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 1-7 of 7 results.

A263942 Positive integers n such that (n+4)^3 - n^3 is a square.

Original entry on oeis.org

6, 28, 110, 416, 1558, 5820, 21726, 81088, 302630, 1129436, 4215118, 15731040, 58709046, 219105148, 817711550, 3051741056, 11389252678, 42505269660, 158631825966, 592022034208, 2209456310870, 8245803209276, 30773756526238, 114849222895680, 428623135056486
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			6 is in the sequence because (6+4)^3 - 6^3 = 28^2.

Links

Crossrefs

Cf. A263943 (21), A263944 (28), A263945 (39), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{5, -5, 1}, {6, 28, 110}, 30] (* Paolo Xausa, Mar 04 2024 *)
  • PARI
    Vec(2*x*(x-3)/((x-1)*(x^2-4*x+1)) + O(x^40))

Formula

a(n) = 5*a(n-1)-5*a(n-2)+a(n-3) for n>3.
G.f.: 2*x*(x-3) / ((x-1)*(x^2-4*x+1)).

A263943 Positive integers n such that (n+21)^3 - n^3 is a square.

Original entry on oeis.org

7, 119, 4564, 32900, 1161895, 8359127, 295119412, 2123188004, 74959171399, 539281396535, 19039334418580, 136975351534532, 4835915983150567, 34791200008377239, 1228303620385828084, 8836827826776286820, 311984283662017185415, 2244519476801168477687
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			7 is in the sequence because (7+21)^3 - 7^3 = 147^2.

Links

Crossrefs

Cf. A263942 (4), A263944 (28), A263945 (39), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{1,254,-254,-1,1},{7,119,4564,32900,1161895},20] (* Harvey P. Dale, Jan 11 2017 *)
  • PARI
    Vec(7*x*(4*x^4+16*x^3-381*x^2-16*x-1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^30))

Formula

a(n) = a(n-1)+254*a(n-2)-254*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: 7*x*(4*x^4+16*x^3-381*x^2-16*x-1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

A263944 Positive integers n such that (n+28)^3 - n^3 is a square.

Original entry on oeis.org

28, 189, 959, 4648, 22323, 107009, 512764, 2456853, 11771543, 56400904, 270233019, 1294764233, 6203588188, 29723176749, 142412295599, 682338301288, 3269279210883, 15664057753169, 75051009555004, 359590990021893, 1722903940554503, 8254928712750664
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			189 is in the sequence because (189+28)^3 - 189^3 = 1862^2.

Links

Crossrefs

Cf. A263942 (4), A263943 (21), A263945 (39), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{6,-6,1},{28,189,959},30] (* Harvey P. Dale, Dec 14 2016 *)
  • PARI
    Vec(7*x*(x-4)*(x+1)/((x-1)*(x^2-5*x+1)) + O(x^40))

Formula

a(n) = 6*a(n-1)-6*a(n-2)+a(n-3) for n>3.
G.f.: 7*x*(x-4)*(x+1) / ((x-1)*(x^2-5*x+1)).
a(n) = 7*(-2+(2^(-1-n)*((5-sqrt(21))^n*(-7+sqrt(21))+(5+sqrt(21))^n*(7+sqrt(21))))/sqrt(21)). - Colin Barker, Mar 05 2016

A263945 Positive integers n such that (n+39)^3 - n^3 is a square.

Original entry on oeis.org

26, 871, 59930, 1155895, 77814386, 1500376111, 101003038370, 1947487061455, 131101866015146, 2527836705417751, 170170121084646410, 3281130096145204615, 220880686066005050306, 4258904336959770197791, 286702960343553470676050, 5528054548243685571553375
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			26 is in the sequence because (26+39)^3 - 26^3 = 507^2.

