A358994 The sum of the numbers that are inside the contour of an n-story Christmas tree drawn at the top of the numerical pyramid containing the positive integers in natural order.

21, 151, 561, 1503, 3310, 6396, 11256, 18466, 28683, 42645, 61171, 85161, 115596, 153538, 200130, 256596, 324241, 404451, 498693, 608515, 735546, 881496, 1048156, 1237398, 1451175, 1691521, 1960551, 2260461, 2593528, 2962110, 3368646, 3815656, 4305741, 4841583, 5425945

Offset: 1

Author

Nicolay Avilov, Dec 25 2022

Keywords

• nonn
• easy

Comments

The numbers of the natural series are written line by line in the form of a numerical pyramid: the first line contains the number 1, the second line contains the next two numbers 2 and 3, the third line contains the next three numbers 4, 5 and 6, etc.; that is, the line starting with the number k contains the k following numbers. In this numerical pyramid, the contour of a "multi-story Christmas tree" is distinguished, each floor of which occupies three lines. The numbers of the sequence are the sum of all the numbers that fall into the contour of the Christmas tree, which has n floors.

Examples

			a(1) = 1 + 2 + 3 + 4 + 5 + 6 = 21;
a(2) = a(1) + (8 + 9 + 12 + 13 + 14 + 17 +18 + 19 + 20) = 151.
		

Links

• Nicolay Avilov, Problem 2128 (in Russian).
• Nicolay Avilov, Explanatory drawing
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001844, A022266, A006137.

Programs

Magma

[n*(27*n^3 + 66*n^2 + 49*n + 26)/8 : n in [1..60]]; // Wesley Ivan Hurt, Jun 14 2025

Python

def a(n): return n*(27*n**3 + 66*n**2 + 49*n + 26) // 8
print([a(n) for n in range(1, 36)]) # Michael S. Branicky, Dec 25 2022

Formula

a(n) = n*(27*n^3 + 66*n^2 + 49*n + 26) / 8.

G.f.: x*(21 + 46*x + 16*x^2 - 2*x^3)/(1 - x)^5. - Stefano Spezia, Dec 25 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Jun 14 2025