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a(n) = (4*n^4 - 4*n^3 - n^2 + 3*n)/2.
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%I #26 May 05 2021 11:04:11

%S 1,17,108,382,995,2151,4102,7148,11637,17965,26576,37962,52663,71267,

%T 94410,122776,157097,198153,246772,303830,370251,447007,535118,635652,

%U 749725,878501,1023192,1185058,1365407,1565595

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

%C Form the infinite matrix:

%C 1 2 4 7 11 ...

%C 3 5 8 12 17 ...

%C 6 9 13 18 24 ...

%C 10 14 19 25 32 ...

%C 15 20 26 33 41 ...

%C ...

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

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

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

%C Let us define the PHI algebra as follows:

%C The basis of the PHI algebra is the PHI(1), PHI(2), PHI(3), ... elements, and the production rules are:

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

%C An element of the PHI algebra is X = Sum_{n>=1} c(n)*PHI(n), where c(n) are real or complex constants.

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

%C For every X elements: UNIT*X = X.

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

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

%C ULTRA*OMEGA = UNIT.

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

%H Harvey P. Dale, <a href="/A135400/b135400.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

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

%F 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

%p 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);

%t 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 *)

%o (PARI) a(n)=n*(4*n^3-4*n^2-n+3)/2 \\ _Charles R Greathouse IV_, Oct 12 2016

%Y Cf. A001844.

%K nonn,easy

%O 1,2

%A _Miklos Kristof_, Dec 11 2007