A005891 -id:A005891 - cp's OEIS frontend

A253411 Indices of centered octagonal numbers (A016754) which are also centered pentagonal numbers (A005891).

1, 76, 646, 108871, 930811, 156991186, 1342228096, 226381180621, 1935491982901, 326441505463576, 2790978097114426, 470728424497295251, 4024588480547018671, 678790061683594287646, 5803453797970703808436, 978814798219318465489561, 8368576352085274344745321

Offset: 1

Colin Barker, Dec 31 2014

Also positive integers y in the solutions to 5*x^2 - 8*y^2 - 5*x + 8*y = 0, the corresponding values of x being A253410.

			76 is in the sequence because the 76th centered octagonal number is 22801, which is also the 96th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..633
Index entries for linear recurrences with constant coefficients, signature (1,1442,-1442,-1,1).

Cf. A005891, A016754, A253410, A253579.

LinearRecurrence[{1,1442,-1442,-1,1},{1,76,646,108871,930811},20] (* Harvey P. Dale, Feb 04 2016 *)

Vec(-x*(x^4+75*x^3-872*x^2+75*x+1)/((x-1)*(x^2-38*x+1)*(x^2+38*x+1)) + O(x^100))

a(n) = a(n-1) + 1442*a(n-2) - 1442*a(n-3) - a(n-4) + a(n-5).

G.f.: -x*(x^4 + 75*x^3 - 872*x^2 + 75*x + 1) / ((x-1)*(x^2 - 38*x + 1)*(x^2 + 38*x + 1)).

A253579 Centered pentagonal numbers (A005891) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 22801, 1666681, 47411143081, 3465632747641, 98584929298781641, 7206305041398228481, 204993755756525779060081, 14984516863488437537571601, 426256225957302372068628976801, 31158234954289838149958560780201, 886340998518823181233611960679431001

Offset: 1

Colin Barker, Jan 04 2015

			22801 is in the sequence because it is the 96th centered pentagonal number and the 76th centered octagonal 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. A005891, A016754, A253410, A253411.

Vec(-x*(x^4+22800*x^3-435482*x^2+22800*x+1)/((x-1)*(x^2-1442*x+1)*(x^2+1442*x+1)) + O(x^100))

a(n) = a(n-1)+2079362*a(n-2)-2079362*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+22800*x^3-435482*x^2+22800*x+1) / ((x-1)*(x^2-1442*x+1)*(x^2+1442*x+1)).

A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 6, 66, 781, 9301, 110826, 1320606, 15736441, 187516681, 2234463726, 26626048026, 317278112581, 3780711302941, 45051257522706, 536834378969526, 6396961290111601, 76226701102369681, 908323451938324566, 10823654722157525106, 128975533213951976701

Offset: 1

Colin Barker, Jan 06 2015

Also positive integers y in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of x being A133272.

			6 is in the sequence because the 6th centered heptagonal number is 106, which is also the 7th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..930
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (13,-13,1).

Cf. A005891, A069099, A133272, A253622.

I:=[1,6]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2)-5: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

RecurrenceTable[{a[1] == 1, a[2] == 6, a[n] == 12 a[n-1] - a[n-2] - 5}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)

Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-12*x+1)) + O(x^100))

a(n) = 13*a(n-1)-13*a(n-2)+a(n-3).
G.f.: -x*(x^2-7*x+1) / ((x-1)*(x^2-12*x+1)).
a(n) = (14-(-7+sqrt(35))*(6+sqrt(35))^n+(6-sqrt(35))^n*(7+sqrt(35)))/28. - Colin Barker, Mar 05 2016
a(n) = 12*a(n-1) - a(n-2) - 5. - Vincenzo Librandi, Mar 05 2016
a(n) = (5*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - Seiichi Manyama, Aug 11 2016

A253622 Centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 106, 15016, 2132131, 302747551, 42988020076, 6103996103206, 866724458635141, 123068769130086781, 17474898492013687726, 2481312517096813570276, 352328902529255513291431, 50028222846637186073812891, 7103655315319951166968139056

Offset: 1

Colin Barker, Jan 06 2015

			106 is in the sequence because it is the 6th centered heptagonal number and the 7th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..465
Index entries for linear recurrences with constant coefficients, signature (143,-143,1).

Cf. A005891, A069099, A133272, A253621.

