A200993 -id:A200993 - cp's OEIS frontend

A352181 a(n) = A200993(n)/2.

0, 5, 495, 48510, 4753490, 465793515, 45643010985, 4472549283020, 438264186724980, 42945417749765025, 4208212675290247475, 412361896760694487530, 40407257669872769530470, 3959498889750770719498535, 387990483937905657741325965, 38019107927025003687930446040

Offset: 0

N. J. A. Sloane, Mar 08 2022

Halves of triangular numbers which are also thirds of triangular numbers.

Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A200993, A200994, A278620.

a(n) = A200994(n)/3.
From Chai Wah Wu, Apr 22 2024: (Start)
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n > 2.
G.f.: -5*x/((x - 1)*(x^2 - 98*x + 1)). (End)
a(n) = 5*A278620(n). - Hugo Pfoertner, Apr 22 2024

A200994 Triangular numbers, T(m), that are three-halves of another triangular number; T(m) such that 2T(m) = 3T(k) for some k.

Original entry on oeis.org

0, 15, 1485, 145530, 14260470, 1397380545, 136929032955, 13417647849060, 1314792560174940, 128836253249295075, 12624638025870742425, 1237085690282083462590, 121221773009618308591410, 11878496669252312158495605, 1163971451813716973223977895

Offset: 0

Charlie Marion, Feb 15 2012

For n > 1, a(n) = 98*a(n-1) - a(n-2) + 15. In general, for m>0, let b(n) be those triangular numbers such that for some triangular number c(n), (m+1)*b(n) = m*c(n). Then b(0) = 0, b(1) = A014105(m) and for n > 1, b(n) = 2*A069129(m+1)*b(n-1) - b(n-2) + A014105(m). Further, c(0) = 0, c(1) = A000384(m+1) and for n>1, c(n) = 2*A069129(m+1)*c(n-1) - c(n-2) + A000384(m+1).

			2*0 = 3*0.
2*15 = 3*10.
2*1485 = 3*990.
2*145530 = 3*97020.

Colin Barker, Table of n, a(n) for n = 0..500
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A001652, A029549, A053141, A075528, A200993-A201008.

A201008 Triangular numbers, T(m), that are five-sixths of another triangular number: T(m) such that 6T(m)=5T(k) for some k.

Original entry on oeis.org

0, 55, 26565, 12804330, 6171660550, 2974727580825, 1433812522297155, 691094661019647940, 333106192798948009980, 160556493834431921162475, 77387896922003387052303025, 37300805759911798127288895630

Offset: 0

Charlie Marion, Dec 20 2011

			6*0 = 5*0;
6*55 = 5*66;
6*26565 = 5*31878;
6*12804330 = 5*15365196.

Vincenzo Librandi, Table of n, a(n) for n = 0..229
Index entries for linear recurrences with constant coefficients, signature (483,-483,1).

Cf. A001652, A029549, A053141, A075528, A200993-A201008.

I:=[0, 55, 26565]; [n le 3 select I[n] else 483*Self(n-1)-483*Self(n-2)+Self(n-3): n in [1..15]]; // Vincenzo Librandi, Dec 22 2011

LinearRecurrence[{483,-483,1},{0,55,26565},30] (* Vincenzo Librandi, Dec 22 2011 *)

makelist(expand(((11-2*sqrt(30))^(2*n+1)+(11+2*sqrt(30))^(2*n+1)-22)/192), n, 0, 11); /* Bruno Berselli, Dec 21 2011 */

concat(0,Vec(55/(1-x)/(1-482*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, Dec 23 2011

For n > 1, a(n) = 482*a(n-1) - a(n-2) + 55. See A200993 for generalization.
From Bruno Berselli, Dec 21 2011: (Start)
G.f.: 55*x/((1-x)*(1-482*x+x^2)).
a(n) = a(-n-1) = 483*a(n-1)-483*a(n-2)+a(n-3).
a(n) = ((11-2*r)^(2*n+1)+(11+2*r)^(2*n+1)-22)/192, where r=sqrt(30). (End)

a(11) corrected by Bruno Berselli, Dec 21 2011

a(6) corrected by Vincenzo Librandi, Dec 22 2011

A278620 Expansion of x/(1 - 99x + 99x^2 - x^3).

Original entry on oeis.org

0, 1, 99, 9702, 950698, 93158703, 9128602197, 894509856604, 87652837344996, 8589083549953005, 841642535058049495, 82472379352138897506, 8081451533974553906094, 791899777950154143899707, 77598096787581131548265193, 7603821585405000737586089208, 745096917272902491151888477192

Offset: 0

Bruno Berselli, Nov 24 2016

Colin Barker, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

First differences: A173205.

Cf. A045502, A108741, A123479, A200993, A200994.

