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 1-8 of 8 results.

A178185 Numerator of Sum_{k=1..n} 1/2^((k^2 + 3*k)/2).

Original entry on oeis.org

1, 9, 145, 4641, 297025, 38019201, 9732915457, 4983252713985, 5102850779120641, 10450638395639072769, 42805814868537642061825, 350665235403060363770470401, 5745299216843741000015387049985
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2 + 3*k)/2) from k=1 to n were studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.
Cf. A036442 (denominators).

Programs

  • Mathematica
    aa = {}; m = 1/2; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa
Numerator[Table[Sum[1/2^((k^2 + 3*k)/2), {k, 1, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 10 2024 *)
  • PARI
    a(n) = numerator(sum(k=1, n, (1/2)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

Formula

a(n) = 2^(n+1)*a(n-1) + 1, a(1) = 1. - Alexandre Herrera, Mar 23 2024
a(n) ~ c * A036442(n+1) = c * 2^(n*(n+3)/2), where c = 2^(1/8) * EllipticTheta[2, 0, 1/Sqrt[2]] - 3 [in Mathematica] = 2^(1/8) * JacobiTheta2(0, 1/sqrt(2)) - 3 [in Maple] = 0.2832651213103077325876855404508588684521230759134794956... - Vaclav Kotesovec, Apr 10 2024

A178186 Sum 3^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

9, 252, 19935, 4802904, 3491587305, 7629089072292, 50039174188071999, 984820941357799304880, 58150721823981417489695049, 10301109611599361435391036962892, 5474411390529830981438591324606714655
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.

Programs

  • Mathematica
    aa = {}; m = 3; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (*Artur Jasinski*)
Table[Sum[3^((k^2+3k)/2),{k,n}],{n,20}] (* Harvey P. Dale, Jul 10 2020 *)
Accumulate[Table[3^((k^2+3k)/2),{k,15}]] (* Harvey P. Dale, Mar 25 2023 *)
  • PARI
    a(n) = sum(k=1, n, 3^((k^2+3*k)/2)); \\ Michel Marcus, Sep 09 2013

A178187 Numerators of sum (1/3)^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

1, 28, 2269, 551368, 401947273, 879058686052, 5767504039187173, 113521782003321126160, 6703347705514109178621841, 1187477935988707898665323267628, 631074461779774914374598062671491949
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.
Cf. A217628 (denominators).

Programs

  • Mathematica
    aa = {}; m = 1/3; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa (*Artur Jasinski*)
Accumulate[Table[(1/3)^((k^2+3k)/2),{k,20}]]//Numerator (* Harvey P. Dale, May 29 2020 *)
  • PARI
    a(n) = numerator(sum(k=1, n, (1/3)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

A178188 a(n) = Sum_{k=1..n} 5^((k^2+3k)/2).

Original entry on oeis.org

25, 3150, 1956275, 6105471900, 95373537112525, 7450675970460940650, 2910390496349340822268775, 5684344796471297836309816409400, 55511156915602623492479419714357425025
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.

Programs

  • Mathematica
    aa = {}; m = 5; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (*Artur Jasinski*)
Table[Sum[5^((k^2+3k)/2),{k,n}],{n,10}] (* Harvey P. Dale, Jan 17 2015 *)
  • PARI
    a(n) = sum(k=1, n, 5^((k^2+3*k)/2)); \\ Michel Marcus, Sep 09 2013

A178189 Numerators of sum (1/5)^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

1, 126, 78751, 246096876, 3845263687501, 300411225586015626, 117348134994537353906251, 229195576161205769348146484376, 2238238048449275091290493011484375001
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.

Programs

  • Mathematica
    aa = {}; m = 1/5; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa (*Artur Jasinski*)
  • PARI
    a(n) = numerator(sum(k=1, n, (1/5)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

A178190 Sum 7^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

49, 16856, 40370463, 678263443312, 79792944561055313, 65712442156478841194856, 378818757978106938161558820799, 15286701010761334171872123930835647200
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.

Programs

  • Maple
    A178190:=n->add(7^((k^2 + 3*k)/2), k=1..n); seq(A178190(n), n=1..10); # Wesley Ivan Hurt, Apr 01 2014
  • Mathematica
    aa = {}; m = 7; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (* Artur Jasinski *)
Table[Sum[7^((k^2 + 3 k)/2), {k, n}], {n, 10}] (* Wesley Ivan Hurt, Apr 01 2014 *)
  • PARI
    a(n) = sum(k=1, n, 7^((k^2+3*k)/2)); \\ Michel Marcus, Sep 09 2013

A178191 Numerators of sum (1/7)^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

1, 344, 825945, 13881657616, 1633163136864785, 1344980069223035633256, 7753542448037025041629822057, 312883404805904029979088478768109600, 88381817680515537538446482833052972519290401
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.

Programs

  • Mathematica
    aa = {}; m = 1/7; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa (*Artur Jasinski*)
Numerator[Accumulate[Table[(1/7)^((n^2+3n)/2),{n,10}]]] (* Harvey P. Dale, Jul 21 2016 *)
  • PARI
    a(n) = numerator(sum(k=1, n, (1/7)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

A178192 Sum 11^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

121, 161172, 2358108863, 379752191692104, 672750374684751701305, 13109994864250305051813161676, 2810243697916419649311518955166566527, 6626407610546884801816680266380777080570215568
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.

Programs

  • Mathematica
    aa = {}; m = 11; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (*Artur Jasinski*)
Table[11^((k^2+3k)/2),{k,10}]//Accumulate (* Harvey P. Dale, Apr 02 2020 *)
  • PARI
    a(n) = sum(k=1, n, 11^((k^2+3*k)/2)); \\ Michel Marcus, Sep 09 2013
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.