LinearRecurrence[{143,-143,1},{1,106,15016},20] (* Harvey P. Dale, Feb 25 2016 *)

Vec(-x*(x^2-37*x+1)/((x-1)*(x^2-142*x+1)) + O(x^100))

a(n) = 143*a(n-1)-143*a(n-2)+a(n-3).
G.f.: -x*(x^2-37*x+1) / ((x-1)*(x^2-142*x+1)).
a(n) = (4+(6+sqrt(35))*(71+12*sqrt(35))^(-n)-(-6+sqrt(35))*(71+12*sqrt(35))^n)/16. - Colin Barker, Mar 07 2016

A253654 Indices of pentagonal numbers (A000326) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 6, 46, 361, 2841, 22366, 176086, 1386321, 10914481, 85929526, 676521726, 5326244281, 41933432521, 330141215886, 2599196294566, 20463429140641, 161108236830561, 1268402465503846, 9986111487200206, 78620489432097801, 618977803969582201, 4873201942324559806

Offset: 1

Colin Barker, Jan 07 2015

Also positive integers x in the solutions to 3*x^2-5*y^2-x+5*y-2 = 0, the corresponding values of y being A253470.

			6 is in the sequence because the 6th pentagonal number is 51, which is also the 5th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Cf. A000326, A005891, A128917, A253470.

LinearRecurrence[{9,-9,1},{1,6,46},30] (* Harvey P. Dale, Nov 12 2017 *)

Vec(-x*(x^2-3*x+1)/((x-1)*(x^2-8*x+1)) + O(x^100))

a(n) = 9*a(n-1)-9*a(n-2)+a(n-3).
G.f.: -x*(x^2-3*x+1) / ((x-1)*(x^2-8*x+1)).
a(n) = (2-(-5+sqrt(15))*(4+sqrt(15))^n+(4-sqrt(15))^n*(5+sqrt(15)))/12. - Colin Barker, Mar 03 2016

A253921 Indices of octagonal numbers (A000567) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 51, 271, 24421, 130461, 11770711, 62881771, 5673458121, 30308883001, 2734595043451, 14608818724551, 1318069137485101, 7041420316350421, 635306589672775071, 3393949983662178211, 306216458153140098961, 1635876850704853547121, 147595697523223854923971

Offset: 1

Colin Barker, Jan 19 2015

Also positive integers x in the solutions to 6*x^2 - 5*y^2 - 4*x + 5*y - 2 = 0, the corresponding values of y being A253922.

			51 is in the sequence because the 51st octagonal number is 7701, which is also the 56th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..745
Index entries for linear recurrences with constant coefficients, signature (1,482,-482,-1,1).

Cf. A000567, A005891, A253922, A253923.

I:=[1,51,271,24421,130461]; [n le 5 select I[n] else Self(n-1)+482*Self(n-2)-482*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Jan 20 2015

CoefficientList[Series[(x^4 + 50 x^3 - 262 x^2 + 50 x + 1)/((1 - x) (x^2 - 22 x + 1) (x^2 + 22 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 20 2015 *)

Vec(-x*(x^4+50*x^3-262*x^2+50*x+1)/((x-1)*(x^2-22*x+1)*(x^2+22*x+1)) + O(x^100))

a(n) = a(n-1)+482*a(n-2)-482*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+50*x^3-262*x^2+50*x+1) / ((x-1)*(x^2-22*x+1)*(x^2+22*x+1)).

A253922 Indices of centered pentagonal numbers (A005891) which are also octagonal numbers (A000567).

Original entry on oeis.org

1, 56, 297, 26752, 142913, 12894168, 68883529, 6214961984, 33201717825, 2995598781880, 16003159107881, 1443872397903936, 7713489488280577, 695943500190915032, 3717885930192129993, 335443323219623141248, 1792013304863118375809, 161682985848358163166264

Offset: 1

Colin Barker, Jan 19 2015

Also positive integers y in the solutions to 6*x^2 - 5*y^2 - 4*x + 5*y - 2 = 0, the corresponding values of x being A253921.

			56 is in the sequence because the 56th centered pentagonal is 7701, which is also the number 51st octagonal number.

Colin Barker, Table of n, a(n) for n = 1..745
Index entries for linear recurrences with constant coefficients, signature (1,482,-482,-1,1).

Cf. A000567, A005891, A253921, A253923.

I:=[1,56,297,26752,142913]; [n le 5 select I[n] else Self(n-1)+482*Self(n-2)-482*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Jan 20 2015

CoefficientList[Series[(55 x^3 + 241 x^2 - 55 x - 1)/((x - 1)(x^2 - 22 x + 1) (x^2 + 22 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 20 2015 *)

Vec(x*(55*x^3+241*x^2-55*x-1)/((x-1)*(x^2-22*x+1)*(x^2+22*x+1)) + O(x^100))

a(n) = a(n-1)+482*a(n-2)-482*a(n-3)-a(n-4)+a(n-5).

