cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 63 results. Next

A253880 Triangular numbers (A000217) that are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 253, 64261, 16322041, 4145734153, 1053000152821, 267457893082381, 67933251842771953, 17254778510170993681, 4382645808331589623021, 1113174780537713593253653, 282742011610770921096804841, 71815357774355276244995175961, 18240818132674629395307677889253
Offset: 1

Views

Author

Colin Barker, Jan 17 2015

Keywords

Examples

			253 is in the sequence because it is the 22nd triangular number and the 9th centered heptagonal number.

Links

Crossrefs

Cf. A000217, A069099, A253878, A253879.
Similar sequences of the type cosh((2*m+1)*arccosh(k))/k are listed in A302329. This is the case k=8.

Programs

  • Mathematica
    LinearRecurrence[{254,-1},{1,253},20] (* Harvey P. Dale, May 17 2017 *)
  • PARI
    Vec(-x*(x-1)/(x^2-254*x+1) + O(x^100))

Formula

a(n) = 254*a(n-1) - a(n-2).
G.f.: -x*(x-1) / (x^2 - 254*x + 1).
a(n) = (1/8)*T(2*n-1, 8), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Jul 08 2022

A322640 Numbers that are sums of consecutive centered heptagonal numbers (A069099).

Original entry on oeis.org

0, 1, 8, 9, 22, 30, 31, 43, 65, 71, 73, 74, 106, 114, 136, 144, 145, 148, 177, 197, 220, 242, 250, 251, 253, 254, 316, 325, 345, 368, 386, 390, 398, 399, 450, 451, 463, 522, 547, 565, 569, 587, 595, 596, 598, 638, 702, 704, 736, 766, 775, 818, 840, 841, 848, 849, 914, 953
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A004126, A069099, A152043, A322610, A322611, A322636, A322638, A320728.

Programs

  • Mathematica
    terms = 58;
nmax = 17; kmax =  8; (* empirical *)
T = Table[(7 n^2 - 7 n + 2)/2, {n, 1, nmax}];
Union[{0}, T, Table[k MovingAverage[T, k], {k, 2, kmax}] // Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 27 2018 *)

A253460 Indices of centered heptagonal numbers (A069099) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 16, 112, 3937, 28321, 999856, 7193296, 253959361, 1827068737, 64504677712, 464068265776, 16383934179361, 117871512438241, 4161454776879856, 29938900091047312, 1056993129393303937, 7604362751613578881, 268472093411122320016, 1931478200009757988336
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers y in the solutions to 4*x^2 - 7*y^2 - 4*x + 7*y = 0, the corresponding values of x being A253459.

Examples

			16 is in the sequence because the 16th centered heptagonal number is 841, which is also the 21st centered square number.

Links

Crossrefs

Cf. A001844, A069099, A253459, A253599.

Programs

  • PARI
    Vec(-x*(x^4+15*x^3-158*x^2+15*x+1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+254*a(n-2)-254*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+15*x^3-158*x^2+15*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).
a(n) = A105040(n) + 1. - Michel Marcus, Mar 12 2024

A133272 Indices of centered heptagonal numbers (A069099) which are also heptagonal numbers (A000566).

Original entry on oeis.org

1, 7, 78, 924, 11005, 131131, 1562562, 18619608, 221872729, 2643853135, 31504364886, 375408525492, 4473397941013, 53305366766659, 635191003258890, 7568986672340016, 90192649064821297, 1074742802105515543
Offset: 1

Views

Author

Richard Choulet, Oct 16 2007

Keywords

Comments

Numbers X such that 140*X^2-140*X+49 is a square.
Also positive integers x in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of y being A253621. - Colin Barker, Jan 06 2015
Also indices of centered pentagonal numbers (A005891) which are also centered heptagonal numbers (A069099). - Colin Barker, Jan 06 2015

Links

Crossrefs

Cf. A000566, A069099, A128919, A253621, A253622.