P:=proc(q) local a,b,c,n; a:=0; b:=1; print(a); print(b);for n from 1 to q do
c:=98*b-a+1; a:=b; b:=c; print(b); od; end: P(100); # Paolo P. Lava, Nov 30 2016

CoefficientList[x/(1 - 99 x + 99 x^2 - x^3) + O[x]^20, x]
LinearRecurrence[{99,-99,1},{0,1,99},20] (* Harvey P. Dale, Aug 22 2020 *)

makelist(coeff(taylor(x/((1-x)*(1-98*x+x^2)), x, 0, n), x, n), n, 0, 20);

concat(0, Vec(1/(1-99*x+99*x^2-x^3) + O(x^20)))

gf = x/((1-x)*(1-98*x+x^2)); print(taylor(gf, x, 0, 20).list())

O.g.f.: x/((1 - x)*(1 - 98*x + x^2)).
E.g.f.: ((5-2*sqrt(6))*exp((5-2*sqrt(6))^2*x) + (5+2*sqrt(6))*exp((5+2*sqrt(6))^2*x) - 10*exp(x))/960.
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>2.
a(n) = 98*a(n-1) - a(n-2) + 1 for n>1.
a(n) = a(-n-1) = ((5+2*sqrt(6))^(2*n+1) + (5-2*sqrt(6))^(2*n+1))/960 - 1/96.
a(n) = floor((5+2*sqrt(6))^(2*n+1)/960).
a(n)*a(n-2) = a(n-1)*(a(n-1)-1) for n>1.
Lim_{i -> infinity} a(i)/a(i-1) = (5 + 2*sqrt(6))^2.
From the closed form: a(n) + a(-n) = A108741(n).
a(n) = A200993(n)/10 = A200994(n)/15.
a(n) = A123479(n)/20 for n>0.
a(n) = A045502(n)/40.

A200998 Triangular numbers, T(m), that are three-quarters of another triangular number: T(m) such that 4T(m)=3T(k) for some k.

Original entry on oeis.org

0, 21, 4095, 794430, 154115346, 29897582715, 5799976931385, 1125165627105996, 218276331681631860, 42344483180609474865, 8214611460706556491971, 1593592278893891349967530, 309148687493954215337208870, 59973251781548223884068553271

Offset: 0

Charlie Marion, Dec 20 2011

For n>1, a(n) = 194*a(n-1) - a (n-2) + 21. See A200993 for generalization.

			4*0 = 3*0.
4*21 = 3*28.
4*4095 = 3*5640.
4*794430 = 3*1059240.

Colin Barker, Table of n, a(n) for n = 0..425
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Cf. A001652, A029549, A053141, A075528, A200993-A201008.

I:=[0,21]; [n le 2 select I[n] else  194*Self(n-1) - Self(n-2) + 21: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016

LinearRecurrence[{195, -195, 1}, {0, 21, 4095}, 30] (* Vincenzo Librandi, Mar 03 2016 *)

concat(0, Vec(21/(1 - 195*x + 195*x^2 - x^3) + O(x^99))) \\ Charles R Greathouse IV, Dec 20 2011

G.f.: (21*x)/(1 - 195*x + 195*x^2 - x^3).
From Colin Barker, Mar 02 2016: (Start)
a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n>2.
a(n) = ((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(-7+4*sqrt(3)+(7+4*sqrt(3))*(97+56*sqrt(3))^n))/128.
(End)

A201003 Triangular numbers, T(m), that are four-fifths of another triangular number: T(m) such that 5T(m) = 4T(k) for some k.

Original entry on oeis.org

0, 36, 11628, 3744216, 1205625960, 388207814940, 125001710784756, 40250162664876528, 12960427376379457296, 4173217365031520372820, 1343763031112773180590780, 432687522800947932629858376, 139324038578874121533633806328, 44861907734874666185897455779276

Offset: 0

Charlie Marion, Dec 20 2011

Also, numbers m such that 8*m+1 and 10*m+1 are squares. Example: 8*1205625960+1 = 98209^2 and 12056259601 = 109801^2. - Bruno Berselli, Mar 03 2016

			5*0 = 4*0;
5*36 = 4*45;
5*11628 = 4*14535;
5*3744216 = 4*4680270.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A001652, A029549, A053141, A075528, A200993-A201008, A298271.

m:=20; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(36*x/((1-x)*(1-322*x+x^2)))); // G. C. Greubel, Jul 15 2018

triNums = Table[(n^2 + n)/2, {n, 0, 4999}]; Select[triNums, MemberQ[triNums, (5/4)#] &] (* Alonso del Arte, Dec 20 2011 *)
CoefficientList[Series[-36 x/((x - 1) (x^2 - 322 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 11 2014 *)
LinearRecurrence[{323,-323,1},{0,36,11628},20] (* Harvey P. Dale, Dec 21 2015 *)

concat(0, Vec(36*x/((1-x)*(1-322*x+x^2)) + O(x^15))) \\ Colin Barker, Mar 02 2016

For n>1, a(n) = 322*a(n-1) - a(n-2) + 36. See A200993 for generalization.
G.f.: 36*x / ((1-x)*(x^2-322*x+1)). - R. J. Mathar, Aug 10 2014
From Colin Barker, Mar 02 2016: (Start)
a(n) = (-18+(9-4*sqrt(5))*(161+72*sqrt(5))^(-n)+(9+4*sqrt(5))*(161+72*sqrt(5))^n)/160.
a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n>2. (End)
a(n) = 36*A298271(n). - Amiram Eldar, Dec 01 2024

A201004 Triangular numbers, T(m), that are five-quarters of another triangular number; T(m) such that 4T(m) = 5T(k) for some k.

