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.

A005009 a(n) = 7*2^n.

Original entry on oeis.org

7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, 28672, 57344, 114688, 229376, 458752, 917504, 1835008, 3670016, 7340032, 14680064, 29360128, 58720256, 117440512, 234881024, 469762048, 939524096, 1879048192, 3758096384
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005

Links

Crossrefs

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), this sequence (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Row sums of (6, 1)-Pascal triangle A093563 and of (1, 6)-Pascal triangle A096956, n>=1.
Cf. A000120, A014311, A118416, A173787, A212191.

Programs

Formula

G.f.: 7/(1-2*x).
a(n) = A118416(n+1,4) for n > 3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 2*a(n-1), for n > 0, with a(0)=7 . - Philippe Deléham, Nov 23 2008
a(n) = 7 * A000079(n). - Omar E. Pol, Dec 16 2008
a(n) = A173787(n+3,n). - Reinhard Zumkeller, Feb 28 2010
Intersection of A014311 and A212191: all terms and their squares are the sum of exactly three distinct powers of 2, A000120(a(n)) = A000120(a(n)^2) = 3. - Reinhard Zumkeller, May 03 2012
G.f.: 2/x/G(0) - 1/x + 9, where G(k)= 1 + 1/(1 - x*(7*k+2)/(x*(7*k+9) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
E.g.f.: 7*exp(2*x). - Stefano Spezia, May 15 2021

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Original entry on oeis.org

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472
Offset: 1

Views

Author

Enoch Haga, Jan 24 2004

Keywords

Comments

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".
The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

Links

Crossrefs

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

Programs

Formula

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Extensions

Edited and extended by Ray Chandler, Feb 07 2004

A110287 a(n) = 17*2^n.

Original entry on oeis.org

17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 8704, 17408, 34816, 69632, 139264, 278528, 557056, 1114112, 2228224, 4456448, 8912896, 17825792, 35651584, 71303168, 142606336, 285212672, 570425344, 1140850688, 2281701376, 4563402752, 9126805504, 18253611008
Offset: 0

Views

Author

Alexandre Wajnberg, Sep 07 2005

Keywords

Comments

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).
17 times powers of 2. - Omar E. Pol, Dec 17 2008

Links

Crossrefs

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A003945 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), this sequence (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A007283.

Programs

Formula

G.f.: 17/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = 17*A000079(n). - Omar E. Pol, Dec 17 2008
a(n) = 2*a(n-1) (with a(0)=17). - Vincenzo Librandi, Dec 26 2010
a(n) = A173786(n+4, n) for n>3. - Reinhard Zumkeller, Feb 28 2010
E.g.f.: 17*exp(2*x). - G. C. Greubel, Jan 05 2023

Extensions

Edited by Omar E. Pol, Dec 16 2008

A110288 a(n) = 19*2^n.

Original entry on oeis.org

19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456, 38912, 77824, 155648, 311296, 622592, 1245184, 2490368, 4980736, 9961472, 19922944, 39845888, 79691776, 159383552, 318767104, 637534208, 1275068416, 2550136832, 5100273664, 10200547328, 20401094656
Offset: 0

Views

Author

Alexandre Wajnberg, Sep 07 2005

Keywords

Comments

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).
19 times powers of 2. - Omar E. Pol, Dec 17 2008

Links

Crossrefs

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), this sequence (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

Programs

Formula

G.f.: 19/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*19. - Omar E. Pol, Dec 17 2008
E.g.f.: 19*exp(2*x). - G. C. Greubel, Jan 04 2023

Extensions

Edited by Omar E. Pol, Dec 16 2008

A247283 Positions of subrecords in A048673.

Original entry on oeis.org

5, 7, 9, 15, 18, 27, 36, 54, 72, 108, 144, 216, 288, 432, 576, 864, 1152, 1728, 2304, 3456, 4608, 6912, 9216, 13824, 18432, 27648, 36864, 55296, 73728, 110592, 147456, 221184, 294912, 442368, 589824, 884736, 1179648, 1769472, 2359296, 3538944, 4718592, 7077888
Offset: 1

Views

Author

Antti Karttunen, Sep 11 2014

Keywords

Comments

Odd bisection seems to be A116453 (i.e. A005010, 9*2^n from a(3)=9 onward).
After terms 7 and 15, even bisection from a(6)=27 onward seems to be A175806 (27*2^n).

