A194880 -id:A194880 - cp's OEIS frontend

A253145 Triangular numbers (A000217) omitting the term 1.

0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275

Offset: 0

Paul Curtz, Mar 23 2015

The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0,      3,    6,  10,   15,  21,  28, 36, ...
-3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
3,     12,   24,  40,   60,  84, ...         essentially  A046092
-9,   -24,  -48, -80, -120, ...              essentially -A033996
15,    48,   96, 160, ...
-33,  -96, -192, ...
63,   192, ...
-129, ...
etc.
First column: (-1)^n*A062510(n).
The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0,   0,  0, 3, 6, 10, 15, 21, ...
0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
0,   3,  0, 1, 1,  1,  1,  1, ...
3,  -3,  1, 0, 0,  0,  0,  0, ...
-6,  4, -1, 0, 0,  0,  0,  0, ...
10, -5,  1, 0, 0,  0,  0,  0, ...
-15, 6, -1, 0, 0,  0,  0,  0, ...
21, -7,  1, 0, 0,  0,  0,  0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.

Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)

a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015

Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015
a(n+1) = 3*A001840(n+1) + A022003(n).
a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018
From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)

A206012 Modular recursion: a(0)=a(1)=a(2)=a(3)=1, thereafter: a(n) equals a(n - 2) + a(n - 3) when n = 0 mod 5, a(n - 1) + a(n - 3) when n = 1 mod 5, a(n - 1) + a(n - 2) when n = 2 mod 5, a(n - 1) + a(n - 4) when n = 3 mod 5, and a(n - 1) + a(n - 2) + a(n - 3) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 3, 5, 8, 16, 13, 21, 34, 50, 105, 84, 134, 218, 323, 675, 541, 864, 1405, 2080, 4349, 3485, 5565, 9050, 13399, 28014, 22449, 35848, 58297, 86311, 180456, 144608, 230919, 375527, 555983, 1162429, 931510, 1487493, 2419003, 3581432, 7487928

Offset: 0

Roger L. Bagula, Mar 19 2012

This sequence was inspired by the work of Paul Curtz on three part sequences. I did a three part version of this that gave a generating polynomial and got even more variance by adding two more modulo sequences.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,6,0,0,0,0,3,0,0,0,0,-1).

Cf. A194880, A195240, A185138, A206568, A110897, A110898.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1))); // Bruno Berselli, Mar 20 2012

a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1;a[n_Integer] := a[n]=If[Mod[n, 5] == 0, a[n - 2] + a[n - 3], If[Mod[n, 5] == 1, a[n - 1] + a[n - 3], If[Mod[n, 5] == 2, a[n - 1] + a[n - 2], If[Mod[n, 5] == 3, a[n - 1] + a[n - 4], a[n - 1] + a[n - 2] + a[n - 3]]]]];b = Table[a[n], {n, 0, 50}];(* FindSequenceFunction gives*);Table[c[n] = b[[n]], {n, 1, 16}];c[n_Integer] := c[n] = -c[-15 + n] + c[-10 + n] + 6 c[-5 + n];d = Table[c[n], {n, 1, Length[b]}]
CoefficientList[Series[(x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1),{x,0,1001}],x] (* Vincenzo Librandi, Apr 01 2012 *)

Vec((x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1)+O(x^99)) \\ Charles R Greathouse IV, Mar 19 2012

G.f.: (x^15 - x^13 + x^12 - 2x^10 - 2x^9 + 2x^8 - x^7 - 3x^6 - 4x^5 + 3x^4 + x^3 + x^2 + x + 1) / (x^15 - 3x^10 - 6x^5 + 1). - Alois P. Heinz, Mar 19 2012

cp's OEIS Frontend

A253145 Triangular numbers (A000217) omitting the term 1.

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