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 1-7 of 7 results.

A171192 G.f. satisfies A(x) = 1/(1 - x*A(2x)^2).

Original entry on oeis.org

1, 1, 5, 53, 1045, 37941, 2596693, 343615093, 89402126741, 46139256172725, 47433024462021589, 97333484052884523765, 399068205440018335950357, 3270764880283567936326235445, 53601302478763156422575938811989
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171193-A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^2) ); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 2^(i+j) * a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Nov 03 2021
a(n) ~ c * 2^(n*(n+1)/2), where c = 1.3216968146657309382653061124105846042506... - Vaclav Kotesovec, Nov 03 2021

A156677 a(n) = 81*n^2 - 118*n + 43.

Original entry on oeis.org

43, 6, 131, 418, 867, 1478, 2251, 3186, 4283, 5542, 6963, 8546, 10291, 12198, 14267, 16498, 18891, 21446, 24163, 27042, 30083, 33286, 36651, 40178, 43867, 47718, 51731, 55906, 60243, 64742, 69403, 74226, 79211, 84358, 89667, 95138, 100771, 106566, 112523, 118642
Offset: 0

Views

Author

Vincenzo Librandi, Feb 15 2009

Keywords

Comments

The identity (6561*n^2 - 9558*n + 3482)^2 - (81*n^2 - 118*n + 43)*(729*n - 531)^2 = 1 can be written as A156773(n)^2 - a(n)*A156771(n)^2 = 1 for n > 0.
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-7; {2, 4, 9n-7, 4, 2, 18n-14}]. For n=1, this collapses to [2; {2, 4}]. - Magus K. Chu, Sep 09 2022

Links

Crossrefs

Cf. A017305, A156771, A156773, A171198.

Programs

  • Magma
    I:=[43, 6, 131]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
  • Mathematica
    LinearRecurrence[{3,-3,1},{43,6,131},40]
  • PARI
    a(n)=81*n^2-118*n+43 \\ Charles R Greathouse IV, Dec 23 2011

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (-43+123*x-242*x^2)/(x-1)^3.
For n > 1: a(n) = A171198(n-2) - A017305(n-2). - Reinhard Zumkeller, Jul 13 2010
E.g.f.: exp(x)*(43 - 37*x + 81*x^2). - Elmo R. Oliveira, Oct 19 2024

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010

A171193 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^3).

Original entry on oeis.org

1, 1, 7, 109, 3207, 174581, 17929279, 3559607005, 1389312382199, 1075527698708485, 1658535837898129263, 5105026337441341642861, 31395991691829167745766311, 385982564381552315528268500501
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192, A171194-A171198.
Cf. A168601.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^3) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^3) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n-1)/2) * 3^n, where c = 0.80142677004566734464115933731029720165641... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (1/2)^i * a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 06 2025

A171194 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^4).

Original entry on oeis.org

1, 1, 9, 185, 7241, 525513, 71973193, 19054326985, 9916177373001, 10235479554015689, 21045100094428458057, 86370025530284981044937, 708236082282948046820221257, 11609413456993946896013575994313
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192, A171193, A171195-A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^4) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^4) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n+3)/2), where c = 0.5726679317239416602436569686037310143000778... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (1/2)^i * a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 06 2025

A171195 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^5).

Original entry on oeis.org

1, 1, 11, 281, 13731, 1245601, 213268203, 70580511385, 45914883339027, 59241954299963729, 152258885235304955131, 781096727709105092232777, 8006263111571482684378716067, 164048440920655457493139473502081
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171194, A171196-A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^5) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^5) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n-1)/2) * 5^n, where c = 0.444871440417987089861554304425221691031547... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} (1/2)^x_1 * Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 06 2025

A171196 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^6).

Original entry on oeis.org

1, 1, 13, 397, 23261, 2532093, 520285021, 206632208765, 161306955003037, 249753449538341821, 770275887324912000733, 4741871606773351738426877, 58325180751309642789169099037, 1434100517517383561901937569640509
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171195, A171197, A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^6) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^6) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n+1)/2) * 3^n, where c = 0.363484431362432363073577975298028185297326... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} (1/2)^x_1 * Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 06 2025

A171197 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^7).

Original entry on oeis.org

1, 1, 15, 533, 36415, 4624621, 1108685495, 513716588981, 467874135168079, 845152554936920445, 3041003426951554000167, 21840734269889733272106629, 313415404907854466274076819391, 8990640466019774671530066108827853
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171196, A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^7) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^7) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n-1)/2) * 7^n, where c = 0.307176924551399606223470587229647816147018... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} (1/2)^x_1 * Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 06 2025
Showing 1-7 of 7 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.