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A253145 Triangular numbers (A000217) omitting the term 1.

%I A253145 #50 May 28 2025 10:51:05
%S A253145 0,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,
%T A253145 253,276,300,325,351,378,406,435,465,496,528,561,595,630,666,703,741,
%U A253145 780,820,861,903,946,990,1035,1081,1128,1176,1225,1275
%N A253145 Triangular numbers (A000217) omitting the term 1.
%C A253145 The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
%C A253145 0,      3,    6,  10,   15,  21,  28, 36, ...
%C A253145 -3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
%C A253145 3,     12,   24,  40,   60,  84, ...         essentially  A046092
%C A253145 -9,   -24,  -48, -80, -120, ...              essentially -A033996
%C A253145 15,    48,   96, 160, ...
%C A253145 -33,  -96, -192, ...
%C A253145 63,   192, ...
%C A253145 -129, ...
%C A253145 etc.
%C A253145 First column: (-1)^n*A062510(n).
%C A253145 The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
%C A253145 An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
%C A253145 b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
%C A253145 The successive differences of b(n) are
%C A253145 0,   0,  0, 3, 6, 10, 15, 21, ...
%C A253145 0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
%C A253145 0,   3,  0, 1, 1,  1,  1,  1, ...
%C A253145 3,  -3,  1, 0, 0,  0,  0,  0, ...
%C A253145 -6,  4, -1, 0, 0,  0,  0,  0, ...
%C A253145 10, -5,  1, 0, 0,  0,  0,  0, ...
%C A253145 -15, 6, -1, 0, 0,  0,  0,  0, ...
%C A253145 21, -7,  1, 0, 0,  0,  0,  0, ...
%C A253145 The inverse binomial transform (first column) is the signed sequence. This is general.
%C A253145 Also generalized hexagonal numbers without 1. - _Omar E. Pol_, Mar 23 2015
%H A253145 Muniru A Asiru, <a href="/A253145/b253145.txt">Table of n, a(n) for n = 0..1000</a>
%H A253145 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A253145 Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
%F A253145 a(n) = (n+1)*(n+2)/2 for n > 0. - _Charles R Greathouse IV_, Mar 23 2015
%F A253145 a(n+1) = 3*A001840(n+1) + A022003(n).
%F A253145 a(n) = A161680(n+2) for n >= 1. - _Georg Fischer_, Oct 30 2018
%F A253145 From _Stefano Spezia_, May 28 2025: (Start)
%F A253145 G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
%F A253145 E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)
%t A253145 Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* _Michael De Vlieger_, Mar 23 2015 *)
%o A253145 (PARI) a(n)=if(n,(n+1)*(n+2)/2,0) \\ _Charles R Greathouse IV_, Mar 23 2015
%o A253145 (GAP) Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # _Muniru A Asiru_, Oct 31 2018
%Y A253145 Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.
%K A253145 nonn,less,easy
%O A253145 0,2
%A A253145 _Paul Curtz_, Mar 23 2015