A118729 Rectangular array where row r contains the 8 numbers 4*r^2 - 3*r, 4*r^2 - 2*r, ..., 4*r^2 + 4*r.

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168

Offset

0,10

Comments

The numbers in row r span the interval ]8*A000217(r-1), 8*A000217(r)].

The first difference between the entries in row r is r.

Partial sums of floor(n/8). - Philippe Deléham, Mar 26 2013

Apart from the initial zeros, the same as A008726. - Philippe Deléham, Mar 28 2013

a(n+7) is the number of key presses required to type a word of n letters, all different, on a keypad with 8 keys where 1 press of a key is some letter, 2 presses is some other letter, etc., and under an optimal mapping of letters to keys and presses (answering LeetCode problem 3014). - Christopher J. Thomas, Feb 16 2024

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

LeetCode, 3014. Minimum Number of Pushes to Type Word I.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Formula

From Philippe Deléham, Mar 26 2013: (Start)

a(8k) = A001107(k).

a(8k+1) = A002939(k).

a(8k+2) = A033991(k).

a(8k+3) = A016742(k).

a(8k+4) = A007742(k).

a(8k+5) = A002943(k).

a(8k+6) = A033954(k).

a(8k+7) = A033996(k). (End)

G.f.: x^8/(((1-x)^2*(1-x^8)). - Philippe Deléham, Mar 28 2013

a(n) = floor(n/8)*(n-3-4*floor(n/8)). - Ridouane Oudra, Jun 04 2019

a(n+7) = (1/2)*(n+(n mod 8))*(floor(n/8)+1). - Christopher J. Thomas, Feb 13 2024

Example

The array starts, with row r=0, as
  r=0:   0  0  0  0  0  0  0  0;
  r=1:   1  2  3  4  5  6  7  8;
  r=2:  10 12 14 16 18 20 22 24;
  r=3:  27 30 33 36 39 42 45 48;

Mathematica

Flatten[Table[4r^2+r(Range[-3, 4]), {r, 0, 6}]] (* or *) LinearRecurrence[ {2, -1, 0, 0, 0, 0, 0, 1, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 2}, 60] (* Harvey P. Dale, Nov 26 2015 *)

Crossrefs

Cf. similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A174738, A218470, A131242.

Sequence in context: A113768 A122936 A330729 * A008726 A302834 A022788

Adjacent sequences: A118726 A118727 A118728 * A118730 A118731 A118732

Keyword

nonn,tabf,easy

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 21 2006

Extensions

Redefined as a rectangular tabf array and description simplified by R. J. Mathar, Oct 20 2010

Status

approved