{"id":61916,"date":"2021-05-27T10:50:39","date_gmt":"2021-05-27T08:50:39","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61916"},"modified":"2026-09-30T20:31:47","modified_gmt":"2026-09-30T17:31:47","slug":"floyds-triangle-generator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/","title":{"rendered":"Floyd\u2019s Triangle Sum Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Floyd&#039;s Triangle Generator<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61916\" data-favorite-title=\"Floyd\u2019s Triangle Sum\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\" data-favorite-excerpt=\"Generate Floyd&#039;s Triangle up to 500 rows and instantly get the sum of every row plus the total sum of all numbers, using the closed-form row sum formula n(n\u00b2 + 1)\/2.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-arithmetic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61916\" data-post=\"61916\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Number of rows (n)<\/label><input type=\"text\" name=\"number_of_rows_n\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"1\" max=\"500\" value=\"5\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61916\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Floyd&#039;s Triangle:<\/span><\/span><span class=\"field-value nowrap\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"61916\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Floyd\u2019s Triangle Sum Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\text{Sum of Row } n = \\frac{n(n^2 + 1)}{2} \\\\\\ \\text{Total Sum up to Row } n = \\frac{T_n (T_n + 1)}{2}, \\quad T_n = \\frac{n(n+1)}{2} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 the row number (the row contains exactly \\(n\\) consecutive integers)<\/li>\n<li>\\(T_n = \\frac{n(n+1)}{2}\\) \u2014 the \\(n\\)-th triangular number, i.e. the last value in row \\(n\\)<\/li>\n<\/ul>\n<p>The first formula gives the sum of the \\(n\\) numbers in row \\(n\\) directly, without building the triangle. The second gives the sum of all numbers from row 1 through row \\(n\\), which is simply the sum of the integers from 1 to \\(T_n\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Floyd\u2019s Triangle Sum \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> you want to see the first 5 rows of Floyd's Triangle with the sum of each row and the grand total. Enter <strong>Number of rows (n) = 5<\/strong>.<\/p>\n<p>The triangle is:<\/p>\n<p>\\[ \\begin{array}{ccccc} 1 \\\\ 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 &amp; 10 \\\\ 11 &amp; 12 &amp; 13 &amp; 14 &amp; 15 \\end{array} \\]<\/p>\n<p>Sum of row 5 using the formula:<\/p>\n<p>\\[ \\frac{n(n^2 + 1)}{2} = \\frac{5(5^2 + 1)}{2} = \\frac{5 \\times 26}{2} = 65 \\]<\/p>\n<p>The last value in row 5 is \\(T_5 = \\frac{5 \\times 6}{2} = 15\\), so the total sum of all 15 numbers is:<\/p>\n<p>\\[ \\frac{15 \\times 16}{2} = 120 \\]<\/p>\n<p><strong>Result:<\/strong> row sums are 1, 5, 15, 34, 65; the sum of row 5 is 65 and the total sum of all values through row 5 is 120 \u2014 exactly \\(1 + 2 + \\dots + 15\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the desired number of rows in the <strong>Number of rows (n)<\/strong> field (any whole number from 1 to 500; the default is 5).<\/li>\n<li>Click the calculate button.<\/li>\n<li>Read the generated triangle under <strong>Floyd's Triangle:<\/strong> each row is printed on its own line.<\/li>\n<li>Check the table of row sums \u2014 for every row \\(n\\) you will see its total, which always equals \\(\\frac{n(n^2+1)}{2}\\).<\/li>\n<li>Find the grand total at the bottom, labeled <strong>Total sum of all values:<\/strong> \u2014 the sum of every number printed in the triangle.<\/li>\n<\/ol>\n<h2>What Is Floyd's Triangle?<\/h2>\n<p>Floyd's Triangle is a triangular array of the natural numbers, named after Robert Floyd. You write the numbers 1, 2, 3, \u2026 consecutively, putting one number in the first row, two in the second, three in the third, and so on. Row \\(n\\) therefore always contains exactly \\(n\\) numbers, ending with the \\(n\\)-th triangular number \\(T_n = \\frac{n(n+1)}{2}\\).<\/p>\n<p>The triangle is a classic exercise in introductory programming and number theory because it combines two simple patterns: the counting numbers and the triangular shape of their arrangement. Although the numbers themselves look unremarkable, their row sums behave very regularly. The sum of row \\(n\\) is \\(\\frac{n(n^2+1)}{2}\\), so the sequence of row sums runs 1, 5, 15, 34, 65, 111, 175, \u2026 \u2014 a pattern you can verify row by row with this calculator.<\/p>\n<p>The total sum of the whole triangle through row \\(n\\) is equally tidy. Since the triangle contains every integer from 1 to \\(T_n\\) exactly once, the total is just \\(\\frac{T_n(T_n+1)}{2}\\). For \\(n = 5\\) that is \\(\\frac{15 \\times 16}{2} = 120\\). This makes it easy to check large triangles without adding hundreds of numbers by hand: generate 100 rows and the total is \\(\\frac{5050 \\times 5051}{2} = 12{,}754{,}275\\).<\/p>\n<p>If you enjoy exploring number arrays like this, compare Floyd's Triangle with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/leibniz-harmonic-triangle-generator\/\">Leibniz Harmonic Triangle<\/a> or a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/magic-square-generator\/\">Magic Square generator<\/a>, two other classic arrangements with surprising sum properties.<\/p>\n<h2>Interpreting the Results<\/h2>\n<p>The row sum tells you how the \"weight\" of the triangle grows as you move downward. Because the row sum formula is roughly \\(\\frac{n^3}{2}\\), the totals grow cubically, not linearly \u2014 row 10 already sums to 505, and row 50 sums to 62,625. If a computed row sum in your own work differs from \\(\\frac{n(n^2+1)}{2}\\), you know immediately that a number was skipped or duplicated when the triangle was built.<\/p>\n<p>The total sum doubles as a quick identity check: it must always equal a triangular number of a triangular number. It also connects to well-known sums of sequences; if you need those sums on their own, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-squares\/\">Sum of Consecutive Squares<\/a> or the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/arithmetic-progression\/\">Arithmetic Progression<\/a> calculators.<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Whole rows only:<\/strong> the input must be a whole number between 1 and 500; the calculator rejects anything outside that range. The upper limit keeps the output readable \u2014 500 rows already print 125,250 numbers.<\/li>\n<li><strong>Starts at 1:<\/strong> the classic triangle begins with 1 in the first row. Variants that start at another number or use a different step are not covered by the formula shown here.<\/li>\n<li><strong>Exact integers:<\/strong> all sums are exact integers, so there are no rounding issues; the formulas always produce whole numbers because \\(n(n^2+1)\\) is even for every \\(n\\).<\/li>\n<li><strong>Display size:<\/strong> for large \\(n\\), the triangle display becomes very long. If you only need the sum of row \\(n\\) or the total, you can jump straight to the summary table and the total line at the bottom without reading every row.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Generate Floyd's Triangle up to 500 rows and instantly get the sum of every row plus the total sum of all numbers, using the closed-form row sum formula n(n\u00b2 + 1)\/2.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[72],"class_list":["post-61916","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-arithmetic-calculators"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61916","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61916"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61916\/revisions"}],"predecessor-version":[{"id":66239,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61916\/revisions\/66239"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}