Programs

  • Mathematica
    LinearRecurrence[{13,-13,1},{1,7,78},25] (* Paolo Xausa, Jan 07 2024 *)
  • PARI
    Vec(x*(6*x-1)/((x-1)*(x^2-12*x+1)) + O(x^100)) \\ Colin Barker, Jan 06 2015

Formula

a(n+2) = 12*a(n+1) - a(n) - 5.
a(n+1) = 6*a(n) - 5/2 + (1/2)*sqrt(140*a(n)^2 - 140*a(n) + 49).
G.f.: x*(-1+6*x)/((-1+x)*(1-12*x+x^2)). - R. J. Mathar, Nov 14 2007
a(n) = 13*a(n-1) - 13*a(n-2) + a(n-3). - Colin Barker, Jan 06 2015

Extensions

More terms from Paolo P. Lava, Jul 14 2008

A253446 Indices of centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 16, 465, 13920, 417121, 12499696, 374573745, 11224712640, 336366805441, 10079779450576, 302057016711825, 9051630721904160, 271246864640412961, 8128354308490484656, 243579382390074126705, 7299253117393733316480, 218734014139421925367681
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers x in the solutions to 7*x^2 - 8*y^2 - 7*x + 8*y = 0, the corresponding values of y being A253447.

Examples

			16 is in the sequence because the 16th centered heptagonal number is 841, which is also the 15th centered octagonal number.

Links

Crossrefs

Cf. A016754, A069099, A253447, A253514.

Programs

  • Mathematica
    LinearRecurrence[{31,-31,1},{1,16,465},20] (* Harvey P. Dale, Oct 04 2023 *)
  • PARI
    Vec(x*(15*x-1)/((x-1)*(x^2-30*x+1)) + O(x^100))

Formula

a(n) = 31*a(n-1)-31*a(n-2)+a(n-3).
G.f.: x*(15*x-1) / ((x-1)*(x^2-30*x+1)).
a(n) = sqrt((-2-(15-4*sqrt(14))^n-(15+4*sqrt(14))^n)*(2-(15-4*sqrt(14))^(1+n)-(15+4*sqrt(14))^(1+n)))/(4*sqrt(7)). - Gerry Martens, Jun 04 2015

A253447 Indices of centered octagonal numbers (A016754) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 15, 435, 13021, 390181, 11692395, 350381655, 10499757241, 314642335561, 9428770309575, 282548466951675, 8467025238240661, 253728208680268141, 7603379235169803555, 227847648846413838495, 6827826086157245351281, 204606934935870946699921
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers y in the solutions to 7*x^2 - 8*y^2 - 7*x + 8*y = 0, the corresponding values of x being A253446.

Examples

			15 is in the sequence because the 15th centered octagonal number is 841, which is also the 16th centered heptagonal number.

Links

Crossrefs

Cf. A016754, A069099, A253446, A253514.

Programs

  • PARI
    Vec(-x*(x^2-16*x+1)/((x-1)*(x^2-30*x+1)) + O(x^100))

Formula

a(n) = 31*a(n-1)-31*a(n-2)+a(n-3).
G.f.: -x*(x^2-16*x+1) / ((x-1)*(x^2-30*x+1)).
a(n) = (8+(4+sqrt(14))*(15+4*sqrt(14))^(-n)-(-4+sqrt(14))*(15+4*sqrt(14))^n)/16. - Colin Barker, Mar 03 2016

A253457 Indices of centered hexagonal numbers (A003215) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 14, 351, 9100, 236237, 6133050, 159223051, 4133666264, 107316099801, 2786084928550, 72330892042487, 1877817108176100, 48750913920536101, 1265645944825762514, 32858043651549289251, 853043488995455758000, 22146272670230300418737, 574950045936992355129150
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers x in the solutions to 6*x^2 - 7*y^2 - 6*x + 7*y = 0, the corresponding values of y being A253458.

Examples

			14 is in the sequence because the 14th centered hexagonal number is 547, which is also the 13th centered heptagonal number.

Links

Crossrefs

Cf. A003215, A069099, A253458, A253546.

