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Search: a176542 -id:a176542
Displaying 1-9 of 9 results found. page 1
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A116476 Numbers n such that T(n) + T(n+1) + ... + T(n+10) is a square, where T(m) = A000217(m) is the m-th triangular number. +10
11
13, 46, 229, 1608, 7335, 20304, 92391, 635710, 2892133, 8001886, 36403981, 250470288, 1139495223, 3152724936, 14343078279, 98684659918, 448958227885, 1242165625054, 5651136440101, 38881505539560, 176888402293623, 489410103548496 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers n such that 11*n^2 + 121*n + 440 = 2*m^2 for some integer m. - Max Alekseyev, Jan 20 2010
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,394,-394,0,0,-1,1).
FORMULA
For n>8, a(n) = 394*a(n-4) - a(n-8) + 2156. - Max Alekseyev, Jan 20 2010
G.f.: x*(2*x^8+7*x^7+15*x^6+33*x^5-605*x^4-1379*x^3-183*x^2-33*x-13)/((x-1)*(x^8-394*x^4+1)). - Colin Barker, Nov 22 2012
EXAMPLE
13 belongs to this sequence since T(13) + T(14) + ... + T(23) = 91 + 105 + 120 + 136 + 153 + 171 + 190 + 210 + 231 + 253 + 276 = 1936 = 44^2.
MATHEMATICA
For[n = 1, n < 100000, n++, If[IntegerQ[Sqrt[Sum[i*(i+1)/2, {i, n, n + 10}]]], Print[n]]] (* Stefan Steinerberger, Mar 30 2006 *)
LinearRecurrence[{1, 0, 0, 394, -394, 0, 0, -1, 1}, {13, 46, 229, 1608, 7335, 20304, 92391, 635710, 2892133}, 30] (* Harvey P. Dale, Sep 01 2017 *)
CROSSREFS
Cf. A176541, A176542, A165517, A202391
KEYWORD
nonn,easy
AUTHOR
Edward Fedorovich (chipramy(AT)012.net.il), Mar 29 2006
EXTENSIONS
Extended by Max Alekseyev, Jan 20 2010
STATUS
approved
A176541 Numbers n such that there exist n consecutive triangular numbers which sum to a square. +10
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 exist, are given in A176542.
Integer n is in the sequence if there exist 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
Table of n, a(n) for n=1..60.
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), so 11 is in the sequence.
CROSSREFS
Cf. A176542, A000217, A000292, A001110, A077415.
KEYWORD
nonn
AUTHOR
Andrew Weimholt, Apr 20 2010
EXTENSIONS
More terms from Max Alekseyev, May 10 2010
STATUS
approved
A257293 Numbers n such that T(n) + T(n+1) + ... + T(n+12) is a square, where T = A000217 (triangular numbers). +10
7
3, 29, 75, 432, 998, 3624, 8310, 44717, 102443, 370269, 848195, 4561352, 10448838, 37764464, 86508230, 465213837, 1065679683, 3851605709, 8822991915, 47447250672, 108688879478, 392826018504, 899858667750, 4839154355357, 11085200027723, 40064402282349 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
It is well known that T(n)+T(n+1) is always a square. T(n)+T(n+1)+T(n+2) is a square for n in A165517. T(n)+T(n+1)+T(n+2)+T(n+3) is a square for n in A202391. There is no sequence of 5, 6, 7, 8, 9 or 10 consecutive T(i)'s which sum to a square, cf. A176541. The next possible length is 11, see A116476. Then comes this sequence, corresponding to length 13.
Positive integers y in the solutions to 2*x^2-13*y^2-169*y-728 = 0. - Colin Barker, May 04 2015
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,102,-102,0,0,-1,1).
FORMULA
G.f.: x*(3*x^8+7*x^7+6*x^6+26*x^5-260*x^4-357*x^3-46*x^2-26*x-3) / ((x-1)*(x^4-10*x^2-1)*(x^4+10*x^2-1)). - Colin Barker, May 04 2015
MATHEMATICA
Select[Range[10^5], IntegerQ[Sqrt[(#^2+13*#+56)*13/2]]&] (* Ivan N. Ianakiev, May 04 2015 *)
LinearRecurrence[{1, 0, 0, 102, -102, 0, 0, -1, 1}, {3, 29, 75, 432, 998, 3624, 8310, 44717, 102443}, 50] (* Vincenzo Librandi, May 05 2015 *)
PROG
(PARI) for(n=0, 10^8, issquare(binomial(n+14, 3)-binomial(n+1, 3))&&print1(n", "))
(PARI) Vec(x*(3*x^8+7*x^7+6*x^6+26*x^5-260*x^4-357*x^3-46*x^2-26*x-3) / ((x-1)*(x^4-10*x^2-1)*(x^4+10*x^2-1)) + O(x^100)) \\ Colin Barker, May 04 2015
(Magma) I:=[3, 29, 75, 432, 998, 3624, 8310, 44717, 102443]; [n le 9 select I[n] else Self(n-1)+102*Self(n-4)-102*Self(n-5)-Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, May 05 2015
CROSSREFS
Cf. A176541, A176542, A000217, A000292, A001110, A077415.
