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 1-4 of 4 results.

A017257 a(n) = 9*n + 8.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 80, 89, 98, 107, 116, 125, 134, 143, 152, 161, 170, 179, 188, 197, 206, 215, 224, 233, 242, 251, 260, 269, 278, 287, 296, 305, 314, 323, 332, 341, 350, 359, 368, 377, 386, 395, 404, 413, 422, 431, 440, 449, 458, 467, 476, 485
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Digital root of any number in this sequence = 8. Any partial sum of digits of any number in this sequence also belongs to this sequence. - Artur Jasinski, Dec 16 2007
Subsequence of A224829: A224823(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2013

Links

Crossrefs

Cf. A013656, A010701, A008591.

Programs

Formula

a(n-1)^2 - A013656(n) * A010701(n)^2 = 1. - Vincenzo Librandi, Nov 19 2010
From Colin Barker, Jan 24 2012: (Start)
a(0)=8, a(1)=17, a(n) = 2*a(n-1)-a(n-2).
G.f.: (8+x)/(1-x)^2. (End)
E.g.f.: exp(x)*(8 + 9*x). - Stefano Spezia, Dec 08 2024

A224829 Numbers m, such that there is no solution m = x + y + 3*z, with triangular numbers x, y, z.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 77, 80, 89, 98, 107, 116, 125, 134, 143, 152, 158, 161, 170, 179, 188, 197, 206, 215, 224, 233, 239, 242, 251, 260, 269, 278, 287, 296, 305, 314, 320, 323, 332, 341, 350, 359, 368, 377, 386, 395, 401, 404, 413, 422, 431, 440
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 21 2013

Keywords

Comments

A224823(a(n)) = 0;
terms not of the form 9*k+8: 77,158,239,320,401,482,563,644,698,... .

Links

Crossrefs

Cf. A017257 (subsequence), A000217.

Programs

  • Haskell
    a224829 n = a224829_list !! n
a224829_list = filter ((== 0) . a224823) [0..]

A224825 Expansion of psi(x) * psi(x^3)^2 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 3, 2, 0, 4, 1, 0, 5, 3, 0, 5, 4, 0, 5, 1, 0, 7, 5, 0, 7, 4, 0, 9, 0, 0, 7, 6, 0, 6, 6, 0, 11, 3, 0, 8, 5, 0, 10, 6, 0, 8, 2, 0, 9, 6, 0, 14, 8, 0, 10, 0, 0, 15, 7, 0, 7, 8, 0, 7, 4, 0, 14, 9, 0, 14, 6, 0, 16, 1, 0, 8, 11, 0, 13, 10, 0, 13, 0, 0, 12
Offset: 0

Views

Author

Michael Somos, Jul 20 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 3*x^3 + 2*x^4 + 4*x^6 + x^7 + 5*x^9 + 3*x^10 + 5*x^12 + 4*x^13 + ...
G.f. = q^7 + q^15 + 3*q^31 + 2*q^39 + 4*q^55 + q^63 + 5*q^79 + 3*q^87 + 5*q^103 + ...

Links

Crossrefs

Cf. A033768, A224823.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(3/2)]^2 / (8 q^(7/8)), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A)^2), n))};

Formula

Expansion of q^(-7/8) * eta(q^2)^2 * eta(q^6)^4 / (eta(q) * eta(q^3)^2) in powers of q.
Euler transform of period 6 sequence [1, -1, 3, -1, 1, -3, ...].
G.f.: (Sum_{k>0} x^(k*(k-1)/2)) * (Sum_{k>0} x^(3 * k*(k-1)/2))^2.
a(3*n + 2) = 0. a(n) = A033768(3*n + 1). a(3*n + 1) = A224823(n).

A224831 Expansion of phi(-x^3)^2 * psi(x) / chi(-x)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 5, 6, 5, 6, 7, 9, 11, 8, 9, 7, 11, 13, 8, 14, 11, 16, 14, 9, 14, 7, 18, 19, 12, 13, 10, 21, 19, 17, 21, 10, 15, 17, 17, 15, 14, 26, 20, 13, 18, 22, 21, 26, 17, 20, 13, 20, 30, 9, 24, 21, 26, 21, 13, 25, 20, 27, 30, 21, 17, 20, 35, 28, 18, 22, 16, 29, 25
Offset: 0

Views

Author

Michael Somos, Jul 21 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 + 3*x + 5*x^2 + 6*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 9*x^7 + 11*x^8 + 8*x^9 + ...
q^5 + 3*q^29 + 5*q^53 + 6*q^77 + 5*q^101 + 6*q^125 + 7*q^149 + 9*q^173 + ...

Links

Crossrefs

Cf. A224823.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 2, 0, q^(1/2)]/(2 q^(1/8) QPochhammer[q, q^2]^2), {q, 0, n}]
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^4 / (eta(x + A)^3 * eta(x^6 + A)^2), n))}

Formula

Expansion of q^(-5/24) * eta(q^2)^4 * eta(q^3)^4 / (eta(q)^3 * eta(q^6)^2) in powers of q.
Euler transform of period 6 sequence [ 3, -1, -1, -1, 3, -3, ...].
a(n) = A224823(3*n).
Showing 1-4 of 4 results.

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

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