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 A235355 #28 Feb 10 2024 03:47:42
%S A235355 0,1,2,7,11,24,33,58,74,115,140,201,237,322,371,484,548,693,774,955,
%T A235355 1055,1276,1397,1662,1806,2119,2288,2653,2849,3270,3495,3976,4232,
%U A235355 4777,5066,5679,6003,6688,7049,7810,8210,9051,9492,10417,10901,11914,12443,13548
%N A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.
%C A235355 Difference table for 0 followed by a(n):
%C A235355 0, 0, 1, 2, 7, 11, 24, 33,...
%C A235355 0, 1, 1, 5, 4, 13, 9, 25,... =A147685(n)
%C A235355 1, 0, 4, -1, 9, -4, 16, -9,... =interleave A000290(n+1),-A000290(n)
%C A235355 -1, 4, -5, 10, -13, 20, -25, 34,...
%C A235355 5, -9, 15, -23, 33, -45, 59, -75,... =(-1)^n*A027688(n+2).
%C A235355 a(-n) = -a(n-1).
%C A235355 From the second row, signature (0,3,0,-3,0,1).
%C A235355 Consider a(n+2k+1)+a(2k-n):
%C A235355 1, 2, 6, 9, 17, 22, 34,...
%C A235355 9, 12, 24, 33, 57, 72, 108,...
%C A235355 35, 40, 60, 75, 115, 140, 200,...
%C A235355 91, 98, 126, 147, 203, 238, 322,...
%C A235355 189, 198, 234, 261, 333, 378, 486,... .
%C A235355 The first column is A005898(n).
%C A235355 The rows are successively divisible by 2*k+1. Hence
%C A235355 1, 2, 6, 9, 17, 22, 34,...
%C A235355 3, 4, 8, 11, 19, 24, 36,...
%C A235355 7, 8, 12, 15, 23, 28, 40,...
%C A235355 13, 14, 18, 21, 29, 34, 46,...
%C A235355 21, 22, 26, 29, 37, 42, 54,...
%C A235355 The first column is A002061(n+1).
%C A235355 The main diagonal is A212965(n).
%C A235355 The first difference of every row is A022998(n+1).
%C A235355 Compare to the (2k+1)-sections of A061037 in A165943.
%H A235355 Harvey P. Dale, <a href="/A235355/b235355.txt">Table of n, a(n) for n = 0..1000</a>
%H A235355 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A235355 a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
%F A235355 a(2n) = 0 followed by A085786(n). a(2n+1) = A081436(n).
%F A235355 a(2n) + a(2n+1) = A005898(n).
%F A235355 a(2n-1) + a(2n) = A061317(n).
%F A235355 a(n) = (-1)*((-1+(-1)^n-2*n)*(2+n+n^2))/16. a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^2+x+1) / ((x-1)^4*(x+1)^3). - _Colin Barker_, Jan 20 2014
%e A235355 a(1)=1, a(2)=2, a(3)=3+4=7, a(4)=5+6=11, a(5)=7+8+9=24, a(6)=10+11+12=33.
%t A235355 LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,2,7,11,24,33},50] (* _Harvey P. Dale_, Nov 22 2014 *)
%o A235355 (PARI) Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Jan 20 2014
%Y A235355 Cf. A075356, A234305.
%K A235355 nonn,easy
%O A235355 0,3
%A A235355 _Paul Curtz_, Jan 07 2014
%E A235355 More terms from _Colin Barker_, Jan 20 2014