Original entry on oeis.org

0, 45, 14535, 4680270, 1507032450, 485259768675, 156252138480945, 50312703331095660, 16200534220474321620, 5216521706289400466025, 1679703788890966475738475, 540859403501184915787322970, 174155048223592651917042257910, 56077384668593332732371819724095

Offset: 0

Charlie Marion, Feb 15 2012

			4*0 = 5*0.
4*45 = 5*36.
4*14535 = 5*11628.
4*4680270 = 5*3744216.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A001652, A029549, A053141, A075528, A200993-A201008, A298271.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(45*x/((1-x)*(1-322*x+x^2)))); // G. C. Greubel, Jul 15 2018

LinearRecurrence[{323, -323, 1}, {0, 45, 14535}, 20] (* T. D. Noe, Feb 15 2012 *)
CoefficientList[Series[-45 x/((x - 1) (x^2 - 322 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 11 2014 *)

concat(0, Vec(45*x/((1-x)*(1-322*x+x^2)) + O(x^15))) \\ Colin Barker, Mar 02 2016

For n > 1, a(n) = 322*a(n-1) - a(n-2) + 45. See A200994 for generalization.
G.f.: 45*x / ((1-x)*(x^2-322*x+1)). - R. J. Mathar, Aug 10 2014
From Colin Barker, Mar 02 2016: (Start)
a(n) = (-18 + (9-4*sqrt(5))*(161+72*sqrt(5))^(-n) + (9+4*sqrt(5))*(161+72*sqrt(5))^n)/128.
a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n > 2. (End)
a(n) = 45*A298271(n). - Amiram Eldar, Dec 01 2024

a(7) corrected by R. J. Mathar, Aug 10 2014

A352182 Twice A200994.

Original entry on oeis.org

0, 30, 2970, 291060, 28520940, 2794761090, 273858065910, 26835295698120, 2629585120349880, 257672506498590150, 25249276051741484850, 2474171380564166925180, 242443546019236617182820, 23756993338504624316991210, 2327942903627433946447955790, 228114647562150022127582676240

Offset: 0

N. J. A. Sloane, Mar 08 2022

Also 3 times A200993 and 6 times A352181.

Numbers that both doubles and triples of triangular numbers.

Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A200993, A200994, A278620, A352181.

From Chai Wah Wu, Apr 22 2024: (Start)
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n > 2.
G.f.: -30*x/((x - 1)*(x^2 - 98*x + 1)). (End)
a(n) = 30*A278620. - Hugo Pfoertner, Apr 22 2024

cp's OEIS Frontend

A352181 a(n) = A200993(n)/2.

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A200994 Triangular numbers, T(m), that are three-halves of another triangular number; T(m) such that 2*T(m) = 3*T(k) for some k.

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A201008 Triangular numbers, T(m), that are five-sixths of another triangular number: T(m) such that 6*T(m)=5*T(k) for some k.

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A278620 Expansion of x/(1 - 99*x + 99*x^2 - x^3).

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A200998 Triangular numbers, T(m), that are three-quarters of another triangular number: T(m) such that 4*T(m)=3*T(k) for some k.

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A201003 Triangular numbers, T(m), that are four-fifths of another triangular number: T(m) such that 5*T(m) = 4*T(k) for some k.

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Magma

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A201004 Triangular numbers, T(m), that are five-quarters of another triangular number; T(m) such that 4*T(m) = 5*T(k) for some k.

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A200994 Triangular numbers, T(m), that are three-halves of another triangular number; T(m) such that 2T(m) = 3T(k) for some k.

A201008 Triangular numbers, T(m), that are five-sixths of another triangular number: T(m) such that 6T(m)=5T(k) for some k.

A278620 Expansion of x/(1 - 99x + 99x^2 - x^3).

A200998 Triangular numbers, T(m), that are three-quarters of another triangular number: T(m) such that 4T(m)=3T(k) for some k.

A201003 Triangular numbers, T(m), that are four-fifths of another triangular number: T(m) such that 5T(m) = 4T(k) for some k.

A201004 Triangular numbers, T(m), that are five-quarters of another triangular number; T(m) such that 4T(m) = 5T(k) for some k.