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A053599 Number of nonempty subsequences {s(k)} of 1..n such that the difference sequence is palindromic. 5
1, 3, 7, 13, 23, 37, 59, 89, 135, 197, 291, 417, 607, 861, 1243, 1753, 2519, 3541, 5075, 7121, 10191, 14285, 20427, 28617, 40903, 57285, 81859, 114625, 163775, 229309, 327611, 458681, 655287, 917429, 1310643, 1834929, 2621359, 3669933, 5242795, 7339945 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equals (n-1)-th row sums of triangle A152202. - Gary W. Adamson, Nov 29 2008

a(n) is the number of positive integers < 2^n such that the binary representation of the odd part is palindromic; i.e., palindromic without the final 0's. - Andrew Woods, May 19 2012

a(n) is the number of ideals of the quotient ring Z_{2^n}[u]/<u^2> for indeterminate u. - Fatih Temiz, Oct 11 2017

LINKS

Andrew Woods, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

FORMULA

a(1)=1, a(2)=3 and, for n > 2, a(n) = 2*a(n-2) + 2*n - 1.

G.f.: x*(1+x)/((1-x)^2*(1-2*x^2)). - Colin Barker, Mar 28 2012

a(n) = 5*2^((n+1)/2) - 2*n - 7 for odd n, 7*2^(n/2) - 2*n - 7 for even n. - Andrew Woods, May 19 2012

EXAMPLE

For n=4 the 13 sequences are 1,2,3,4,12,13,14,23,24,34,123,234,1234.

MATHEMATICA

t={1, 1}; Do[AppendTo[t, t[[-2]]+t[[-1]]]; AppendTo[t, 2*t[[-2]]], {n, 40}]; Nest[Accumulate, t, 2] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

CROSSREFS

Cf. A027383 (first differences), A016116 (second differences).

Cf. A152202.

Sequence in context: A058682 A081995 A291141 * A212146 A136851 A155339

Adjacent sequences:  A053596 A053597 A053598 * A053600 A053601 A053602

KEYWORD

nonn

AUTHOR

John W. Layman, Jan 19 2000

EXTENSIONS

Corrected by T. D. Noe, Nov 08 2006

STATUS

approved

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Last modified September 20 10:04 EDT 2020. Contains 337264 sequences. (Running on oeis4.)