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%I #27 Mar 09 2024 16:26:59
%S 2,13,43,102,200,347,553,828,1182,1625,2167,2818,3588,4487,5525,6712,
%T 8058,9573,11267,13150,15232,17523,20033,22772,25750,28977,32463,
%U 36218,40252,44575,49197,54128,59378,64957,70875,77142,83768,90763,98137
%N a(n) = (1/6)*(n+1)*(10*n^2 + 17*n + 12).
%C A floretion-generated sequence which arises from a particular transform of the centered square numbers: A001844. Specifically, (a(n)) is the jesfor-transform of the sequence A001844 with respect to the floretion given in the program code. The sequence relates centered square numbers, triangular numbers and centered octahedral numbers to (n+1)^3. Note: this was made possible in part by the formula already given for A085786.
%C Floretion Algebra Multiplication Program, FAMP Code: 4jesforseq[ + .25'j + .25'k + .25j' + .25k' + .25'ij' + .25'ik' + .25'ji' + .25'ki' + e ], vesforseq = A001844, ForType: 1A, LoopType: tes.
%H Vincenzo Librandi, <a href="/A102296/b102296.txt">Table of n, a(n) for n = 0..2000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (x+1)*(3x+2)/(x-1)^4;
%F a(n) = 2*A001844(n+1)_0 - 2*A001845(n+1)_0 + A085786(n+1)_1 ( "_" denotes offset ) (n+1)^3 = 2*A001845(n+1) - 2*A001844(n+1) - A000217(n+1) - a(n).
%t LinearRecurrence[{4, -6, 4, -1}, {2, 13, 43, 102}, 50] (* _Paolo Xausa_, Mar 09 2024 *)
%o (Magma) [(1/6)*(n+1)*(10*n^2+17*n+12): n in [0..50]]; // _Vincenzo Librandi_, May 30 2011
%o (PARI) a(n) = (n+1)*(10*n^2+17*n+12)/6
%Y Cf. A001844, A001845, A085786, A000217.
%K easy,nonn
%O 0,1
%A _Creighton Dement_, Feb 19 2005
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