Problem 56293. Heronian Triangles with Consecutive Integral Sides and Integral Area

Created by David Hill

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<div style = "text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; "><div style="block-size: 72.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36.25px; transform-origin: 407px 36.25px; vertical-align: baseline; "><div style="block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Hero's area formula for any triangle with sides a, b, c is: </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="195.5" height="21" style="width: 195.5px; height: 21px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">.</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Heronian triangles are triangles having integral sides and integral area. Find Heronian triangles with consecutive sides. For example, H(1) would have sides 3,4,5 with area 6. Output the sum of the digits of the nth Heronian triangle's area. S(1)=6.</span></span></div></div></div>

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Problem 56293. Heronian Triangles with Consecutive Integral Sides and Integral Area

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