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.

A073712 Self-convolution of A073711.

Original entry on oeis.org

1, 2, 3, 6, 7, 12, 16, 26, 31, 42, 59, 72, 104, 116, 184, 186, 303, 282, 497, 406, 783, 612, 1224, 840, 1856, 1232, 2784, 1656, 4136, 2376, 6008, 3138, 8735, 4362, 12345, 5754, 17693, 7756, 24432, 10170, 34471, 13302, 46771, 17688, 65144, 22296, 87008
Offset: 0

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Author

Paul D. Hanna, Aug 05 2002

Keywords

Comments

The g.f. G(x) of A073711 satisfies: G(x) = G(x^2) + x*G(x^2)^2.
The terms of this sequence found at odd-indexed positions are equal to twice that of A194279, which equals the self-convolution cube of A073711.

Links

Crossrefs

Cf. A073711, A194279.

Programs

  • Haskell
    a073712 n = a073712_list !! n
a073712_list = map (g a073711_list) [1..] where
g xs k = sum $ zipWith (*) xs $ reverse $ take k xs
-- Reinhard Zumkeller, Dec 20 2012
  • Mathematica
    nmax = 46; max = 2*nmax+1; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[ f[x] - f[x^2] - x*f[x^2]^2, {x, 0, max}], x]; sol = Solve[ Thread[ coes == 0]] // First; Table[ a[2*n+1], {n, 0, nmax}] /. sol (* Jean-François Alcover, Mar 06 2013 *)
  • PARI
    a(n)=local(A=1); for(i=0,#binary(n), A=subst(A,x,x^2+x*O(x^n))+x*subst(A,x,x^2+x*O(x^n))^2);polcoeff(A^2,n)
for(n=0,65,print1(a(n),", ")) \\ Paul D. Hanna, Dec 21 2012

Formula

a(n) = A073711(2*n+1) for n>=0.
a(2*n+1) = 2*A194279(n) for n>=0, where A194279 equals the self-convolution cube of A073711.

Extensions

Name changed and entry revised by Paul D. Hanna, Dec 21 2012

A194279 Self-convolution cube of A073711.

Original entry on oeis.org

1, 3, 6, 13, 21, 36, 58, 93, 141, 203, 306, 420, 616, 828, 1188, 1569, 2181, 2877, 3878, 5085, 6651, 8844, 11148, 14928, 18196, 24864, 29118, 40540, 45858, 65280, 70884, 103425, 108357, 161853, 162852, 250395, 242963, 382644, 356970, 579637
Offset: 0

Views

Author

Paul D. Hanna, Dec 21 2012

Keywords

Comments

The g.f. G(x) of A073711 satisfies: G(x) = G(x^2) + x*G(x^2)^2.

Links

Crossrefs

Cf. A073711, A073712.

Programs

  • PARI
    {a(n)=local(A=1); for(i=0, #binary(n), A=subst(A, x, x^2+x*O(x^n))+x*subst(A, x, x^2+x*O(x^n))^2); polcoeff(A^3, n)}
for(n=0, 65, print1(a(n), ", "))

Formula

a(n) = A073711(4*n+3)/2.
a(n) = A073712(2*n+1)/2, where A073712 equals the self-convolution of A073711.

A211604 a(n) = A073711(2^n-1) for n>=0.

Original entry on oeis.org

1, 1, 2, 6, 26, 186, 3138, 206850, 91058098, 534571085778, 126075037515248882, 6062019374259400059294162, 470527304983253008023608694415844658, 1285056632958988628362087081869760004715744193806354
Offset: 0

Views

Author

Paul D. Hanna, Feb 18 2013

Keywords

Comments

A073711(2^n) = 1 for n>=0.
The g.f. G(x) for A073711 satisfies: G(x) = G(x^2) + x*G(x^2)^2.
What is the rate of growth of this sequence?

Crossrefs

Cf. A073711.

A374571 Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2.

Original entry on oeis.org

1, -1, -1, 2, -1, 1, 2, -6, -1, 5, 1, 0, 2, -8, -6, 22, -1, -11, 5, -30, 1, 33, 0, 0, 2, -16, -8, 52, -6, -40, 22, -114, -1, 125, -11, 90, 5, -149, -30, 154, 1, -123, 33, -360, 0, 552, 0, 144, 2, -440, -16, 256, -8, -360, 52, -552, -6, 1176, -40, 576, 22, -1360, -114, 470, -1, -235, 125, -1710, -11, 3387, 90, 486, 5, -3353, -149, 1864, -30, -2152, 154, -2250, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 11 2024

