A195312 Multiples of 9 and odd numbers interleaved.

0, 1, 9, 3, 18, 5, 27, 7, 36, 9, 45, 11, 54, 13, 63, 15, 72, 17, 81, 19, 90, 21, 99, 23, 108, 25, 117, 27, 126, 29, 135, 31, 144, 33, 153, 35, 162, 37, 171, 39, 180, 41, 189, 43, 198, 45, 207, 47, 216, 49, 225, 51, 234, 53, 243, 55, 252, 57, 261, 59, 270, 61

Offset: 0

Author

Omar E. Pol, Sep 14 2011

Keywords

• nonn
• easy
• mult

Comments

Partial sums give the generalized 13-gonal (or tridecagonal) numbers A195313.

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 13-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 9 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, this sequence.

Cf. A195313, A175885.

Programs

Magma

/* By definition */ &cat[[9*n,2*n+1]: n in [0..33]]; // Bruno Berselli, Sep 16 2011

Mathematica

With[{nn=30},Riffle[9Range[0,nn],Range[1,2nn+1,2]]] (* Harvey P. Dale, Sep 24 2011 *)

PARI

a(n)=(7*(-1)^n+11)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

Formula

From Bruno Berselli, Sep 15 2011: (Start)

G.f.: x*(1+9*x+x^2)/((1-x)^2*(1+x)^2).

a(n) = (7*(-1)^n+11)*n/4.

a(n) + a(n-1) = A175885(n).

Sum_{i=0..n} a(i) = A195313(n). (End)

Multiplicative with a(2^e) = 9*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018

Dirichlet g.f.: zeta(s-1) * (1 + 7/2^s). - Amiram Eldar, Oct 25 2023

E.g.f.: x*(cosh(x) + 9*sinh(x)/2). - Stefano Spezia, Jun 12 2025