Cf. A116476 (length 11).
KEYWORD
nonn,easy
AUTHOR
M. F. Hasler, May 04 2015
STATUS
approved
A202391 Indices of the smallest of four consecutive triangular numbers summing up to a square. +10
6
5, 39, 237, 1391, 8117, 47319, 275805, 1607519, 9369317, 54608391, 318281037, 1855077839, 10812186005, 63018038199, 367296043197, 2140758220991, 12477253282757, 72722761475559, 423859315570605, 2470433131948079 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers n such that A000217(n) + A000217(n + 1) + A000217(n + 2) + A000217(n + 3) is a square (=A075870(n+1)^2).
LINKS
Table of n, a(n) for n=1..20.
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).
FORMULA
a(n) = A002315(n) - 2.
G.f. x*(1+x)*(x-5) / ( (x-1)*(1-6*x+x^2) ). - R. J. Mathar, Dec 19 2011
a(n+2) = 6*a(n+1) - a(n) + 8; a(n+3) = 7*a(n+2) - 7*a(n+1) + a(n); a(n+1) = (-4 + (7 + 5*r)*(3 + 2*r)^n + (7 - 5*r)*(3 - 2*r)^n)/2 where r = sqrt(2). - Paul Weisenhorn, Jan 13 2013
CROSSREFS
Cf. A176541, A176542, A165517, A116476, A075870
KEYWORD
nonn
AUTHOR
Max Alekseyev, Dec 18 2011
STATUS
approved
A257707 Numbers n such that T(n) + T(n+1) + ... + T(n+22) is a square, where T = A000217 (triangular numbers). +10
4
56, 470, 1094, 7856, 128534, 201539, 3293081, 23435699, 53805155, 382911281, 6256309475, 9809462822, 160274811896, 1140616029542, 2618697452438, 18636292598096, 304494582579398, 477426555904883, 7800575092244921, 55513782134933123, 127452004956911987 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers y in the solutions to 2*x^2-23*y^2-529*y-4048 = 0.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,48670,-48670,0,0,0,0,-1,1).
FORMULA
G.f.: x*(10*x^12 +3*x^11 +66*x^10 +414*x^9 +624*x^8 +6762*x^7 -366022*x^6 -73005*x^5 -120678*x^4 -6762*x^3 -624*x^2 -414*x -56) / ((x -1)*(x^12 -48670*x^6 +1)).
MATHEMATICA
LinearRecurrence[{1, 0, 0, 0, 0, 48670, -48670, 0, 0, 0, 0, -1, 1}, {56, 470, 1094, 7856, 128534, 201539, 3293081, 23435699, 53805155, 382911281, 6256309475, 9809462822, 160274811896}, 50] (* Vincenzo Librandi, May 05 2015 *)
PROG
(PARI) Vec(x*(10*x^12 +3*x^11 +66*x^10 +414*x^9 +624*x^8 +6762*x^7 -366022*x^6 -73005*x^5 -120678*x^4 -6762*x^3 -624*x^2 -414*x -56) / ((x -1)*(x^12 -48670*x^6 +1)) + O(x^100))
CROSSREFS
Cf. A176541, A176542, A000217, A000292, A001110, A077415.
Cf. A116476 (length 11), A257293 (length 13), A257708 (length 25), A257709 (length 27), A257710 (length 37).
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 04 2015
STATUS
approved
A257708 Numbers n such that T(n) + T(n+1) + ... + T(n+24) is a square, where T = A000217 (triangular numbers). +10
4
25, 55, 208, 382, 1273, 2287, 7480, 13390, 43657, 78103, 254512, 455278, 1483465, 2653615, 8646328, 15466462, 50394553, 90145207, 293721040, 525404830, 1711931737, 3062283823, 9977869432, 17848298158, 58155284905, 104027505175, 338953840048, 606316732942 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers y in the solutions to 2*x^2-25*y^2-625*y-5200 = 0.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).
FORMULA
G.f.: x*(x^2+4*x+5)*(2*x^2-2*x-5) / ((x-1)*(x^2-2*x-1)*(x^2+2*x-1)).
MATHEMATICA
LinearRecurrence[{1, 6, -6, -1, 1}, {25, 55, 208, 382, 1273}, 50] (* Vincenzo Librandi, May 05 2015 *)
PROG
(PARI) Vec(x*(x^2+4*x+5)*(2*x^2-2*x-5)/((x-1)*(x^2-2*x-1)*(x^2+2*x-1)) + O(x^100))
CROSSREFS
Cf. A176541, A176542, A000217, A000292, A001110, A077415.
Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257709 (length 27), A257710 (length 37).
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 04 2015
STATUS
approved
A257709 Numbers n such that T(n) + T(n+1) + ... + T(n+26) is a square, where T = A000217 (triangular numbers). +10
4
8, 14, 39, 53, 103, 112, 206, 264, 509, 647, 1141, 1230, 2160, 2734, 5159, 6525, 11415, 12296, 21502, 27184, 51189, 64711, 113117, 121838, 212968, 269214, 506839, 640693, 1119863, 1206192, 2108286, 2665064, 5017309, 6342327, 11085621, 11940190, 20870000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers y in the solutions to 2*x^2-27*y^2-729*y-6552 = 0.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,10,-10,0,0,0,0,-1,1).
