^(n+2) + (7+3*n)*(-1)^n)/27.

a(n) = ((6*n+4)*2^(n) + (-1)^(n-1)*(3*n+4))/27. - Artur Jasinski, Feb 09 2007

E.g.f.: (1/27)*(4*(5+6*x)*exp(2*x) + (7-3*x)*exp(-x)). - G. C. Greubel, Sep 28 2022

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007