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A278930 a(n) is the least positive integer that differs (in absolute value) by an (n+1)-st power from the reverse of its binary representation. 2
2, 36, 100, 2081, 8257, 32897, 131329, 524801, 2098177, 8390657, 33558529, 134225921, 536887297, 2147516417, 8590000129, 34359869441, 137439215617, 549756338177, 2199024304129, 8796095119361, 35184376283137, 140737496743937, 562949970198529, 2251799847239681 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The numbers whose binary representation is a palindrome are excluded by definition because 0 is not a power of a positive number.

It might be thought that the first term should be 1 instead of 2, since by prepending its binary representation (itself) with a zero we get 01 with reverse 10 (decimal 2), and their difference in absolute value is abs(1-2)=1, which is itself its 1st power 1^1. However, leading zeros are ignored. Another alternative interpretation is to consider 1 as a palindrome, which also excludes it from this sequence.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

FORMULA

For n>3, a(n) = 1+2*(2^n+4^(n+1)).

From Colin Barker, Dec 02 2016: (Start)

a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>6.

G.f.: x*(2 + 22*x - 124*x^2 + 1869*x^3 - 5198*x^4 + 3432*x^5) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).

(End)

EXAMPLE

2 in binary is 10, its binary reverse 01 or simply 1 is the decimal number 1, subtracting them gives abs(2-1)=1, and since 1 is its own square, a(1)=2.

36 in binary is 100100, its binary reverse 1001 is the decimal number 9, subtracting them abs(36-9)=27=3^3, a third power, therefore a(2)=36.

100 in binary is 1100100, its binary reverse 10011 is the decimal number 19, subtracting them abs(100-19)=81=3^4, a fourth power, therefore a(3)=100.

For n>3 if we represent zeros with dots and place the binary representation for each term followed by its reverse, up to n=12 we obtain the graph:

1.....1....1

1....1.....1,

1......1.....1

1.....1......1,

1.......1......1

1......1.......1,

1........1.......1

1.......1........1,

1.........1........1

1........1.........1,

1..........1.........1

1.........1..........1,

1...........1..........1

1..........1...........1,

1............1...........1

1...........1............1,

1.............1............1

1............1.............1;

which illustrates better why the absolute value should be part of the definition, and how the difference is an (n+1)th power: From the first two rows for a(4) we have abs(2081-2113) = abs(-32) = 2^5.

MATHEMATICA

Rest@ CoefficientList[Series[x (2 + 22 x - 124 x^2 + 1869 x^3 - 5198 x^4 + 3432 x^5)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 24}], x] (* Michael De Vlieger, Dec 07 2016 *)

PROG

(PARI) a(n)=if(n>3, 1+2*(2^n+4^(n+1)), [2, 36, 100][n]);

(PARI) Vec(x*(2 + 22*x - 124*x^2 + 1869*x^3 - 5198*x^4 + 3432*x^5) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Dec 02 2016

CROSSREFS

Inspired by: A278328.

Cf. A283050.

Sequence in context: A081310 A187298 A069067 * A258356 A145450 A196558

Adjacent sequences:  A278927 A278928 A278929 * A278931 A278932 A278933

KEYWORD

nonn,easy,base

AUTHOR

R. J. Cano, Dec 01 2016

EXTENSIONS

More terms from Colin Barker, Dec 02 2016

STATUS

approved

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Last modified May 22 06:54 EDT 2022. Contains 353933 sequences. (Running on oeis4.)