This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A176541 Numbers n such that there exist n consecutive triangular numbers which sum to a square. 9
 0, 1, 2, 3, 4, 11, 13, 22, 23, 25, 27, 32, 37, 39, 46, 47, 48, 49, 50, 52, 59, 66, 71, 73, 83, 94, 98, 100, 104, 107, 109, 111, 118, 121, 128, 143, 146, 147, 148, 157, 167, 176, 179, 181, 183, 191, 192, 193, 194, 200, 214, 219, 227, 239, 241, 242, 243, 244, 253, 263 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Numbers n such that there exists some x>=0, such that A000292(x+n) - A000292(x) is a square. Terms of this sequence, for which only a finite number of solutions x exists, are given in A176542. Integer n is in the sequence if there exists non-degenerate solutions to the diophantine equation: 8x^2 - n*y^2 - A077415(n) = 0. A degenerate solution is one involving triangular numbers with negative indexes. The sum of n consecutive triangular numbers starting at the j'th is sum_{k=j..j+n-1} A000217(k) = n*(n^2+3*j*n+3*j^2-1)/6, see A143037. - R. J. Mathar, May 06 2015 LINKS EXAMPLE 0 is in the sequence because the sum of 0 consecutive triangular numbers is 0 (a square) 1 is in the sequence because there exist triangular numbers which are squares (cf. A001110) 2 is in the sequence because ANY 2 consecutive triangular numbers sum to a square. 3 is in the sequence because there are infinitely many solutions (cf. A165517). 4 is in the sequence because there infinitely many solutions (cf. A202391). 5 is NOT in the sequence because no 5 consecutive triangular numbers sum to a square. For n=8, solutions to the Diophantine equation exist, but start at A000217(-2) and A000217(-6): 1 + 0 + 0 + 1 + 3 + 6 + 10 + 15 = 36 and 15 + 10 + 6 + 3 + 1 + 0 + 0 + 1 = 36. There are no non-degenerate solutions for n=8. Hence, 8 is not included in the sequence. For n=11, there exist infinitely many solutions (cf. A116476). CROSSREFS Cf. A176542, A000217, A000292, A001110, A077415. Sequence in context: A155768 A138985 A184806 * A171376 A141704 A061919 Adjacent sequences:  A176538 A176539 A176540 * A176542 A176543 A176544 KEYWORD nonn AUTHOR Andrew Weimholt, Apr 20 2010 EXTENSIONS More terms from Max Alekseyev, May 10 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.