A165188 Interleaving of A014125 and zero followed by A014125.

1, 0, 3, 1, 6, 3, 11, 6, 18, 11, 27, 18, 39, 27, 54, 39, 72, 54, 94, 72, 120, 94, 150, 120, 185, 150, 225, 185, 270, 225, 321, 270, 378, 321, 441, 378, 511, 441, 588, 511, 672, 588, 764, 672, 864, 764, 972, 864, 1089, 972, 1215, 1089, 1350, 1215, 1495, 1350

Offset

1,3

Comments

This sequence convolved with A000217 (without initial term 0) yields A164680.

See A164680 for similar convolutions.

A165188 convolved with A000217 yields sequence A164680. This is to be expected since A000217 can be associated with partition 1+1+1, A164680 with partition 1+1+1+2+2+2+3 and A165188 with partition 2+2+2+3 by observing their unreduced generating functions and verified by generating the sequences by converting the partitions into finite sequences and using Euler's Transform. Thus partition 1+1+1 yields the finite sequence (3); partition 2+2+2+3 yields the finite sequence (0,3,1); and, when combined, partition 1+1+1+2+2+2+3 yields (3,3,1). - Alford Arnold, Sep 24 2009

Links

Table of n, a(n) for n=1..56.

Index entries for linear recurrences with constant coefficients, signature (0,3,1,-3,-3,1,3,0,-1).

Formula

a(n) = -a(n-1)+2*a(n-2)+3*a(n-3)-3*a(n-5)-2*a(n-6)+a(n-7)+a(n-8)+1 for n > 8; a(1)=1, a(2)=0, a(3)=3, a(4)=1, a(5)=6, a(6)=3, a(7)=11, a(8)=6. - Klaus Brockhaus, Sep 15 2009

G.f.: x/((1-x)^4*(1+x)^3*(1+x+x^2)). - Klaus Brockhaus, Sep 15 2009

a(n) = (2*n^3+21*n^2+63*n+49)/288-(-1)^n*(9+7*n+n^2)/32+A057078(n)/9. - R. J. Mathar, Sep 17 2009

Euler transform of length 3 sequence [ 0, 3, 1]. - Michael Somos, Feb 01 2015

G.f.: x / ((1 - x^2)^3 * (1 - x^3)). - Michael Somos, Feb 01 2015

a(n) = -a(-7 - n) for all n in Z.

a(n+3) - a(n) = 0 if n odd else (n+6) * (n+4) / 8. - Michael Somos, Feb 01 2015

a(2*n + 1) = a(2*n + 4) = A014125(n) for all n in Z. - Michael Somos, Feb 01 2015

Example

A014125 begins 1,3,6,11,18,27,..., thus this sequence begins 1,0,3,1,6,3,11,6,18,11,27,18,... .
G.f. = x + 3*x^3 + x^4 + 6*x^5 + 3*x^6 + 11*x^7 + 6*x^8 + 18*x^9 + 11*x^10 + ...

Mathematica

a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -7 - n]; s SeriesCoefficient[ x / ((1 - x^2)^3 (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Feb 01 2015 *)
LinearRecurrence[{0, 3, 1, -3, -3, 1, 3, 0, -1}, {1, 0, 3, 1, 6, 3, 11, 6, 18}, 60] (* Harvey P. Dale, Apr 14 2018 *)

Prog

(PARI) /* first computes u = A014125 as second bisection of A001400, then interleaves */ {m=28; u=vector(m, n, polcoeff(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))+O(x^(2*n)), 2*n-1)); vector(2*m, k, if(k%2==1, u[(k+1)/2], if(k==2, 0, u[k/2-1])))} /* Klaus Brockhaus, Sep 15 2009 */
(PARI) {a(n) = my(s=1); if( n<0, s=-1; n=-7-n); s * polcoeff( x / ((1 - x^2)^3 * (1 - x^3)) + x * O(x^n), n)}; /* Michael Somos, Feb 01 2015 */
(Magma) I:=[1, 0, 3, 1, 6, 3, 11, 6]; [n le 8 select I[n] else -Self(n-1)+2*Self(n-2)+3*Self(n-3)-3*Self(n-5)-2*Self(n-6)+Self(n-7)+Self(n-8)+1: n in [1..60]]; // Vincenzo Librandi, Jun 24 2015

Crossrefs

Cf. A014125, A000217, A164680, A001400.

Sequence in context: A226132 A121443 A008795 * A294778 A132180 A207630

Adjacent sequences: A165185 A165186 A165187 * A165189 A165190 A165191

Keyword

nonn

Author

Alford Arnold, Sep 13 2009

Extensions

Edited and extended by Klaus Brockhaus, Sep 15 2009

Linear recurrence link and signature correct by Michel Marcus, Jun 25 2015

Status

approved