A350090 a(n) is the number of indices i in the range 0 <= i <= n-1 such that A003215(n) - A003215(i) is an oblong number (A002378), where A003215 are the hex numbers.

0, 1, 1, 1, 1, 3, 1, 2, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 3, 3, 3, 3, 5, 1, 1, 1, 5, 1, 1, 3, 1, 3, 1, 7, 1, 3, 3, 1, 1, 3, 7, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 7, 1, 3, 7, 1, 7, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 7, 5, 3, 3, 1, 5, 3, 3, 7, 3, 1, 1, 3, 3, 3, 7, 1, 3, 1, 3, 1

Offset: 0

Author

Klaus Purath and Michel Marcus, Dec 14 2021

Keywords

• nonn

Comments

There are very few even terms in the data (3 up to 10000). They are obtained for indices coming from A001921. For odd terms see A350120.

a(n) = 1 for n in A111251.

Examples

			For n=5, the 5 numbers hex(5)-hex(i), for i=0 to 4, are (90, 84, 72, 54, 30) out of which 90, 72 and 30 are oblong, so a(5) = 3.
		

Links

• Michel Marcus, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A002378, A003215.

Cf. also A001921, A111251, A350120.

Programs

Mathematica

obQ[n_] := IntegerQ @ Sqrt[4*n + 1]; hex[n_] := 3*n*(n + 1) + 1; a[n_] := Module[{h = hex[n]}, Count[Range[0, n - 1], ?(obQ[h - hex[#]] &)]]; Array[a, 100, 0] (* _Amiram Eldar, Dec 14 2021 *)

PARI

hex(n) = 3*n*(n+1)+1; \\ A003215
isob(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
a(n) = my(h=hex(n)); sum(k=0, n-1, isob(h - hex(k)));

PARI

a(n) = numdiv(3*n*n + 3*n + 1) - 1; \\ Jinyuan Wang, Dec 19 2021

Formula

a(n) = A000005(A003215(n)) - 1. - Jinyuan Wang, Dec 19 2021

Extensions

Edited by N. J. A. Sloane, Dec 25 2021