This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135400 #26 May 05 2021 11:04:11
%S A135400 1,17,108,382,995,2151,4102,7148,11637,17965,26576,37962,52663,71267,
%T A135400 94410,122776,157097,198153,246772,303830,370251,447007,535118,635652,
%U A135400 749725,878501,1023192,1185058,1365407,1565595
%N A135400 a(n) = (4*n^4 - 4*n^3 - n^2 + 3*n)/2.
%C A135400 Form the infinite matrix:
%C A135400 1 2 4 7 11 ...
%C A135400 3 5 8 12 17 ...
%C A135400 6 9 13 18 24 ...
%C A135400 10 14 19 25 32 ...
%C A135400 15 20 26 33 41 ...
%C A135400 ...
%C A135400 The diagonal elements are b(n) = 1, 5, 13, 25, 41, ... = 2*n*(n-1)+1 = A001844(n-1).
%C A135400 M(n,m) = ((n+m)^2-n-3*m+2)/2.
%C A135400 a(n) = M(n,b(n)) = M(1,1), M(2,5), M(3,13), M(4,25), M(5,41), ...
%C A135400 Let us define the PHI algebra as follows:
%C A135400 The basis of the PHI algebra is the PHI(1), PHI(2), PHI(3), ... elements, and the production rules are:
%C A135400 PHI(M(n,m))*PHI(n) = PHI(m) and every other production is zero.
%C A135400 An element of the PHI algebra is X = Sum_{n>=1} c(n)*PHI(n), where c(n) are real or complex constants.
%C A135400 UNIT = Sum_{n>=1} PHI(b(n)) = PHI(1) + PHI(5) + PHI(13) + PHI(25)+ ...
%C A135400 For every X elements: UNIT*X = X.
%C A135400 OMEGA = Sum_{n>=1} PHI(n) = PHI(1) + PHI(2) + PHI(3) + ...
%C A135400 ULTRA = Sum_{n>=1} PHI(a(n)) = PHI(1) + PHI(17) + PHI(108) + PHI(382) + ...
%C A135400 ULTRA*OMEGA = UNIT.
%C A135400 The PHI algebra is a nonassociative, but universal algebra; every finite or countable algebra can be modeled in the PHI algebra.
%H A135400 Harvey P. Dale, <a href="/A135400/b135400.txt">Table of n, a(n) for n = 1..1000</a>
%H A135400 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A135400 G.f.: (2*x^4 + 33*x^3 + 12*x^2 + x)/(1-x)^5.
%F A135400 E.g.f.: (1/2)*(4*x^4 + 20*x^3 + 15*x^2 + 2*x)*exp(x).
%F A135400 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 A135400 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 A135400 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 A135400 (PARI) a(n)=n*(4*n^3-4*n^2-n+3)/2 \\ _Charles R Greathouse IV_, Oct 12 2016
%Y A135400 Cf. A001844.
%K A135400 nonn,easy
%O A135400 1,2
%A A135400 _Miklos Kristof_, Dec 11 2007