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

Author

Paul Curtz, Mar 23 2015

Keywords

• nonn
• less
• easy

Comments

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

Links

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

Crossrefs

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

Programs

GAP

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

Mathematica

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

PARI

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

Formula

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)