A176346 A dual remainder symmetrical triangle sequence T(n, m) = 1 + 2*(n+1) - (m+1)*floor((n+1)/(m+1)) - (n-m+1)*floor((n+1)/( n-m+1)), read by rows.

1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 2, 3, 3, 2, 1, 1, 3, 4, 7, 4, 3, 1, 1, 2, 5, 4, 4, 5, 2, 1, 1, 3, 3, 5, 9, 5, 3, 3, 1, 1, 2, 4, 6, 5, 5, 6, 4, 2, 1, 1, 3, 5, 7, 6, 11, 6, 7, 5, 3, 1, 1, 2, 3, 4, 7, 6, 6, 7, 4, 3, 2, 1, 1, 3, 4, 5, 8, 7, 13, 7, 8, 5, 4, 3, 1

Offset

0,5

Comments

This sequence comes from computability functions.

Row sums are : {1, 2, 5, 6, 13, 12, 23, 24, 33, 36, 55,...}.

References

N. J. Cutland, "Computability, An introduction to recursive function theory", Cambridge University Press, London, 1980, page 37.

Martin Davis, "Computability and Unsolvability", Dover Press, New York, page 43.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n, m) = 1 + 2*(n+1) - (m+1)*floor((n+1)/(m+1)) - (n-m+1)*floor((n+1)/( n-m+1)).

Example

Triangle begins as:
  1;
  1, 1;
  1, 3, 1;
  1, 2, 2, 1;
  1, 3, 5, 3, 1;
  1, 2, 3, 3, 2, 1;
  1, 3, 4, 7, 4, 3, 1;
  1, 2, 5, 4, 4, 5, 2, 1;
  1, 3, 3, 5, 9, 5, 3, 3, 1;
  1, 2, 4, 6, 5, 5, 6, 4, 2, 1;
  1, 3, 5, 7, 6, 11, 6, 7, 5, 3, 1;

Mathematica

T[n_, m_]:= 3 +2*n -(m+1)*Floor[(n+1)/(m+1)] -(n-m+1)*Floor[(n+1)/(n-m+1 )]; Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 26 2019 *)

Prog

(PARI) {T(n, m) = 1 + 2*(n+1) - (m+1)*floor((n+1)/(m+1)) - (n-m+1)* floor((n+1)/(n-m+1))}; \\ G. C. Greubel, Apr 26 2019
(Magma) [[1 + 2*(n+1) - (m+1)*Floor((n+1)/(m+1)) - (n-m+1)*Floor((n+1)/( n-m+1)): m in [0..n]]: n in [0..12]]; // G. C. Greubel, Apr 26 2019
(Sage) [[1 + 2*(n+1) - (m+1)*floor((n+1)/(m+1)) - (n-m+1)*floor((n+1)/( n-m+1)) for m in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 26 2019

Crossrefs

Cf. A176298.

Sequence in context: A356400 A153066 A126209 * A338878 A073166 A050169

Adjacent sequences: A176343 A176344 A176345 * A176347 A176348 A176349

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 15 2010

Extensions

Edited by G. C. Greubel, Apr 26 2019

Status

approved