A072229 Witt index of the standard bilinear form <1,1,1,...,1> over the 2-adic rationals.

0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20, 21, 22, 23, 24, 24, 24, 24, 25, 26, 27, 28, 28, 28, 28, 29, 30, 31, 32, 32, 32, 32, 33, 34, 35, 36, 36, 36, 36, 37, 38, 39, 40, 40, 40, 40, 41, 42

Offset

0,6

Comments

There is another interesting bilinear form over Q_2 : it is <1, ..., 1, 2>. It has Witt index 0, 0, 0, 1, 1, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, ...

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 1, -1).

Formula

a(n) = 4 floor(n/7) + [0,0,0,0,1,2,3][n%7 + 1]. [Formula corrected by Franklin T. Adams-Watters, Apr 13 2009]

From R. J. Mathar, Apr 16 2009: (Start)

a(n) = a(n-1) + a(n-7) - a(n-8).

G.f.: x^4*(1+x)*(1+x^2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2). (End)

Maple

for n from 0 to 120 do printf("%d, ", 4*floor(n/7)+op( (n mod 7)+1, [0, 0, 0, 0, 1, 2, 3]) ) ; od: # R. J. Mathar, Apr 16 2009

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 0, 0, 0, 1, 2, 3, 4}, 80] (* Harvey P. Dale, Jun 21 2012 *)

Prog

(Haskell)
a072229 n = a072229_list !! n
a072229_list = [0, 0, 0, 0, 1, 2, 3, 4] ++ zipWith (+)
               (zipWith (-) (tail a072229_list) a072229_list)
               (drop 7 a072229_list)
-- Reinhard Zumkeller, Nov 02 2015
(PARI) a(n)=n\7*4 + [0, 0, 0, 0, 1, 2, 3][n%7 + 1] \\ Charles R Greathouse IV, Feb 09 2017

Crossrefs

Sequence in context: A140427 A194816 A178770 * A120509 A029106 A064004

Adjacent sequences: A072226 A072227 A072228 * A072230 A072231 A072232

Keyword

nonn,nice,easy

Author

Gaël Collinet, Jul 05 2002

Extensions

More terms from R. J. Mathar, Apr 16 2009

Status

approved