A135400 a(n) = (4*n^4 - 4*n^3 - n^2 + 3*n)/2.

1, 17, 108, 382, 995, 2151, 4102, 7148, 11637, 17965, 26576, 37962, 52663, 71267, 94410, 122776, 157097, 198153, 246772, 303830, 370251, 447007, 535118, 635652, 749725, 878501, 1023192, 1185058, 1365407, 1565595

Offset: 1

Author

Miklos Kristof, Dec 11 2007

Keywords

• nonn
• easy

Comments

Form the infinite matrix:

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_{n>=1} c(n)*PHI(n), where c(n) are real or complex constants.

UNIT = Sum_{n>=1} PHI(b(n)) = PHI(1) + PHI(5) + PHI(13) + PHI(25)+ ...

For every X elements: UNIT*X = X.

OMEGA = Sum_{n>=1} PHI(n) = PHI(1) + PHI(2) + PHI(3) + ...

ULTRA = Sum_{n>=1} PHI(a(n)) = PHI(1) + PHI(17) + PHI(108) + PHI(382) + ...

ULTRA*OMEGA = UNIT.

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001844.

Programs

Maple

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);

Mathematica

Table[2n^4-2n^3-n^2/2+(3n)/2,{n,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,17,108,382,995},30] (* Harvey P. Dale, May 25 2012 *)

PARI

a(n)=n*(4*n^3-4*n^2-n+3)/2 \\ Charles R Greathouse IV, Oct 12 2016

Formula

G.f.: (2*x^4 + 33*x^3 + 12*x^2 + x)/(1-x)^5.

E.g.f.: (1/2)*(4*x^4 + 20*x^3 + 15*x^2 + 2*x)*exp(x).

a(1)=1, a(2)=17, a(3)=108, a(4)=382, a(5)=995, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, May 25 2012