A099429 -id:A099429 - cp's OEIS frontend

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

Wolfdieter Lang, Aug 02 2002

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

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

Second (m=1) column of triangle A073370.

Cf. A001045, A062111, A099429, A127976.

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

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

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

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

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

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

A245962 Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).

Original entry on oeis.org

1, 1, 3, 2, 4, 3, 7, 8, 2, 11, 15, 5, 18, 30, 15, 2, 29, 56, 35, 7, 47, 104, 80, 24, 2, 76, 189, 171, 66, 9, 123, 340, 355, 170, 35, 2, 199, 605, 715, 407, 110, 11, 322, 1068, 1410, 932, 315, 48, 2, 521, 1872, 2730, 2054, 832, 169, 13, 843, 3262, 5208, 4396, 2079, 532, 63, 2

Offset: 0

Emeric Deutsch, Aug 14 2014

Number of entries in row n is 1 + floor(n/2).
The entries in row n are the coefficients of the cube polynomial of the Lucas cube Lambda(n).
For n >= 1, sum of entries in row n = A014551(n) = 2^n + (-1)^n (the Jacobsthal-Lucas numbers).
Sum_{k >= 0} k*T(n,k) = A099429(n).
T(n,0) = A000032(n) (n >= 1; the Lucas numbers); T(n,1) = A099920(n-1); T(n,2) = A245961(n).
As communicated by the authors, Theorem 5.2 and Corollary 5.3 of the Klavzar et al. paper contains a typo: 2nd binomial should be binomial(n - a - 1, a) resp. binomial(n - i - 1, i).

			Row 4 is 7, 8, 2. Indeed, the Lucas cube Lambda(4) is the graph <><> obtained by identifying a vertex of a square with a vertex of another square; it has 7 vertices (i.e., Q(0)s), 8 edges (i.e., Q(1)s), and 2 squares (i.e., Q(2)s).
Triangle starts:
   1;
   1;
   3,  2;
   4,  3;
   7,  8,  2;
  11, 15,  5;

Paolo Xausa, Table of n, a(n) for n = 0..10200 (rows 0..200 of the triangle, flattened).
Sandi Klavzar and Michel Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
Sandi Klavzar and Michel Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105.
Jun Wan and Zuo-Ru Zhang, A proof of the only mode of a unimodal sequence, arXiv:2402.12858 [math.CO], 2024.

Cf. A000032, A014551, A099429, A099920, A245961.

T := proc (n, k) options operator, arrow: add((2*binomial(n-i, i)-binomial(n-i-1, i))*binomial(i, k), i = k .. floor((1/2)*n)) end proc: for n from 0 to 20 do seq(T(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form

A245962[n_, k_] := Sum[(2*Binomial[n-i, i]-Binomial[n-i-1, i])*Binomial[i, k], {i, k, n/2}]; Table[A245962[n, k], {n, 0, 15}, {k, 0, n/2}] (* Paolo Xausa, Feb 29 2024 *)

T(n,k) = Sum_{i = k..floor(n/2)} (2*binomial(n - i, i) - binomial(n - i - 1, i))*binomial(i, k).
G.f.: (1+(1+t)*z^2)/(1-z-(1+t)*z^2).
The generating polynomial of row n (i.e., the cube polynomial of Lambda(n)) is Sum_{i = 0..floor(n/2)} (2*binomial(n - i, i) - binomial(n - i - 1))(1+x)^i.
The generating polynomial of row n (i.e., the cube polynomial of Lambda(n)) is ((1+w)/2)^n + ((1-w)/2)^n, where w = sqrt(5 + 4x).
The generating function of column k (k >= 1) is z^(2k)(2-z)/(1-z-z^2)^(k+1).

cp's OEIS Frontend

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

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