A076645 Irregular triangle read by rows of ways to write 0 as 1 +- 2 +- 3 +- 4 +- ... +- k for some k, where signs alternate except that there is one instance of two consecutive positive terms.

1, 2, -3, 1, -2, 3, 4, -5, 6, -7, 1, -2, 3, -4, 5, 6, -7, 8, -9, 10, -11, 1, -2, 3, -4, 5, -6, 7, 8, -9, 10, -11, 12, -13, 14, -15, 1, -2, 3, -4, 5, -6, 7, -8, 9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11

Offset

1,2

Comments

Derived from an attempt to demonstrate that the fundamental state of a ferromagnetic systems defined with the Hamiltonian defined in "A measure for the complexity of Boolean functions" has energy equal to -0.5.

Rows are sorted by increasing k, then by increasing position of the two consecutive positive terms. - Sean A. Irvine, Apr 10 2025

References

L. Franco, Long-range ferromagnetic model in the hypercell. In preparation

Links

Table of n, a(n) for n=1..89.

L. Franco, A measure for the complexity of Boolean functions related to their implementation in neural networks, arXiv:cond-mat/0111169 [cond-mat.dis-nn], 2001.

Sean A. Irvine, Java program (github)

Example

Triangle begins:
  0 = 1+2-3,
  0 = 1-2+3+4-5+6-7,
  0 = 1-2+3-4+5+6-7+8-9+10-11,
  ...

Crossrefs

Sequence in context: A124171 A276146 A210530 * A382502 A350604 A011448

Adjacent sequences: A076642 A076643 A076644 * A076646 A076647 A076648

Keyword

sign,tabf

Author

Leonardo Franco (Leonardo.Franco(AT)psy.ox.ac.uk), Oct 23 2002

Extensions

Edited by Joel B. Lewis, Nov 15 2006

Revised by Sean A. Irvine, Apr 10 2025

Status

approved