Keywords

Comments

Conjecture: for n > 0, a(n) is odd iff n = A003714(k) for some k > 0, where A003714 lists Fibbinary numbers whose binary representation contains no two adjacent 1's.
Conjectures. For n > 0, we have the following occurrences:
a(n) = 0 iff n = 11 * 2^k or n = 23 * 2^k,
a(n) = 1 iff n = 5 * 2^k,
a(n) = 2 iff n = 3 * 2^k,
a(n) = 5 iff n = 9 * 2^k,
a(n) = 22 iff n = 15 * 2^k,
a(n) = 33 iff n = 21 * 2^k,
a(n) = 42 iff n = 131 * 2^k,
a(n) = 52 iff n = 27 * 2^k,
a(n) = 90 iff n = 35 * 2^k,
a(n) = 125 iff n = 33 * 2^k,
a(n) = 144 iff n = 47 * 2^k,
a(n) = 154 iff n = 39 * 2^k,
a(n) = 256 iff n = 51 * 2^k,
a(n) = 470 iff n = 63 * 2^k,
a(n) = -1 iff n = 2^k,
a(n) = -6 iff n = 7 * 2^k,
a(n) = -8 iff n = 13 * 2^k,
a(n) = -11 iff n = 17 * 2^k,
a(n) = -16 iff n = 25 * 2^k,
a(n) = -30 iff n = 19 * 2^k,
a(n) = -40 iff n = 29 * 2^k,
a(n) = -114 iff n = 31 * 2^k,
a(n) = -123 iff n = 41 * 2^k,
a(n) = -149 iff n = 37 * 2^k,
a(n) = -235 iff n = 65 * 2^k,
a(n) = -360 iff n = 43 * 2^k or n = 53 * 2^k,
etc., each of which hold for k >= 0.

Examples

			G.f.: A(x) = 1 - x - x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 6*x^7 - x^8 + 5*x^9 + x^10 + 2*x^12 - 8*x^13 - 6*x^14 + 22*x^15 - x^16 - 11*x^17 + 5*x^18 - 30*x^19 + x^20 + ...
where A(x^2) = (1 - sqrt(1 - 4*x*A(x)))/(2*x).
RELATED SERIES.
Let B(x) = Series_Reversion(x*A(x)), then
B(x) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 90*x^6 + 320*x^7 + 1152*x^8 + 4257*x^9 + 15934*x^10 + 60486*x^11 + 231894*x^12 + ... + A374570(n)*x^n + ...
where B(x)^2 = B( B(x)*C(x) ), and C(x) begins:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ,,,
where C(x) = (1 - sqrt(1 - 4*x))/2 is the Catalan function.
SPECIFIC VALUES.
A(t) = 4/5 at t = 0.1786763406278486221896028296025274247659944115...
A(t) = 3/4 at t = 0.2209727374872302749773868295900473238254186343...
A(t) = 2/3 at t = 0.2927920532546611624693565662579476873870699464...
A(t) = 3/5 at t = 0.3532836501852252091389612952989266014287213872...
A(t) = 1/2 at t = 0.4540878993396162878365437853450173746622109652...
A(t) = 2/5 at t = 0.5753264646036491718800481741299163550606457682...
A(t) = 1/3 at t = 0.6711059159867924708010090309770441047524321152...
A(t) = 1/4 at t = 0.8063263233032142016966341297674341884930955548...
A(t) = 1/5 at t = 0.8884702348196434968520432792716046325517863531...
A(1/2) = 0.4596569887547343191321148479065626411948116168891503813...
where A(1/4) = (1 - sqrt(1 - 2*A(1/2))).
A(1/3) = 0.6215166290026409046430206750366100166629591510407086872...
where A(1/9) = (3/2) * (1 - sqrt(1 - (4/3)*A(1/3))).
A(1/4) = 0.7159471484203487850228006105062270686816491955635126263...
where A(1/16) = 2 * (1 - sqrt(1 - A(1/4))).
A(1/5) = 0.7747713037551020088783260174094983351988173792698848600...
where A(1/25) = (5/2) * (1 - sqrt(1 - (4/5)*A(1/5))).
A(1/6) = 0.8141931617547219509824463958597943246122338043286847588...
where A(1/36) = 3 * (1 - sqrt(1 - (2/3)*A(1/6))).
A(1/8) = 0.8630723739180924020163457579861333293488991044015651008...
where A(1/64) = 4 * (1 - sqrt(1 - (1/2)*A(1/8))).
A(1/10) = 0.891911395101161792043000371010714789952867553398091597...
where A(1/100) = 5 * (1 - sqrt(1 - (2/5)*A(1/10))).

Links

Crossrefs

Cf. A073711, A374570, A000108, A003714.

Programs

  • PARI
    {a(n) = my(A = 1+x); for(i=0,#binary(n), A = subst(A,x,x^2) - x*subst(A^2,x,x^2) + x*O(x^n) ); polcoeff(A,n)}
for(n=0, 80, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n, where B(x) is the g.f. of A374570 and C(x) = x + C(x)^2 is the g.f. of A000108, satisfies the following formulas.
(1) A(x) = A(x^2) - x*A(x^2)^2.
(2) A(x^2) = (1 - sqrt(1 - 4*x*A(x))) / (2*x).
(3) A(x^2) = (1/x) * C(x*A(x)).
(4) x^2 = B( x * C(x*A(x)) ).
(5) A(B(x)) = x / B(x).
(6) A(B(x)^2) = C(x) / B(x).
(7) B(x)^2 = B( B(x)*C(x) ).
Showing 1-4 of 4 results.

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