Links

Crossrefs

Cf. A263942 (4), A263943 (21), A263944 (28), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{1,1298,-1298,-1,1},{26,871,59930,1155895,77814386},20] (* Harvey P. Dale, Mar 25 2020 *)
  • PARI
    Vec(13*x*(5*x^4+65*x^3-1947*x^2-65*x-2)/((x-1)*(x^2-36*x-1)*(x^2+36*x-1)) + O(x^30))

Formula

a(n) = a(n-1)+1298*a(n-2)-1298*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: 13*x*(5*x^4+65*x^3-1947*x^2-65*x-2) / ((x-1)*(x^2-36*x-1)*(x^2+36*x-1)).

A263946 Positive integers n such that (n+52)^3 - n^3 is a square.

Original entry on oeis.org

26, 2626, 132522, 6624722, 331104826, 16548617826, 827099787722, 41338440769522, 2066094938689626, 103263408493713026, 5161104329746962922, 257951953078854434322, 12892436549612974754426, 644363875527569883288226, 32205301339828881189658122
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			26 is in the sequence because (26+52)^3 - 26^3 = 676^2.

Links

Crossrefs

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{51,-51,1},{26,2626,132522},20] (* Harvey P. Dale, Feb 05 2019 *)
  • PARI
    Vec(26*x*(3*x^2-50*x-1)/((x-1)*(x^2-50*x+1)) + O(x^30))

Formula

a(n) = 51*a(n-1)-51*a(n-2)+a(n-3) for n>3.
G.f.: 26*x*(3*x^2-50*x-1) / ((x-1)*(x^2-50*x+1)).
a(n) = 26*(-6-(6+sqrt(39))*(25+4*sqrt(39))^(-n)+(-6+sqrt(39))*(25+4*sqrt(39))^n)/6. - Colin Barker, Mar 03 2016

A263947 Positive integers n such that (n+57)^3 - n^3 is a square.

Original entry on oeis.org

551, 13471, 67002512, 1560515752, 7745359676111, 180392503180711, 895348087775371352, 20853012581126608912, 103500448242912021166871, 2410566548172681237123151, 11964444815088795735075876992, 278656671814812593067838694872, 1383065891631134161140389210648831
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			551 is in the sequence because (551+57)^3 - 551^3 = 7581^2.

Links

Crossrefs

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263946 (52), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{1, 115598, -115598, -1, 1}, {551, 13471, 67002512, 1560515752, 7745359676111}, 15] (* Paolo Xausa, Mar 05 2024 *)
  • PARI
    Vec(19*x*(32*x^4+680*x^3-173397*x^2-680*x-29)/((x-1)*(x^2-340*x+1)*(x^2+340*x+1)) + O(x^20))

Formula

a(n) = a(n-1)+115598*a(n-2)-115598*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: 19*x*(32*x^4+680*x^3-173397*x^2-680*x-29) / ((x-1)*(x^2-340*x+1)*(x^2+340*x+1)).

A263948 Positive integers n such that (n+61)^3 - n^3 is a square.

Original entry on oeis.org

244, 267607, 260678620, 253900737919, 247299058084132, 240869028673236295, 234606186628674096844, 228506184907299897119407, 222564789493523471120235220, 216777876460506953571212014519, 211141429107744279254889381935932, 205651535173066467487308686793612895
Offset: 1

Views

Author

Colin Barker, Oct 30 2015

Keywords

Examples

			244 is in the sequence because (244+61)^3 - 244^3 = 3721^2.

Links

Crossrefs

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263946 (52), A263947 (57), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

Programs

  • Mathematica
    LinearRecurrence[{975, -975, 1}, {244, 267607, 260678620}, 15] (* Paolo Xausa, Mar 05 2024 *)
  • PARI
    Vec(61*x*(5*x^2-487*x-4)/((x-1)*(x^2-974*x+1)) + O(x^15))

Formula

a(n) = 975*a(n-1)-975*a(n-2)+a(n-3) for n>3.
G.f.: 61*x*(5*x^2-487*x-4) / ((x-1)*(x^2-974*x+1)).
Showing 1-7 of 7 results.

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