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A104563 A floretion-generated sequence relating to centered square numbers.

%I A104563 #33 Dec 15 2024 06:14:28
%S A104563 0,1,3,5,8,13,19,25,32,41,51,61,72,85,99,113,128,145,163,181,200,221,
%T A104563 243,265,288,313,339,365,392,421,451,481,512,545,579,613,648,685,723,
%U A104563 761,800,841,883,925,968,1013,1059,1105,1152,1201,1251
%N A104563 A floretion-generated sequence relating to centered square numbers.
%C A104563 Floretion Algebra Multiplication Program, FAMP Code: a(n) = 1vesrokseq[A*B] with A = - .5'i - .5i' + .5'ii' + .5e, B = + .5'ii' - .5'jj' + .5'kk' + .5e. RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + Math.signum(Y[sqa.Findk()])*p (internal program code). Note: many slight variations of the "RokType" already exist, such that it has become difficult to assign them all names.
%H A104563 Colin Barker, <a href="/A104563/b104563.txt">Table of n, a(n) for n = 0..1000</a>
%H A104563 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F A104563 G.f.: x*(1 + x^3)/((1 + x^2)*(1 - x)^3).
%F A104563 FAMP result: 2*a(n) + 2*A004525(n+1) = A104564(n) + a(n+1).
%F A104563 Superseeker results:
%F A104563 a(2*n+1) = A001844(n) = 2*n*(n+1) + 1 (Centered square numbers);
%F A104563 a(n+1) - a(n) = A098180(n) (Odd numbers with two times the odd numbers repeated in order between them);
%F A104563 a(n) + a(n+2) = A059100(n+1) = A010000(n+1);
%F A104563 a(n+2) - a(n) = A047599(n+1) (Numbers that are congruent to {0, 3, 4, 5} mod 8);
%F A104563 a(n+2) - 2*a(n+1) + a(n) = A007877(n+3) (Period 4 sequence with initial period (0, 1, 2, 1));
%F A104563 Coefficients of g.f.*(1-x)/(1+x) = convolution of this with A280560 gives A004525;
%F A104563 Coefficients of g.f./(1+x) = convolution of this with A033999 gives A054925.
%F A104563 a(n) = (1/2)*(n^2 + 1 - cos(n*Pi/2)). - _Ralf Stephan_, May 20 2007
%F A104563 From _Colin Barker_, Apr 29 2019: (Start)
%F A104563 a(n) = (2 - (-i)^n - i^n + 2*n^2) / 4 where i=sqrt(-1).
%F A104563 a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4. (End)
%F A104563 a(n) = A011848(n-1)+A011848(n+2). - _R. J. Mathar_, Sep 11 2019
%F A104563 Sum_{n>=1} 1/a(n) = Pi^2/48 + (Pi/2) * tanh(Pi/2) + (Pi/(4*sqrt(2)) * tanh(Pi/(2*sqrt(2)))). - _Amiram Eldar_, Dec 14 2024
%t A104563 LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 3, 5, 8}, 60] (* _Amiram Eldar_, Dec 14 2024 *)
%o A104563 (PARI) concat(0, Vec(x*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x^2)) + O(x^40))) \\ _Colin Barker_, Apr 29 2019
%Y A104563 Cf. A001844, A004525, A104564, A098180, A059100, A010000, A047599, A007877.
%K A104563 nonn,easy
%O A104563 0,3
%A A104563 _Creighton Dement_, Mar 15 2005
%E A104563 Stephan's formula corrected by _Bruno Berselli_, Apr 29 2019