A331729 -id:A331729 - cp's OEIS frontend

A330909 Floor of area of triangle whose sides are consecutive Ulam numbers (A002858).

0, 2, 5, 11, 23, 43, 70, 100, 141, 227, 361, 478, 670, 826, 1044, 1183, 1405, 1668, 1960, 2272, 2545, 2889, 3351, 3819, 4267, 4523, 4955, 5669, 6558, 7474, 8203, 8914, 9633, 10813, 12245, 13611, 13972, 14587, 15473, 16798, 17987, 19298, 20229, 21909, 23166

Offset: 1

Frank M Jackson, May 01 2020

nonn

It has been proved that three consecutive Ulam numbers U(n) for n > 1 satisfy the triangle inequality. See Wikipedia link below.

			a(2) = 2 because the triangle with sides (2, 3, 4) has area 3*sqrt(15)/4 = 2.9047...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Wikipedia, Ulam number.

Cf. A002858, A331729.

lst1 = ReadList["https://oeis.org/A002858/b002858.txt", {Number,Number}]; lst={}; Do[{a, b, c}={lst1[[n]][[2]], lst1[[n+1]][[2]], lst1[[n+2]][[2]]}; s = (a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; AppendTo[lst, Floor@A], {n, 1, 50}]; lst

Given a triangle with sides a, b and c, the area A = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.

A347212 a(n) is the next Ulam number (A002858) after 2^(n-1).

Original entry on oeis.org

1, 2, 3, 6, 11, 18, 36, 69, 131, 258, 522, 1025, 2054, 4101, 8194, 16394, 32817, 65542, 131085, 262183, 524313, 1048588, 2097164, 4194310, 8388624, 16777218, 33554433, 67108881, 134217730, 268435473, 536870927, 1073741861

Offset: 0

Frank M Jackson, Sep 02 2021

This sequence appears to be a "complete" (sic) sequence as defined in the Wikipedia link.

			a(0) = 1 because 1 is the next Ulam number after 2^(-1) = 1/2.
a(6) = 36 because 36 is the next Ulam number after 2^5 = 32.

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A002858, A331729.

ulams={1, 2}; Do[AppendTo[ulams, n=Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n-ulams], n/2, 1, 1]]!= 2]; n], {10^4}]; lst=ulams; nextu[n_] := Module[{m = 1}, While[lst[[m]]<=n, m++]; lst[[m]]]; Join[{1}, Table[nextu[2^(n - 1)], {n, 1, 18}]]

a(22)-a(31) from Amiram Eldar, Sep 02 2021

cp's OEIS Frontend

A330909 Floor of area of triangle whose sides are consecutive Ulam numbers (A002858).

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