

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
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OFFSET

1,2


COMMENTS

Equals (n1)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(n2) + 2*n  1.
G.f.: x*(1+x)/((1x)^2*(12*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