G.f.: x*(55*x^3+241*x^2-55*x-1) / ((x-1)*(x^2-22*x+1)*(x^2+22*x+1)).

A253923 Octagonal numbers (A000567) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 7701, 219781, 1789106881, 51059956641, 415648888795141, 11862351246525781, 96564381140875635681, 2755885166244302532001, 22434030154994860543881301, 640252753580346501593005701, 5211918753572151610134715970401, 148744800214537374776845967930881

Offset: 1

Colin Barker, Jan 19 2015

			7701 is in the sequence because it is the 51st octagonal number and the 56th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..373
Index entries for linear recurrences with constant coefficients, signature (1,232322,-232322,-1,1).

Cf. A000567, A005891, A253921, A253922.

I:=[1,7701,219781,1789106881,51059956641]; [n le 5 select I[n] else Self(n-1)+232322*Self(n-2)-232322*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Jan 20 2015

CoefficientList[Series[(x^4 + 7700 x^3 - 20242 x^2 + 7700 x + 1) / ((1 - x) (x^2 - 482 x + 1) (x^2 + 482 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 20 2015 *)

Vec(-x*(x^4+7700*x^3-20242*x^2+7700*x+1)/((x-1)*(x^2-482*x+1)*(x^2+482*x+1)) + O(x^100))

a(n) = a(n-1)+232322*a(n-2)-232322*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+7700*x^3-20242*x^2+7700*x+1) / ((x-1)*(x^2-482*x+1)*(x^2+482*x+1)).

A254626 Indices of triangular numbers (A000217) that are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 3, 23, 61, 421, 1103, 7563, 19801, 135721, 355323, 2435423, 6376021, 43701901, 114413063, 784198803, 2053059121, 14071876561, 36840651123, 252509579303, 661078661101, 4531100550901, 11862575248703, 81307300336923, 212865275815561, 1459000305513721

Offset: 1

Colin Barker, Feb 03 2015

Also positive integers x in the solutions to x^2 - 5*y^2 + x + 5*y - 2 = 0, the corresponding values of y being A254627.

			3 is in the sequence because the 3rd triangular number is 6, which is also the 2nd centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-1,1).

Cf. A000217, A005891, A254627, A254628.

LinearRecurrence[{1,18,-18,-1,1},{1,3,23,61,421},30] (* Harvey P. Dale, Jun 15 2024 *)

Vec(-x*(x+1)^2*(x^2+1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))

a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(x+1)^2*(x^2+1) / ((x-1)*(x^2-4*x-1)*(x^2+4*x-1)).
a(n) = (-2 + (2-r)^n - (-2-r)^n*(-2+r) + 2*(-2+r)^n + r*(-2+r)^n + (2+r)^n)/4 where r = sqrt(5). - Colin Barker, Nov 25 2016

A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 11, 231, 5061, 111101, 2439151, 53550211, 1175665481, 25811090361, 566668322451, 12440892003551, 273132955755661, 5996484134620981, 131649518005905911, 2890292911995309051, 63454794545890893201, 1393115187097604341361, 30585079321601404616731

Offset: 1

Colin Barker, Feb 07 2015

Also positive integers y in the solutions to 5*x^2 - 6*y^2 - 5*x + 6*y = 0, the corresponding values of x being A133285.

The numbers (as opposed to the indices) are A133141.

			11 is in the sequence because the 11th centered hexagonal number is 331, which is also the 12th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..746
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (23,-23,1).

Cf. A003215, A005891, A133141, A133285.

LinearRecurrence[{23,-23,1},{1,11,231},20] (* Harvey P. Dale, Mar 01 2022 *)

Vec(-x*(x^2-12*x+1)/((x-1)*(x^2-22*x+1)) + O(x^100))

a(n) = 23*a(n-1)-23*a(n-2)+a(n-3).
G.f.: -x*(x^2-12*x+1) / ((x-1)*(x^2-22*x+1)).
a(n) = 1/2+1/24*(11+2*sqrt(30))^(-n)*(6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n)). - Colin Barker, Mar 03 2016

cp's OEIS Frontend

A253411 Indices of centered octagonal numbers (A016754) which are also centered pentagonal numbers (A005891).

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A253579 Centered pentagonal numbers (A005891) which are also centered octagonal numbers (A016754).

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A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

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A253622 Centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

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A253654 Indices of pentagonal numbers (A000326) which are also centered pentagonal numbers (A005891).

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A253921 Indices of octagonal numbers (A000567) which are also centered pentagonal numbers (A005891).

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A253922 Indices of centered pentagonal numbers (A005891) which are also octagonal numbers (A000567).

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