Programs

  • PARI
    Vec(x*(13*x-1)/((x-1)*(x^2-26*x+1)) + O(x^100))

Formula

a(n) = 27*a(n-1)-27*a(n-2)+a(n-3).
G.f.: x*(13*x-1) / ((x-1)*(x^2-26*x+1)).
a(n) = sqrt((-2-(13-2*sqrt(42))^n-(13+2*sqrt(42))^n)*(2-(13-2*sqrt(42))^(1+n)-(13+2*sqrt(42))^(1+n)))/(4*sqrt(6)). - Gerry Martens, Jun 04 2015

A253458 Indices of centered heptagonal numbers (A069099) which are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 13, 325, 8425, 218713, 5678101, 147411901, 3827031313, 99355402225, 2579413426525, 66965393687413, 1738520822446201, 45134575989913801, 1171760454915312613, 30420637251808214125, 789764808092098254625, 20503464373142746406113, 532300308893619308304301
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers y in the solutions to 6*x^2 - 7*y^2 - 6*x + 7*y = 0, the corresponding values of x being A253457.

Examples

			13 is in the sequence because the 13th centered heptagonal number is 547, which is also the 14th centered hexagonal number.

Links

Crossrefs

Cf. A003215, A069099, A253457, A253546.

Programs

  • Mathematica
    LinearRecurrence[{27,-27,1},{1,13,325},20] (* Harvey P. Dale, Oct 13 2022 *)
  • PARI
    Vec(-x*(x^2-14*x+1)/((x-1)*(x^2-26*x+1)) + O(x^100))

Formula

a(n) = 27*a(n-1)-27*a(n-2)+a(n-3).
G.f.: -x*(x^2-14*x+1) / ((x-1)*(x^2-26*x+1)).
a(n) = 1/2+(13+2*sqrt(42))^(-n)*(7+sqrt(42)-(-7+sqrt(42))*(13+2*sqrt(42))^(2*n))/28. - Colin Barker, Mar 03 2016

A253459 Indices of centered square numbers (A001844) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 21, 148, 5208, 37465, 1322685, 9515836, 335956656, 2416984753, 85331667813, 613904611300, 21673907667720, 155929354285321, 5505087215932941, 39605442083860108, 1398270478939299168, 10059626359946181985, 355155196563366055605, 2555105489984246363956
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers x in the solutions to 4*x^2 - 7*y^2 - 4*x + 7*y = 0, the corresponding values of y being A253460.

Examples

			21 is in the sequence because the 21st centered square number is 841, which is also the 16th centered heptagonal number.

Links

Crossrefs

Cf. A001844, A069099, A253460, A253599.

Programs

  • PARI
    Vec(x*(20*x^3+127*x^2-20*x-1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+254*a(n-2)-254*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(20*x^3+127*x^2-20*x-1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

A253476 Indices of centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 15, 70, 1596, 7645, 175491, 840826, 19302360, 92483161, 2123084055, 10172306830, 233519943636, 1118861268085, 25685070715851, 123064567182466, 2825124258799920, 13535983528803121, 310737983397275295, 1488835123601160790, 34178353049441482476
Offset: 1

Views

Author

Colin Barker, Jan 02 2015

Keywords

Comments

Also positive integers x in the solutions to 3*x^2 - 7*y^2 - 3*x + 7*y = 0, the corresponding values of y being A253477.

Examples

			15 is in the sequence because the 15th centered triangular number is 316, which is also the 10th centered heptagonal number.

Links

Crossrefs

Cf. A005448, A069099, A253477, A253689.

Programs

  • Mathematica
    LinearRecurrence[{1,110,-110,-1,1},{1,15,70,1596,7645},30] (* Harvey P. Dale, Jun 14 2016 *)
  • PARI
    Vec(x*(14*x^3+55*x^2-14*x-1)/((x-1)*(x^4-110*x^2+1)) + O(x^100))

Formula

a(n) = a(n-1)+110*a(n-2)-110*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(14*x^3+55*x^2-14*x-1) / ((x-1)*(x^4-110*x^2+1)).
Showing 1-10 of 63 results. Next

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