A195161 Multiples of 8 and odd numbers interleaved.

0, 1, 8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59

Offset: 0

Author

Omar E. Pol, Sep 10 2011

Keywords

• nonn
• easy
• mult

Comments

A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.

Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.

The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (- 4*log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 12-gonal numbers. - Omar E. Pol, Jul 27 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Column 8 of A195151.

Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, this sequence, A195312.

Cf. A144433.

Programs

Magma

&cat[[8*n, 2*n+1]: n in [0..30]]; // Vincenzo Librandi, Sep 27 2011

Maple

a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # Johannes W. Meijer, Jul 03 2016

Mathematica

With[{nn=30},Riffle[8*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,8,3},60] (* Harvey P. Dale, Nov 24 2013 *)

PARI

concat(0, Vec(x*(1+8*x+x^2)/((1-x)^2*(1+x)^2) + O(x^99))) \\ Altug Alkan, Jul 04 2016

Formula

a(2n) = 8n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]

a(n) = (6*(-1)^n+10)*n/4. - Vincenzo Librandi, Sep 27 2011

a(n) = 2*a(n-2)-a(n-4). G.f.: x*(1+8*x+x^2)/((1-x)^2*(1+x)^2). - Colin Barker, Aug 11 2012

From Ilya Gutkovskiy, Jul 03 2016: (Start)

a(m*2^k) = m*2^(k+2), k>0.

E.g.f.: x*(4*sinh(x) + cosh(x)).

Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)

Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018

a(n) = A144433(n-1) for n > 1. - Georg Fischer, Oct 14 2018