A259167 -id:A259167 - cp's OEIS frontend

A259156 Positive triangular numbers (A000217) that are pentagonal numbers (A000326) divided by 2.

6, 105, 58311, 1008910, 559902916, 9687554415, 5376187741821, 93019896484620, 51622154137063026, 893177036357767525, 495675918647891434531, 8576285810087387291130, 4759480119234899417304336, 82349495455282056411663435, 45700527609217585557064800441

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000217 and A193866 (even pentagonal numbers divided by 2). - Michel Marcus, Jun 20 2015

			6 is in the sequence because 6 is the 3rd triangular number, and 2*6 is the 3rd pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..502
Index entries for linear recurrences with constant coefficients, signature (1,9602,-9602,-1,1).

Cf. A000217, A000326, A193866, A259157-A259167.

LinearRecurrence[{1, 9602, -9602, -1, 1}, {6, 105, 58311, 1008910, 559902916}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-x*(x^3+594*x^2+99*x+6)/((x-1)*(x^2-98*x+1)*(x^2+98*x+1)) + O(x^20))

G.f.: -x*(x^3+594*x^2+99*x+6) / ((x-1)*(x^2-98*x+1)*(x^2+98*x+1)).

A259158 Positive triangular numbers (A000217) that are heptagonal numbers (A000566) divided by 2.

Original entry on oeis.org

171, 355817826, 739874066553765, 1538466018777014327028, 3199027777735470191011289295, 6651936797966044301302839516949686, 13831784604089070782667939513452521937961, 28761287297921206463992005998270360082445758120

Offset: 1

Colin Barker, Jun 19 2015

			171 is in the sequence because 171 is the 18th triangular number, and 2*171 is the 12th heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..158
Index entries for linear recurrences with constant coefficients, signature (2079363,-2079363,1).

Cf. A000217, A000566, A259156, A259157, A259159-A259167.

LinearRecurrence[{2079363, -2079363, 1}, {171, 355817826, 739874066553765}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-171*x*(1443*x+1) / ((x-1)*(x^2-2079362*x+1)) + O(x^20))

G.f.: -171*x*(1443*x+1) / ((x-1)*(x^2-2079362*x+1)).

A259157 Positive triangular numbers (A000217) that are hexagonal numbers (A000384) divided by 2.

Original entry on oeis.org

3, 3570, 4119885, 4754343828, 5486508657735, 6331426236682470, 7306460390622912753, 8431648959352604634600, 9730115592632515125415755, 11228544962248963102125146778, 12957731156319710787337293966165, 14953210525847983999624135111807740

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000217 and A033991 (even hexagonal numbers divided by 2). - Michel Marcus, Jun 20 2015

			3 is in the sequence because 3 is the 2nd triangular number, and 2*3 is the 2nd hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..327
Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

Cf. A000217, A000384, A033991, A259156, A259158-A259167.

LinearRecurrence[{1155, -1155, 1}, {3, 3570, 4119885}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-3*x*(35*x+1)/((x-1)*(x^2-1154*x+1)) + O(x^20))

G.f.: -3*x*(35*x+1) / ((x-1)*(x^2-1154*x+1)).

a(n) = 1155*a(n-1) - 1155*a(n-2) + a(n-3). - Wesley Ivan Hurt, Aug 04 2025

A259159 Positive squares (A000290) that are heptagonal numbers (A000566) divided by 2.

Original entry on oeis.org

9, 938961, 97353360225, 10093791093915321, 1046544448101974957529, 108507821458015176452612289, 11250307943363385076857772396401, 1166454428075294670080752381151042025, 120940328000452394039949183305644566845481

Offset: 1

Colin Barker, Jun 19 2015

			9 is in the sequence because 9 is the 3rd square, and 2*9 is the 3rd heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (103683,-103683,1).

Cf. A000290, A000566, A259156-A259158, A259160-A259167.

LinearRecurrence[{103683, -103683, 1},{9, 938961, 97353360225}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-9*x*(x^2+646*x+1)/((x-1)*(x^2-103682*x+1)) + O(x^20))

G.f.: -9*x*(x^2+646*x+1) / ((x-1)*(x^2-103682*x+1)).

A259160 Positive squares (A000290) that are octagonal numbers (A000567) divided by 2.

Original entry on oeis.org

4, 39204, 376437604, 3614553835204, 34706945549192004, 333256087548787788004, 3199924917936514791223204, 30725678728770327476537417604, 295027963953727766493197492611204, 2832858479158015285097354847515364004, 27201106821847298813777034752645032556004

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000290 and A033579 (even octagonal numbers divided by 2). - Michel Marcus, Jun 20 2015

			4 is in the sequence because 4 is the 2nd square, and 2*4 is the 2nd octagonal number.

Colin Barker, Table of n, a(n) for n = 1..251
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Cf. A000290, A000567, A033579, A259156-A259159, A259161-A259167.

LinearRecurrence[{9603, -9603, 1}, {4, 39204, 376437604}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-4*x*(x^2+198*x+1)/((x-1)*(x^2-9602*x+1)) + O(x^20))

G.f.: -4*x*(x^2+198*x+1) / ((x-1)*(x^2-9602*x+1)).

A259161 Positive pentagonal numbers (A000326) that are triangular numbers (A000217) divided by 2.

Original entry on oeis.org

5, 48510, 465793515, 4472549283020, 42945417749765025, 412361896760694487530, 3959498889750770719498535, 38019107927025003687930446040, 365059470355795195660737423378045, 3505300996337237541709397051345542550, 33657899801770684519698434826282476187555

Offset: 1

Colin Barker, Jun 19 2015

			5 is in the sequence because 5 is the 2nd pentagonal number, and 2*5 is the 4th triangular number.

