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

0,2

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).

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

Mathematica

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

Prog

(PARI) a(n)=if(n, (n+1)*(n+2)/2, 0) \\ Charles R Greathouse IV, Mar 23 2015
(GAP) Concatenation([0], List([1..50], n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Crossrefs

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

Sequence in context: A105338 A105339 A089594 * A161680 A000217 A179865

Adjacent sequences: A253142 A253143 A253144 * A253146 A253147 A253148

Keyword

nonn,easy

Author

Paul Curtz, Mar 23 2015

Status

approved