A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

1, 5, 10, 17, 26, 36, 49, 62, 79, 95, 116, 135, 160, 182, 211, 236, 269, 297, 334, 365, 406, 440, 485, 522, 571, 611, 664, 707, 764, 810, 871, 920, 985, 1037, 1106, 1161, 1234, 1292, 1369, 1430, 1511, 1575, 1660, 1727, 1816, 1886, 1979, 2052, 2149, 2225, 2326

Offset: 0

Author

Bruno Berselli, Oct 20 2011 - based on remarks and sequences by Omar E. Pol

Keywords

• nonn
• easy

Comments

For an origin of this sequence, see the numerical spiral illustrated in the Links section.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Bruno Berselli, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A195020 (vertices of the numerical spiral in link).

Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].

Programs

Magma

[(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];

Mathematica

Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* Harvey P. Dale, Jun 19 2020 *)

PARI

for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));

Formula

O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).

E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - G. C. Greubel, Aug 19 2017

a(n) = A195020(n) + n + 1.

a(n) - a(-n-1) = A047336(n+1).

a(n+1) - a(-n) = A113804(n+1).

a(n+2) - a(n) = A047385(n+3).

a(n+4) - a(n) = A113803(n+4).

a(2*n) + a(2*n-1) = A069127(n+1).

a(2*n) - a(2*n-1) = A016813(n).

a(2*n+1) - a(2*n) = A016777(n+1).

a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).