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 1-3 of 3 results.

A196661 Expansion of g.f. (1-2*x)/(1-7*x).

Original entry on oeis.org

1, 5, 35, 245, 1715, 12005, 84035, 588245, 4117715, 28824005, 201768035, 1412376245, 9886633715, 69206436005, 484445052035, 3391115364245, 23737807549715, 166164652848005, 1163152569936035, 8142067989552245, 56994475926865715, 398961331488060005
Offset: 0

Views

Author

Philippe Deléham, Oct 05 2011

Keywords

Links

Crossrefs

Cf. A002001, A193577 (which is the same except for the initial 1), A193722.

Programs

Formula

a(0) = 1, a(n) = 5*7^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*2^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (5*exp(7*x) + 2)/7.
a(n) = 7*a(n-1). (End)

A270471 Expansion of g.f. (1-3*x)/(1-7*x).

Original entry on oeis.org

1, 4, 28, 196, 1372, 9604, 67228, 470596, 3294172, 23059204, 161414428, 1129900996, 7909306972, 55365148804, 387556041628, 2712892291396, 18990246039772, 132931722278404, 930522055948828, 6513654391641796, 45595580741492572, 319169065190448004, 2234183456333136028
Offset: 0

Views

Author

Colin Barker, Mar 17 2016

Keywords

Comments

After 1, is this A208704?

Links

Crossrefs

Cf. A208704.
Cf. A000420 (powers of 7), A083076 (partial sums).
Cf. A193577: (1-2*x)/(1-7*x); A169634: (1-4*x)/(1-7*x).

Programs

  • Mathematica
    CoefficientList[Series[(1 - 3 x)/(1 - 7 x), {x, 0, 21}], x] (* Michael De Vlieger, Mar 18 2016 *)
Join[{1},NestList[7#&,4,20]] (* Harvey P. Dale, Dec 21 2019 *)
  • PARI
    Vec((1-3*x)/(1-7*x) + O(x^30))

Formula

G.f.: (1-3*x)/(1-7*x).
a(n) = 7*a(n-1) for n>1.
a(n) = 4*7^(n-1) for n>0.
E.g.f.: (4*exp(7*x) + 3)/7. - Elmo R. Oliveira, Mar 25 2025

A324265 a(n) = 5*343^n.

Original entry on oeis.org

5, 1715, 588245, 201768035, 69206436005, 23737807549715, 8142067989552245, 2792729320416420035, 957906156902832072005, 328561811817671400697715, 112696701453461290439316245, 38654968598537222620685472035, 13258654229298267358895116908005, 4547718400649305704101025099445715
Offset: 0

Views

Author

Stefano Spezia, Feb 20 2019

Keywords

Comments

x = a(n) and y = A324266(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 7^(6*n+1) = 4*y^3 (see Theorem 2.1 in Chakraborty, Hoque and Sharma).

Examples

			For a(0) = 5 and A324266(0) = 2, 5^2 + 7 = 32 = 4*2^3.

Links

Crossrefs

Cf. A324266 (2*49^n), A000290 (n^2), A000578 (n^3), A193577 (5*7^n).

Programs

  • GAP
    List([0..20], n->5*343^n);
  • Magma
    [5*343^n: n in [0..20]];
  • Maple
    a:=n->5*343^n: seq(a(n), n=0..20);
  • Mathematica
    5*343^Range[0,20]
  • PARI
    a(n) = 5*343^n;

Formula

O.g.f.: 5/(1 - 343*x).
E.g.f.: 5*exp(343*x).
a(n) = 343*a(n-1) for n > 0.
a(n) = (1/25)*(A193577(n))^3.
Showing 1-3 of 3 results.

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

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