A247643 a(n) = ( 10*n*(n+1)+(2*n+1)*(-1)^n+7 )/8.

1, 3, 9, 15, 27, 37, 55, 69, 93, 111, 141, 163, 199, 225, 267, 297, 345, 379, 433, 471, 531, 573, 639, 685, 757, 807, 885, 939, 1023, 1081, 1171, 1233, 1329, 1395, 1497, 1567, 1675, 1749, 1863, 1941, 2061, 2143, 2269, 2355, 2487, 2577, 2715, 2809, 2953, 3051

Offset: 0

Author

N. J. A. Sloane, Sep 23 2014

Keywords

• nonn
• easy

Comments

From Paul Curtz, Jan 01 2020: (Start)

In the following pentagonal spiral of odd numbers

101

99 61 63

97 59 31 33 65

95 57 29 11 13 35 67

93 55 27 9 1 3 15 37 69

91 53 25 7 5 17 39 71

89 51 23 21 19 41 73

87 49 47 45 43 75

85 83 81 79 77

the terms of this sequence appear on the x axis. A062786 and A172043 are in the spiral as well. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

A diagonal of triangle in A247646.

Cf. A005408, A062786, A085787, A090772, A172043.

Programs

Maple

f:=n->(10*n*(n+1)+(2*n+1)*(-1)^n+7)/8;

Mathematica

Table[(10 n (n + 1) + (2 n + 1) (-1)^n + 7)/8, {n, 0, 60}] (* Vincenzo Librandi, Sep 26 2014 *)

PARI

Vec(-(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 25 2014

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Sep 25 2014

G.f.: -(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 25 2014

From Paul Curtz, Jan 01 2020: (Start)

a(n) = 1 + 2*A085787(n).

a(n+1) = a(n-1) + A090772(n+1). (End)

E.g.f.: (1/4)*((1 + x)*(4 + 5*x)*cosh(x) + (3 + x*(11 + 5*x))*sinh(x)). - Stefano Spezia, Jan 01 2020

Extensions

More terms from Colin Barker, Sep 25 2014