FORMULA
G.f.: x*(2*x^12+x^11+6*x^10+2*x^9+5*x^8+2*x^7-14*x^6-9*x^5-50*x^4-14*x^3-25*x^2-6*x-8) / ((x-1)*(x^12-10*x^6+1)).
MATHEMATICA
LinearRecurrence[{1, 0, 0, 0, 0, 10, -10, 0, 0, 0, 0, -1, 1}, {8, 14, 39, 53, 103, 112, 206, 264, 509, 647, 1141, 1230, 2160}, 50] (* Vincenzo Librandi, May 05 2015 *)
Position[Total/@Partition[Accumulate[Range[70000]], 27, 1], _?(IntegerQ[ Sqrt[ #]]&)]//Flatten (* The program generates the first 22 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Jul 27 2021 *)
PROG
(PARI) Vec(x*(2*x^12+x^11+6*x^10+2*x^9+5*x^8+2*x^7-14*x^6-9*x^5-50*x^4-14*x^3-25*x^2-6*x-8) / ((x-1)*(x^12-10*x^6+1)) + O(x^100))
(Magma) I:=[8, 14, 39, 53, 103, 112, 206, 264, 509, 647, 1141, 1230, 2160]; [n le 13 select I[n] else Self(n-1)+10*Self(n-6)-10*Self(n-7)-Self(n-12)+Self(n-13): n in [1..40]]; // Vincenzo Librandi, May 05 2015
CROSSREFS
Cf. A176541, A176542, A000217, A000292, A001110, A077415.
Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257708 (length 25), A257710 (length 37).
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 04 2015
STATUS
approved
A257710 Numbers n such that T(n) + T(n+1) + ... + T(n+36) is a square, where T = A000217 (triangular numbers). +10
4
5, 32, 291, 661, 4102, 8515, 13685, 113558, 182368, 377701, 2290342, 5027232, 30483491, 63130838, 101378488, 840238915, 1349295285, 2794368792, 16944086651, 37191598501, 225516999142, 467042067835, 749998177365, 6216087516438, 9982086472888, 20672740082341 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers y in the solutions to 2*x^2-37*y^2-1369*y-16872 = 0.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,7398,-7398,0,0,0,0,0,0,-1,1).
FORMULA
G.f.: x*(5*x^16 +27*x^15 +10*x^14 +27*x^13 +259*x^12 +370*x^11 +3441*x^10 +4413*x^9 -31820*x^8 -99873*x^7 -5170*x^6 -4413*x^5 -3441*x^4 -370*x^3 -259*x^2 -27*x -5) / ((x -1)*(x^8 -86*x^4 -1)*(x^8 +86*x^4 -1)).
MATHEMATICA
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 7398, -7398, 0, 0, 0, 0, 0, 0, -1, 1}, {5, 32, 291, 661, 4102, 8515, 13685, 113558, 182368, 377701, 2290342, 5027232, 30483491, 63130838, 101378488, 840238915, 1349295285}, 50] (* Vincenzo Librandi, May 05 2015 *)
PROG
(PARI) Vec(x*(5*x^16 +27*x^15 +10*x^14 +27*x^13 +259*x^12 +370*x^11 +3441*x^10 +4413*x^9 -31820*x^8 -99873*x^7 -5170*x^6 -4413*x^5 -3441*x^4 -370*x^3 -259*x^2 -27*x -5) / ((x -1)*(x^8 -86*x^4 -1)*(x^8 +86*x^4 -1)) + O(x^100))
CROSSREFS
Cf. A176541, A176542, A000217, A000292, A001110, A077415.
Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257708 (length 25), A257709 (length 27).
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 04 2015
STATUS
approved
A254443 Numbers n such that T(n) + T(n+1) + ... + T(n+21) is a square, where T(m) = A000217(m) is the m-th triangular number. +10
1
35, 75, 911, 1707, 18383, 34263, 366947, 683751, 7320755, 13640955, 146048351, 272135547, 2913646463, 5429070183, 58126881107, 108309268311, 1159623975875, 2160756296235, 23134352636591, 43106816656587, 461527428756143, 859975576835703, 9207414222486467 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers y in the solutions to 2*x^2-22*y^2-484*y-3542 = 0.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-1,1).
FORMULA
G.f.: x*(9*x^4+4*x^3-136*x^2-40*x-35) / ((x-1)*(x^4-20*x^2+1)).
PROG
(PARI) Vec(x*(9*x^4+4*x^3-136*x^2-40*x-35)/((x-1)*(x^4-20*x^2+1)) + O(x^100))
CROSSREFS
Cf. A176541, A176542, A000217, A000292, A001110, A077415.
Cf. A116476 (length 11), A257293 (length 13).
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 04 2015
STATUS
approved
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