A195817 Multiples of 10 and odd numbers interleaved.

0, 1, 10, 3, 20, 5, 30, 7, 40, 9, 50, 11, 60, 13, 70, 15, 80, 17, 90, 19, 100, 21, 110, 23, 120, 25, 130, 27, 140, 29, 150, 31, 160, 33, 170, 35, 180, 37, 190, 39, 200, 41, 210, 43, 220, 45, 230, 47, 240, 49, 250, 51, 260, 53, 270, 55, 280, 57, 290, 59, 300

Offset: 0

Author

Omar E. Pol, Sep 29 2011

Keywords

• nonn
• easy
• mult

Comments

A008592 and A005408 interleaved.

Partial sums give the generalized 14-gonal (or tetradecagonal) numbers A195818.

a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 14-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 10 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, A195161, A195312, this sequence.

Cf. A091998, A195152.

Programs

Magma

[(2*(-1)^n+3)*n: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

Mathematica

With[{nn=30},Riffle[10*Range[0,nn],Range[1,2*nn+1,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,10,3},70] (* Harvey P. Dale, Nov 24 2013 *)

PARI

a(n) = (2*(-1)^n+3)*n; \\ Andrew Howroyd, Jul 23 2018

Formula

a(n) = (2*(-1)^n+3)*n. - Vincenzo Librandi, Sep 30 2011

From Bruno Berselli, Sep 30 2011: (Start)

G.f.: x*(1+10*x+x^2)/((1-x)^2*(1+x)^2).

a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).

a(n) * a(n+1) = a(n(n+1)).

a(n) + a(n+1) = A091998(n+1). (End)

a(0)=0, a(1)=1, a(2)=10, a(3)=3, a(n)=2*a(n-2)-a(n-4). - Harvey P. Dale, Nov 24 2013

Multiplicative with a(2^e) = 5*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018

Dirichlet g.f.: zeta(s-1) * (1 + 2^(3-s)). - Amiram Eldar, Oct 25 2023