1,2

Form the infinite matrice:

1. . .2. . .4. . .7. . .11. . .

3. . .5. . .8. . 12. . .17. . .

6. . .9. . 13. ..18. . .24. . .

10. ..14. .19. ..25. . .32. . .

15. ..20. .26. ..33. . .41. . .

. . . . . . . . . . . . . . . . . . .

The diagonal elements are b(n) = 1, 5, 13, 25, 41, . . . =2*n*(n-1)+1 = A001844(n-1)

M(n,m) = ((n+m)^2-n-3*m+2)/2

a(n) = M(n,b(n)) = M(1,1), M(2,5), M(3,13), M(4,25), M(5,41), . . .

Let us define the PHI algebra as follows:

The basis of the PHI algebra is the PHI(1), PHI(2), PHI(3), . . . elements,

and the production rules are:

PHI(M(n,m))*PHI(n) = PHI(m), and every other production is zero.

An element of the PHI algebra is X = Sum(c(n)*PHI(n), n=1,2,3,. . .), where c(n) are real or complex constants.

UNIT = Sum(PHI(b(n)), n=1, 2, 3, . . .) = PHI(1) + PHI(5) + PHI(13) + PHI(25)+ . . .

For every X elements: UNIT*X = X.

OMEGA = Sum(PHI(n), n=1, 2, 3, . . .) = PHI(1) + PHI(2) + PHI(3) + . . .

ULTRA = Sum(PHI(a(n), n=1, 2, 3, . . .) = PHI(1) + PHI(17) + PHI(108) + + PHI(382) + . . .

ULTRA*OMEGA = UNIT.

The PHI algebra is nonassociative, but universal algebra, every finite or countable algebra can be modelled in the PHI algebra.

G.f.: G(x)=(2*x^4+33*x^3+12*x^2+x)/(1-x)^5 E.g.f: E(x)=(2*x^4+10*x^3+15/2*x^2+x)*exp(x)

seq(2*n^4-2*n^3-1/2*n^2+3/2*n, n=1..30); for n from 1 to 30 do b[n]:=2*n*(n-1)+1 od: seq(((n+b[n])^2-n-3*b[n]+2)/2, n=1..30);

Cf. A001844.

Adjacent sequences: A135397 A135398 A135399 this_sequence A135401 A135402 A135403

Sequence in context: A125327 A126485 A080441 this_sequence A052254 A141921 A108649

nonn

Miklos Kristof (kristmikl(AT)freemail.hu), Dec 11 2007