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 1-6 of 6 results.

A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900
Offset: 0

Views

Author

Creighton Dement, Feb 17 2005

Keywords

Comments

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16, 30, 49, ... = 0, a(n).
( 25, 12, 4, 0, 1, 6, 16, 30, ...
-13, -8, -4 1, 5, 10, 14, 19, ...
5, 4, 5, 4, 5, 4, 5, 4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Links

Crossrefs

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.
Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

Programs

  • Magma
    [(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011
  • Mathematica
    aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)
  • PARI
    a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

Formula

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2011

A157823 a(n) = A156591(n) + A156591(n+1).

Original entry on oeis.org

-5, -1, -2, -4, -8, -16, -32, -64, -128, -256, -512, -1024, -2048, -4096, -8192, -16384, -32768, -65536, -131072, -262144, -524288, -1048576, -2097152, -4194304, -8388608, -16777216, -33554432, -67108864, -134217728, -268435456, -536870912, -1073741824
Offset: 0

Views

Author

Paul Curtz, Mar 07 2009

Keywords

Comments

A156591 = 2,-7,6,-8,4,-12,... a(n) is companion to A154589 = 4,-1,-2,-4,-8,.For this kind ,companion of sequence b(n) is first differences a(n), second differences being b(n). Well known case: A131577 and A011782. a(n)+b(n)=A000079 or -A000079. a(n)=A154570(n+2)-A154570(n) ,A154570 = 1,3,-4,2,-6,-2,-14,. See sequence(s) identical to its p-th differences (A130785,A130781,A024495,A000749,A138112(linked to Fibonacci),A139761).

Links

Programs

  • PARI
    Vec(-(9*x-5)/(2*x-1) + O(x^100)) \\ Colin Barker, Feb 03 2015

Formula

a(n) = 2*a(n-1) for n>1. G.f.: -(9*x-5) / (2*x-1). - Colin Barker, Feb 03 2015

Extensions

Edited by Charles R Greathouse IV, Oct 11 2009

A158916 Inverse binomial transform of A153130.

Original entry on oeis.org

1, 1, 1, 1, -8, 19, -35, 64, -125, 253, -512, 1027, -2051, 4096, -8189, 16381, -32768, 65539, -131075, 262144, -524285, 1048573, -2097152, 4194307, -8388611, 16777216, -33554429, 67108861, -134217728, 268435459, -536870915, 1073741824, -2147483645
Offset: 0

Views

Author

Paul Curtz, Mar 30 2009

Keywords

Comments

a(n)= A154589(n)+ A099838(n+4).

Links

Programs

  • Mathematica
    LinearRecurrence[{-3,-3,-2},{1,1,1,1},40] (* Harvey P. Dale, Feb 04 2019 *)

Formula

a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n > 3.
G.f.: (4*x+7*x^2+9*x^3+1)/((2*x+1)*(1+x+x^2)). [R. J. Mathar, May 17 2009]

Extensions

Edited and extended by R. J. Mathar, May 17 2009

A156067 a(0)=1. a(n)= -2^(n-1)-3*(-1)^n, n>1.

Original entry on oeis.org

1, 2, -5, -1, -11, -13, -35, -61, -131, -253, -515, -1021, -2051, -4093, -8195, -16381, -32771, -65533, -131075, -262141, -524291, -1048573, -2097155, -4194301, -8388611, -16777213, -33554435, -67108861, -134217731, -268435453, -536870915, -1073741821, -2147483651
Offset: 0

Views

Author

Paul Curtz, Feb 03 2009

Keywords

Comments

The main diagonal of the array of A153130 and its successive differences.
A154589 is the second upper diagonal of the array.

Links

Programs

  • Mathematica
    Join[{1},LinearRecurrence[{1,2},{2,-5},40]] (* Harvey P. Dale, Dec 11 2011 *)

Formula

a(n)= +a(n-1) +2*a(n-2), n>2.
G.f.: x*(-2+7*x) / ( (1+x)*(2*x-1) ).
a(n) == A153130(n) (mod 9).
a(n+1)-2*a(n) = (-1)^n*9, n>0.
a(n) = A154589(n)-3*(-1)^n.
a(n)+a(n+3) = -A005010(n-1) = -9*A131577(n).
a(2*n)+a(2*n+1) = -3*2^(2n-1) = -A002023(n-2).

A173114 a(0)=a(1)=1, a(n) = 2*a(n-1)- A010686(n), n>1.

Original entry on oeis.org

1, 1, 1, -3, -7, -19, -39, -83, -167, -339, -679, -1363, -2727, -5459, -10919, -21843, -43687, -87379, -174759, -349523, -699047, -1398099, -2796199, -5592403, -11184807, -22369619, -44739239, -89478483, -178956967, -357913939, -715827879, -1431655763
Offset: 0

Views

Author

Paul Curtz, Feb 10 2010

Keywords

Comments

The sequence in the first row and successive differences in followup rows defines the array
1, 1, 1, -3, -7, -19, -39, -83, -167, -339, ..
0, 0, -4, -4, -12, -20, -44, -84, -172, -340, ..
0, -4, 0, -8, -8, -24, -40, -88, -168, -344, ..
-4, 4, -8, 0, -16, -16, -48, -80, -176, -336, ..
8, -12, 8, -16, 0, -32, -32, -96, -160, -352, ..
The first two subdiagonals show essentially the powers of 2.

Links

Formula

a(n) = 3 + 2*( (-1)^n-2^n )/3 = 3-A078008(n+1), n>0. - R. J. Mathar, Jun 30 2010
a(n+2)-a(n)= A154589(n+2) = -2^(n+1), n>0.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3), n>3.
G.f.: (-x-2*x^2-4*x^3+1)/( (1-x)*(1-2*x)*(1+x) ).
a(n) + A173078(n) = 2^n.
a(n) - a(n-1) = -4*A001045(n-2) = -A097074(n-1), n>1.

Extensions

Edited and extended by R. J. Mathar, Jun 30 2010

A158935 a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n>3. a(0)=4, a(1)=4, a(2)=-5, a(3)=4.

Original entry on oeis.org

4, 4, -5, 4, -5, 13, -32, 67, -131, 256, -509, 1021, -2048, 4099, -8195, 16384, -32765, 65533, -131072, 262147, -524291, 1048576, -2097149, 4194301, -8388608, 16777219, -33554435, 67108864, -134217725, 268435453, -536870912, 1073741827, -2147483651
Offset: 0

Views

Author

Paul Curtz, Mar 31 2009

Keywords

Comments

The third column of the array of differences described in A153130. The first two columns are in A158916 and A158987. Taking differences like in A158926 keeps the recurrence.
Also the inverse binomial transform of A153130 if the first two items of A153130 are omitted.

Links

Programs

  • Mathematica
    Join[{4},LinearRecurrence[{-3,-3,-2},{4,-5,4},50]] (* Harvey P. Dale, May 25 2011 *)

Formula

a(n)= A154589(n) + A099838(n+2).
G.f.: (4+16*x+19*x^2+9*x^3)/((2*x+1)*(1+x+x^2)). - R. J. Mathar, Apr 08 2009

Extensions

Partially edited and extended by R. J. Mathar, Apr 08 2009
Edited by N. J. A. Sloane, Apr 08 2009
Showing 1-6 of 6 results.

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

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