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.

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A306472 a(n) = 37*27^n.

Original entry on oeis.org

37, 999, 26973, 728271, 19663317, 530909559, 14334558093, 387033068511, 10449892849797, 282147106944519, 7617971887502013, 205685240962554351, 5553501505988967477, 149944540661702121879, 4048502597865957290733, 109309570142380846849791, 2951358393844282864944357
Offset: 0

Views

Author

Stefano Spezia, Feb 18 2019

Keywords

Comments

x = a(n) and y = A002042(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 3^(6*n+1) = 4*y^3 (see Theorem 2.1 in Chakraborty, Hoque and Sharma).

Examples

			For a(0) = 37 and A002042(0) = 7, 37^2 + 3 = 1372 = 4*7^3.

Links

Crossrefs

Cf. A002042 (7*4^n), A009971 (27^n), A000290 (n^2), A000578 (n^3).

Programs

  • GAP
    List([0..20], n->37*27^n);
  • Magma
    [37*27^n: n in [0..20]];
  • Maple
    a:=n->37*27^n: seq(a(n), n=0..20);
  • Mathematica
    37*27^Range[0,20]
  • PARI
    a(n) = 37*27^n;

Formula

O.g.f.: 37/(1 - 27*x).
E.g.f.: 37*exp(27*x).
a(n) = 27*a(n-1) for n > 0.
a(n) = 37*A009971(n).

A308124 a(n) = (2 + 7*4^n)/3.

Original entry on oeis.org

3, 10, 38, 150, 598, 2390, 9558, 38230, 152918, 611670, 2446678, 9786710, 39146838, 156587350, 626349398, 2505397590, 10021590358, 40086361430, 160345445718, 641381782870, 2565527131478, 10262108525910, 41048434103638, 164193736414550, 656774945658198, 2627099782632790
Offset: 0

Views

Author

Paul Curtz, Jul 23 2019

Keywords

Comments

Consider A092808 and its differences:
1, 0, 3, 1, 11, 5, 43, 21, 171, ...
-1, 3, -2, 10, -6, 38, -22, 150, ... = b(n).
a(n) is the second bisection of b(n). The first is A047849.
a(n) mod 9 is the period 9 sequence: repeat [3, 1, 2, 6, 4, 5, 0, 7, 8].
b(n) + b(n+1) = A135520(n).

Links

Crossrefs

Cf. A000302, A001045, A002042, A092808, A047849, A135520, A141495.

Programs

  • Mathematica
    LinearRecurrence[{5,-4},{3,10},30] (* Paolo Xausa, Nov 13 2023 *)
(2+7*4^Range[0,30])/3 (* Harvey P. Dale, Aug 15 2025 *)
  • PARI
    a(n) = (2 + 7*4^n)/3; \\ Stefano Spezia, Jul 23 2019
  • PARI
    Vec((3 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^40)) \\ Colin Barker, Jul 23 2019

Formula

a(n) = 4*a(n-1) - 2 for n=1,2,... , a(0) = 3.
a(n+1) = a(n) + A002042(n).
Binomial transform of A141495(n+1) = 3, 7, 21, ....
From Colin Barker, Jul 23 2019: (Start)
G.f.: (3 - 5*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n>1.
(End)
a(n+2) = a(n) + 35*A000302(n) for n=0,1,2, ... .

Extensions

a(14)-a(25) from Stefano Spezia, Jul 23 2019

A383414 Array read by ascending antidiagonals: A(n,k) = 4^n*(8*k + 7).

Original entry on oeis.org

7, 28, 15, 112, 60, 23, 448, 240, 92, 31, 1792, 960, 368, 124, 39, 7168, 3840, 1472, 496, 156, 47, 28672, 15360, 5888, 1984, 624, 188, 55, 114688, 61440, 23552, 7936, 2496, 752, 220, 63, 458752, 245760, 94208, 31744, 9984, 3008, 880, 252, 71, 1835008, 983040, 376832, 126976, 39936, 12032, 3520, 1008, 284, 79
Offset: 0

Views

Author

Stefano Spezia, Apr 26 2025

Keywords

Examples

			The array begins as:
      7,    15,    23,     31,     39,     47, ...
     28,    60,    92,    124,    156,    188, ...
    112,   240,   368,    496,    624,    752, ...
    448,   960,  1472,   1984,   2496,   3008, ...
   1792,  3840,  5888,   7936,   9984,  12032, ...
   7168, 15360, 23552,  31744,  39936,  48128, ...
  28672, 61440, 94208, 126976, 159744, 192512, ...
  ...

References

  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 12.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 246-247.

Crossrefs

Cf. A000302, A004215, A383415 (antidiagonal sums).
Row n=0 gives A004771.
Column k=0 gives A002042.

Programs

  • Mathematica
    A[n_,k_]:=4^n(8k+7); Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

Formula

A(n,k) = A000302(n)*A004771(k).
G.f.: (7 + y)/((1 - 4*x)*(1 - y)^2).
E.g.f.: exp(4*x+y)*(7 + 8*y).
Previous Showing 11-13 of 13 results.

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