Colin Barker, Table of n, a(n) for n = 1..251
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Cf. A000217, A000326, A074378, A259156-A259160, A259162-A259167.

LinearRecurrence[{9603, -9603, 1}, {5, 48510, 465793515}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-5*x*(99*x+1)/((x-1)*(x^2-9602*x+1)) + O(x^20))

G.f.: -5*x*(99*x+1) / ((x-1)*(x^2-9602*x+1)).

A259162 Positive hexagonal numbers (A000384) that are pentagonal numbers (A000326) divided by 2.

Original entry on oeis.org

6, 58311, 559902916, 5376187741821, 51622154137063026, 495675918647891434531, 4759480119234899417304336, 45700527609217585557064800441, 438816461344227137284036796530846, 4213515616126741362983735763224383551, 40458176507232509223142693514443734326556

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000384 and A193866 (even pentagonal numbers divided by 2). - Michel Marcus, Jun 20 2015

			6 is in the sequence because 6 is the 2nd hexagonal number, and 2*6 is the 3rd pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..251
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Cf. A000326, A000384, A193866, A259156-A259161, A259163-A259167.

LinearRecurrence[{9603, -9603, 1}, {6, 58311, 559902916}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-x*(x^2+693*x+6)/((x-1)*(x^2-9602*x+1)) + O(x^20))

G.f.: -x*(x^2+693*x+6) / ((x-1)*(x^2-9602*x+1)).

A259163 Positive heptagonal numbers (A000566) that are triangular numbers (A000217) divided by 2.

Original entry on oeis.org

18, 189, 37727235, 393298308, 78448579122960, 817809556618215, 163122994382238923193, 1700522115268371779430, 339191755844562643229618814, 3536001066647854270462804353, 705302447816298343956844397692383, 7352626249945315029422809413582264

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000566 and A074378 (even triangular numbers divided by 2). - Michel Marcus, Jun 20 2015

			18 is in the sequence because 18 is the 3rd heptagonal number, and 2*18 is the 8th triangular number.

Colin Barker, Table of n, a(n) for n = 1..317
Index entries for linear recurrences with constant coefficients, signature (1,2079362,-2079362,-1,1).

Cf. A000217, A000566, A074378, A259156-A259162, A259164-A259167.

LinearRecurrence[{1, 2079362, -2079362, -1, 1}, {18, 189, 37727235, 393298308, 78448579122960}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-9*x*(2*x^4+19*x^3+33170*x^2+19*x+2)/((x-1)*(x^2-1442*x+1)*(x^2+1442*x+1)) + O(x^20))

G.f.: -9*x*(2*x^4+19*x^3+33170*x^2+19*x+2) / ((x-1)*(x^2-1442*x+1)*(x^2+1442*x+1)).

A259164 Positive heptagonal numbers (A000566) that are squares (A000290) divided by 2.

Original entry on oeis.org

18, 1877922, 194706720450, 20187582187830642, 2093088896203949915058, 217015642916030352905224578, 22500615886726770153715544792802, 2332908856150589340161504762302084050, 241880656000904788079898366611289133690962

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000566 and A001105 (even squares divided by 2). - Michel Marcus, Jun 20 2015

			18 is in the sequence because 18 is the 3rd heptagonal number, and 2*18 is the 6th square.

Colin Barker, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (103683,-103683,1).

Cf. A000290, A000566, A001105, A259156-A259163, A259165-A259167.

LinearRecurrence[{103683, -103683, 1}, {18, 1877922, 194706720450}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-18*x*(x^2+646*x+1)/((x-1)*(x^2-103682*x+1)) + O(x^20))

G.f.: -18*x*(x^2+646*x+1) / ((x-1)*(x^2-103682*x+1)).

A259165 Positive heptagonal numbers (A000566) that are pentagonal numbers (A000326) divided by 2.

Original entry on oeis.org

540, 125482435, 29152330292086, 6772727677992549813, 1573453639577432827392256, 365547896447135621647431177175, 84924818396817988252797073116286650, 19729903659220000770419185998874515952681, 4583690677832384200588508141377728222042497188

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000566 and A193866 (even pentagonal numbers divided by 2). - Michel Marcus, Jun 20 2015

			540 is in the sequence because 540 is the 15th heptagonal number, and 2*540 is the 27th pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..186
Index entries for linear recurrences with constant coefficients, signature (232323,-232323,1).

Cf. A000326, A000566, A193866, A259156-A259164, A259166, A259167.

LinearRecurrence[{232323, -232323, 1}, {540, 125482435, 29152330292086}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-x*(x^2+28015*x+540)/((x-1)*(x^2-232322*x+1)) + O(x^20))

G.f.: -x*(x^2+28015*x+540) / ((x-1)*(x^2-232322*x+1)).

cp's OEIS Frontend

A259156 Positive triangular numbers (A000217) that are pentagonal numbers (A000326) divided by 2.

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A259158 Positive triangular numbers (A000217) that are heptagonal numbers (A000566) divided by 2.

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A259157 Positive triangular numbers (A000217) that are hexagonal numbers (A000384) divided by 2.

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A259159 Positive squares (A000290) that are heptagonal numbers (A000566) divided by 2.

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A259160 Positive squares (A000290) that are octagonal numbers (A000567) divided by 2.

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A259161 Positive pentagonal numbers (A000326) that are triangular numbers (A000217) divided by 2.

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A259162 Positive hexagonal numbers (A000384) that are pentagonal numbers (A000326) divided by 2.

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