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 1-5 of 5 results.

A163403 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152
Offset: 1

Views

Author

Klaus Brockhaus, Jul 26 2009

Keywords

Comments

a(n+1) is the number of palindromic words of length n using a two-letter alphabet. - Michael Somos, Mar 20 2011

Examples

			x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + 16*x^8 + 16*x^9 + 32*x^10 + ...

Links

Crossrefs

Equals A016116 without initial 1. Unsigned version of A152166.
Partial sums are in A136252.
Binomial transform is A078057, second binomial transform is A007070, third binomial transform is A102285, fourth binomial transform is A163350, fifth binomial transform is A163346.
Cf. A000079 (powers of 2), A009116, A009545, A051032.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

Programs

  • Magma
    [ n le 2 select n else 2*Self(n-2): n in [1..43] ];
  • Mathematica
    LinearRecurrence[{0, 2}, {1, 2}, 50] (* Paolo Xausa, Feb 02 2024 *)
  • PARI
    {a(n) = if( n<1, 0, 2^(n\2))} /* Michael Somos, Mar 20 2011 */
  • Sage
    def A163403():
    x, y = 1, 1
    while True:
        yield x
        x, y = x + y, x - y
a = A163403(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

Formula

a(n) = 2^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 2*x)/(1 - 2*x^2).
a(n) = A051032(n) - 1.
G.f.: x / (1 - 2*x / (1 + x / (1 + x))) = x * (1 + 2*x / (1 - x / (1 - x / (1 + 2*x)))). - Michael Somos, Jan 03 2013
From R. J. Mathar, Aug 06 2009: (Start)
a(n) = A131572(n).
a(n) = A060546(n-1), n > 1. (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = |A009116(n-1)| + |A009545(n-1)|. - Bruno Berselli, May 30 2011
E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Feb 05 2023

A163350 a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

Original entry on oeis.org

1, 6, 34, 188, 1028, 5592, 30344, 164464, 890896, 4824672, 26124832, 141453248, 765878336, 4146681216, 22451153024, 121555687168, 658129355008, 3563255219712, 19292230787584, 104452273224704, 565526954771456
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Keywords

Comments

Binomial transform of A102285. Fourth binomial transform of A163403. Inverse binomial transform of A163346.

Links

Crossrefs

Cf. A102285, A163403, A163346.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009
  • Mathematica
    LinearRecurrence[{8,-14},{1,6},30] (* Harvey P. Dale, May 08 2014 *)
  • PARI
    Vec((1-2*x)/(1-8*x+14*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
a(n) = ((1+sqrt(2))*(4+sqrt(2))^n+(1-sqrt(2))*(4-sqrt(2))^n)/2.
G.f.: (1-2*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009
New name from G. C. Greubel, Dec 19 2016

A069514 Numbers n such that sigma(reversal(n)) = reversal(sigma(n)). Ignore leading 0's.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 41, 124, 194, 333, 421, 491, 1324, 4231, 13324, 17054, 17571, 42331, 45071, 120530, 138465, 386650, 564831, 1130324, 1216360, 1333324, 1727571, 1757271, 1757571, 1787871, 2249422, 4230311, 4233331, 4369634
Offset: 1

Views

Author

Joseph L. Pe, Apr 15 2002

Keywords

Comments

For an arithmetical function f, call the arguments n such that f(reverse(n)) = reverse(f(n)) the "palinpoints" of f. This sequence is the sequence of palinpoints of f(n) = sigma(n).
If n is in the sequence and 10 doesn't divide n then the reversal of n is also in the sequence. - Farideh Firoozbakht, Aug 31 2004
Comments from Farideh Firoozbakht, Jan 16 2005. "The largest term that I found is M=(58*100^687 - 157)/33; the length of M is 1375. I proved the following facts about this sequence:
"I : If p=(58*100^n - 157)/99 is prime then 3*p is in the sequence, the sequence A102285 gives such n's.
"II : If p=(59*100^n - 257)/99 is prime then 3*p is in the sequence, I found only two primes of this form the first for n=3 and the second for n=27, next such n is greater than 3400.
"III : If both numbers p=10^n - 3 & q=5*10^n - 9 are primes then both numbers 2*p & q are in the sequence, q is reversal of 2*p. I found only two such n's, n=1 & 2.
"IV : If both numbers p=(10^n-7)/3 & q=(127*10^(n-1)-7)/3 are primes then both numbers 4*p & q are in the sequence, q is the reversal of 4*p, the sequence A102287 are these terms of A069514, I found only four such n's, n=2,3,4 & 6."

Examples

			Let f(n) = sigma(n). Then f(194) = 294, f(491) = 492, so f(reverse(194)) = reverse(f(194)). Therefore 194 belongs to the sequence.

Crossrefs

Cf. A102285, A102286, A102287.

Programs

  • Mathematica
    rev[n_] := FromDigits[Reverse[IntegerDigits[n]]]; f[n_] := DivisorSigma[1, n]; Select[Range[10^6], f[rev[ # ]] == rev[f[ # ]] &]

Extensions

More terms from Farideh Firoozbakht, Aug 31 2004

A164537 a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 28.

Original entry on oeis.org

5, 28, 154, 840, 4564, 24752, 134120, 726432, 3933776, 21300160, 115328416, 624425088, 3380802880, 18304471808, 99104534144, 536573667840, 2905125864704, 15728975567872, 85160042437120, 461074681546752
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A102285 without initial term 1. Fourth binomial transform of A164682. Inverse binomial transform of A164538.

Links

Crossrefs

Cf. A102285, A164682, A164538.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+4*r)*(4+r)^n+(5-4*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 21 2009

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 28.
G.f.: (5-12*x)/(1-8*x+14*x^2).
a(n) = ((5+4*sqrt(2))*(4+sqrt(2))^n + (5-4*sqrt(2))*(4-sqrt(2))^n)/2.

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 21 2009

A102589 Numbers of the forms 4*p or q where both numbers p=(10^n-7)/3 and q=(127*10^(n-1)-7)/3 are primes.

Original entry on oeis.org

124, 421, 1324, 4231, 13324, 42331, 1333324, 4233331
Offset: 1

Views

Author

Farideh Firoozbakht, Jan 22 2005

Keywords

Comments

This sequence is a subsequence of A069514 (If m is in the sequence then sigma(reversal(m)) = reversal(sigma(m))), so see Comments on A069514. There exists only four known n's such that both numbers p=(10^n-7)/3 and q=(127*10^(n-1)-7)/3 are primes, n=2,3, 4 and 6.

Examples

			124 and 421 are in the sequence because 124=4*31 and both numbers
31=(10^2-7)/3 and 421=(127*10^(2-1)-7)/3 are primes. It seems that
there is no further term.

Crossrefs

Cf. A069514, A102285.
Showing 1-5 of 5 results.

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