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A187468 Sum of the squares modulo 2^n of the odd numbers less than 2^n. 2
1, 2, 4, 40, 208, 928, 3904, 16000, 64768, 260608, 1045504, 4188160, 16764928, 67084288, 268386304, 1073643520, 4294770688, 17179475968, 68718690304, 274876334080, 1099508482048, 4398040219648, 17592173461504, 70368719011840, 281474926379008 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

There is a simple formula for this case; the sum of the squares of the odd numbers less than 2^n is A016131(n-1).

Can the general case for m^n, m > 2 be calculated with a formula of the same kind?

From R. J. Mathar, Aug 26 2013: (Start)

For n>=3, the sum of the squares of the even numbers less than 2^n (each square mod 2^n) are 8 times the sequence 1, 2, 12, 56, 304, 1376, 6336, 27008 etc. and appear to obey a(n)= +6*a(n-1) -48*a(n-3) +64*a(n-4). For n>=1, the sum of the squares of the odd numbers less than 3^n (modulo 3^n) start as 2 times 1, 12, 144, 1404, 13689, 126360,.. and apparently obey a(n)= +12*a(n-1) -324*a(n-3) +729*a(n-4). For n>=1, the sum of the squares of the odd numbers less than 4^n (modulo 4^n) start as 2 times 1, 28, 688, 13504, 238336,... and seem to obey a(n)= +28*a(n-1) -224*a(n-2) +512*a(n-3).

(End)

LINKS

Table of n, a(n) for n=1..25.

Index entries for linear recurrences with constant coefficients, signature (6,-8).

FORMULA

For n>2 the sum of all r_j = (c_j)^2 mod 2^n for a particular n is given by [2^(n-1)]*[2^(n-1) - 3].

a(n) = 2^(n-2)*(2^n-6) for n>2. a(n) = 6*a(n-1)-8*a(n-2) for n>4. G.f.: x*(32*x^3-4*x+1) / ((2*x-1)*(4*x-1)). - Colin Barker, Aug 19 2013

EXAMPLE

For n=5, 2^5=32. The c_j, numbers prime to 32 are the odd numbers

less than 32.  The r_j = (c_j)^2 mod 32 are 1,9,25,17,17,25,9,1,1,9,25,17,17,25,9,1=4*52=208.

From the formula, for n=5, [2^(5-1)]*[2^(5-1) - 3]=16*13=208;

MATHEMATICA

Join[{1, 2}, Table[2^(n - 1) (2^(n - 1) - 3), {n, 3, 20}]]

CROSSREFS

Sequence in context: A184952 A098337 A326483 * A238719 A158213 A012596

Adjacent sequences:  A187465 A187466 A187467 * A187469 A187470 A187471

KEYWORD

nonn,easy

AUTHOR

J. M. Bergot, Mar 22 2011

EXTENSIONS

Heavily edited by Olivier Gérard, Mar 23 2011

More terms from Colin Barker, Aug 19 2013

STATUS

approved

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Last modified September 22 06:08 EDT 2020. Contains 337289 sequences. (Running on oeis4.)