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

1,1

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.

Links

Table of n, a(n) for n=1..35.

Nicolay Avilov, Problem 2128 (in Russian).

Nicolay Avilov, Explanatory drawing

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

Example

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

Prog

(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

Crossrefs

Cf. A001844, A022266, A006137.

Sequence in context: A041850 A239123 A241697 * A239569 A219599 A221694

Adjacent sequences: A358991 A358992 A358993 * A358995 A358996 A358997

Keyword

nonn,easy

Author

Nicolay Avilov, Dec 25 2022

Status

approved