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 A226379 #28 Mar 24 2024 02:32:14
%S A226379 0,-15,-3,2,3,6,-7,5,12,15,20,9,21,30,35,42,33,45,56,63,72,65,77,90,
%T A226379 99,110,105,117,132,143,156,153,165,182,195,210,209,221,240,255,272,
%U A226379 273,285,306,323,342,345,357,380,399,420,425,437
%N A226379 a(5n) = 2*n*(2*n+1), a(5n+1) = (2*n-3)*(2*n+5), a(5n+2) = (2*n-1)*(2*n+3), a(5n+3) = (2*n+2)*(2*n+1), a(5n+4) = (2*n+1)*(2*n+3).
%C A226379 The sequence is the fifth row of the following array:
%C A226379 0, 6, 20, 42, 72, 110, 156, 210, 272, ... A002943
%C A226379 0, 3, 6, 15, 20, 35, 42, 63, 72, ... bisections A002943, A000466
%C A226379 0, 2, 3, 6, 12, 15, 20, 30, 35, ... A226023 (trisections A002943, A000466, A002439)
%C A226379 0, -3, 2, 3, 6, 5, 12, 15, 20, ... A214297 (quadrisections A078371)
%C A226379 0, -15, -3, 2, 3, 6, -7, 5, 12, ... a(n)
%C A226379 0, -63, -15, -3, 2, 3, 6, -55, -7, ...
%C A226379 The principle of construction is that (i) the lower left triangular portion has constant values down the diagonals (6, 3, 2, -3, -15, ...), defined from row 4 on by the negated values of A024036. (ii) The extension along the rows is defined by maintaining bisections, trisections, quadrisections etc of the form (2*n+x)*(2*n+y) with some constants x and y. In the fifth line this needs the quintisections shown in the NAME.
%C A226379 Each row in the array has the subsequences of the previous row plus another subsequence of the format (2*n+1)*(2*n+y) shuffled in; the first A002943, the second also A000466, the third also A002439, the fourth also A078371, and the fifth (2*n+3)*(2*n-5).
%C A226379 Only the first three rows are monotonically increasing everywhere.
%C A226379 a(n) is divisible by A226203(n).
%C A226379 Numerators of: 0, -15/4, -3/4, 2/9, 3/16, 6/25, -7/36, 5/36, 12/49, 15/64, 20/81, ... = a(n)/A226096(n). A permutation of A225948(n+1)/A226008(n+1).
%C A226379 Is the sequence increasing monotonically from 221 on?
%H A226379 G. C. Greubel, <a href="/A226379/b226379.txt">Table of n, a(n) for n = 0..5000</a>
%H A226379 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,2,-2,0,0,0,-1,1).
%F A226379 4*a(n) = A226096(n) - period 5: repeat [1, 64, 16, 1, 4].
%F A226379 G.f.: x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9) / ( (x^4+x^3+x^2+x+1)^2 *(x-1)^3 ). - _R. J. Mathar_, Jun 13 2013
%F A226379 a(n) = a(n-1)+2*a(n-5)-2*a(n-6)-a(n-10)+a(n-11) for n > 10. - _Wesley Ivan Hurt_, Oct 03 2017
%t A226379 CoefficientList[Series[x*(15 - 12*x - 5*x^2 - x^3 - 3*x^4 - 17*x^5 + 12*x^6 + 3*x^7 - x^8 + x^9)/((x^4 + x^3 + x^2 + x + 1)^2*(x - 1)^3), {x, 0, 80}], x] (* _Wesley Ivan Hurt_, Oct 03 2017 *)
%o A226379 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( -x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9)/((1-x^5)^2*(1-x)) )); // _G. C. Greubel_, Mar 23 2024
%o A226379 (SageMath)
%o A226379 def A226379_list(prec):
%o A226379 P.<x> = PowerSeriesRing(ZZ, prec)
%o A226379 return P( -x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9)/((1-x^5)^2*(1-x)) ).list()
%o A226379 A226379_list(50) # _G. C. Greubel_, Mar 23 2024
%Y A226379 Cf. A000466, A002439, A002939, A002943, A024036, A078371, A145923.
%Y A226379 Cf. A214297, A225948, A226008, A226023, A226096, A226203.
%K A226379 sign,easy
%O A226379 0,2
%A A226379 _Paul Curtz_, Jun 05 2013