A227356 Partial sums of A129361.

1, 2, 5, 10, 20, 36, 65, 112, 193, 324, 544, 900, 1489, 2442, 4005, 6534, 10660, 17336, 28193, 45760, 74273, 120408, 195200, 316216, 512257, 829458, 1343077, 2174130, 3519412, 5696124, 9219105, 14919408, 24144289

Offset: 1

Author

Kival Ngaokrajang, Jul 08 2013

Keywords

• nonn

Comments

Sum of labeled numbers of boxes arranged as Pyramid type-II with base Fibonacci(n).

Let us call a Pyramid "type-I" when each row starts with the same number as the leftmost base number, and "type-II" when each column has the same number as the base.

The Pyramid arrangements are related to other sequences as follows:

Base Number Type-I Type-II

----------- ------ -------

Natural A002623 A034828

Odd A000292 A128624

Fibonacci A129696 a(n)

1 A002620 A002620

1,0 A008805

See illustration in links.

Links

• Kival Ngaokrajang, Illustration for some small n.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,1,-1,0,1).

Crossrefs

Cf. A002623, A034828, A002620, A000292, A128624, A129696, A008805.

Programs

Mathematica

LinearRecurrence[{2,1,-3,1,-1,0,1},{1,2,5,10,20,36,65},40] (* Harvey P. Dale, Jun 30 2025 *)

Formula

For n >=2, a(n) = a(n-1) + A129361(n-1).

G.f. -x*(1+x)*(x^2-x+1) / ( (x-1)*(x^2+x-1)*(x^4+x^2-1) ). - Joerg Arndt, Jul 10 2013

a(n) = 2 + A000045(n+4) - A096748(n+6). - R. J. Mathar, Jul 20 2013