Examples

			The fourth (A246360(4) = 5) and the fifth (A246360(5) = 8) record of A048673 (1, 2, 3, 5, 4, 8, ...) occur at A029744(4) = 4 and A029744(5) = 6 respectively. In range between, the maximum must occur at 5, thus a(4-3) = a(1) = 5. (All the previous records of A048673 are in consecutive positions, 1, 2, 3, 4, thus there are no previous subrecords).
The ninth (A246360(9) = 68) and the tenth (A246360(10) = 122) record of A048673 occur at A029744(9) = 24 and A029744(10) = 32 respectively. For n in range 25 .. 31 the values of A048673 are: 25, 26, 63, 50, 16, 53, 19, of which 63 is the maximum, and because it occurs at n = 27, we have a(9-3) = a(6) = 27.

Links

Crossrefs

A247284 gives the corresponding values.
Cf. A005010 (A116453), A175806, A029744, A048673, A064216, A246360, A245449.

Programs

  • PARI
    \\ Compute A245449, A246360, A247283 and A247284 at the same time:
default(primelimit,(2^31)+(2^30));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
n = 0; i2 = 0; i3 = 0; ir = 0; prevmax = 0; submax = 0; while(n < 2^32, n++; a_n = A048673(n); if((A048673(a_n) == n), i2++; write("b245449.txt", i2, " ", n)); if((a_n > prevmax), if(submax > 0, i3++; write("b247283.txt", i3, " ", submaxpt); write("b247284.txt", i3, " ", submax)); prevmax = a_n; submax = 0; ir++; write("b029744_empirical.txt", ir, " ", n); write("b246360_empirical.txt", ir, " ", a_n), if((a_n > submax), submax = a_n; submaxpt = n)));
  • Scheme
    (definec (A247283 n) (max_pt_in_range A048673 (+ (A029744 (+ n 3)) 1) (- (A029744 (+ n 4)) 1)))
(define (max_pt_in_range intfun lowlim uplim) (let loop ((i (+ 1 lowlim)) (maxnow (intfun lowlim)) (maxpt lowlim)) (cond ((> i uplim) maxpt) (else (let ((v (intfun i))) (if (> v maxnow) (loop (+ 1 i) v i) (loop (+ 1 i) maxnow maxpt)))))))

Formula

a(n) = A064216(A247284(n)).
Conjectures from Chai Wah Wu, Jul 30 2020: (Start)
a(n) = 2*a(n-2) for n > 6.
G.f.: x*(3*x^5 - x^3 + x^2 - 7*x - 5)/(2*x^2 - 1). (End)

A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683
Offset: 0

Views

Author

Paul Curtz, Jan 20 2009

Keywords

Comments

Deleting column k=0 and reading by antidiagonals yields A036561.
Deleting column k=0 and reading the antidiagonals downwards yields A175840.

Examples

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021

Links

Crossrefs

Cf. A001045, A004054, A046936.

Programs

  • Magma
    t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021
  • Maple
    T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011
  • Mathematica
    t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)
  • Sage
    def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

Formula

For the square array:
T(n,k) = 2^n*3^(k-1), k>0.
T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.
Rows:
T(0,k) = A140429(k) = A000244(k-1).
T(1,k) = A025192(k).
T(2,k) = A003946(k).
T(3,k) = A080923(k+1).
T(4,k) = A257970(k+3).
Columns:
T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).
T(n,1) = A000079(n).
T(n,2) = A007283(n).
T(n,3) = A005010(n).
T(n,4) = A175806(n).
T(0,k) - T(k+1,0) = 4*A094705(k-2).
From G. C. Greubel, Mar 25 2021: (Start)
For the antidiagonal triangle:
t(n, k) = T(n-k, k).
t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).
Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).
Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

Extensions

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011

A159274 a(0)=209; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1))).

Original entry on oeis.org

209, 223, 237, 252, 267, 283, 299, 316, 333, 351, 369, 388, 407, 427, 447, 468, 489, 511, 533, 556, 579, 603, 627, 652, 677, 703, 729, 756, 783, 810, 838, 866, 895, 924, 954, 984, 1015, 1046, 1078, 1110, 1143, 1176, 1210, 1244, 1279, 1314, 1350, 1386
Offset: 0

Views

Author

Philippe Deléham, Apr 07 2009

Keywords

Comments

Row 13 in square array A159016. This sequence contains infinitely many squares.
The squares in the sequence are (A175806(k))^2, k = 0,1,2,... [Vincenzo Librandi, Dec 05 2010]

Crossrefs

Cf. A159016, A175806.

Programs

  • Magma
    [n eq 1 select 209 else Self(n-1)+Floor(Sqrt (Self(n-1))):n in [1..30]]; // Marius A. Burtea, Jan 07 2020
  • Mathematica
    NestList[#+Floor[Sqrt[#]]&,209,50] (* Harvey P. Dale, Jun 12 2017 *)
Showing 1-7 of 7 results.

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

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