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.

A082248 a(n) = A082247(n)/7.

Original entry on oeis.org

493827, 1877359453117, 2887460463218, 3897561473319, 4907662483420, 5917763493521, 6927864503622, 7937965513723, 8948066523824, 9958167533925, 10968268544026, 11978369554127, 12988470564228, 14586157729300872444, 15587158730301873445, 16588159731302874446
Offset: 1

Views

Author

Amarnath Murthy, Apr 11 2003

Keywords

Comments

From 2nd terms onwards some pattern is visible which would change when three or more digit numbers occur.

Crossrefs

Cf. A082242, A082243, A082247.

A077925 Expansion of 1/((1-x)*(1+2*x)).

Original entry on oeis.org

1, -1, 3, -5, 11, -21, 43, -85, 171, -341, 683, -1365, 2731, -5461, 10923, -21845, 43691, -87381, 174763, -349525, 699051, -1398101, 2796203, -5592405, 11184811, -22369621, 44739243, -89478485, 178956971, -357913941, 715827883, -1431655765, 2863311531, -5726623061
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

a(n+1) is the reflection of a(n) through a(n-1) on the numberline. - Floor van Lamoen, Aug 31 2004
If a zero is added as the (new) a(0) in front, the sequence represents the inverse binomial transform of A001045. Partial sums are in A077898. - R. J. Mathar, Aug 30 2008
a(n) = A077953(2*n+3). - Reinhard Zumkeller, Oct 07 2008
Related to the Fibonacci sequence by an INVERT transform: if A(x) = 1+x^2*g(x) is the generating function of the a(n) prefixed with 1, 0, then 1/A(x) = 2+(x+1)/(x^2-x+1) is the generating function of 1, 0, -1, 1, -2, 3, ..., the signed Fibonacci sequence A000045 prefixed with 1. - Gary W. Adamson, Jan 07 2011
Also: Gaussian binomial coefficients [n+1,1], or q-integers, for q=-2, diagonal k=1 in the triangular (or column r=1 in the square) array A015109. - M. F. Hasler, Nov 04 2012
With a leading zero, 0, 1, -1, 3, -5, 11, -21, 43, -85, 171, -341, 683, ... we obtain the Lucas U(-1,-2) sequence. - R. J. Mathar, Jan 08 2013
Let m = a(n). Then 18*m^2 - 12*m + 1 = A000225(2n+3). - Roderick MacPhee, Jan 17 2013

Examples

			G.f. = 1 - x + 3*x^2 - 5*x^3 + 11*x^4 - 21*x^5 + 43*x^6 - 85*x^7 + ...

Links

Crossrefs

Cf. A001045 (unsigned version).
Cf. A014983, A014985, A014986. - Zerinvary Lajos, Dec 16 2008
Cf. A232600, A232601, A232602.

Programs

Formula

G.f.: 1/(1+x-2*x^2).
a(n) = (1-(-2)^(n+1))/3. - Vladeta Jovovic, Apr 17 2003
a(n) = Sum_{k=0..n} (-2)^k. - Paul Barry, May 26 2003
a(n+1) - a(n) = A122803(n). - R. J. Mathar, Aug 30 2008
a(n) = Sum_{k=0..n} A112555(n,k)*(-2)^k. - Philippe Deléham, Sep 11 2009
a(n) = A082247(n+1) - 1. - Philippe Deléham, Oct 07 2009
G.f.: Q(0)/(3*x), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k + 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k - 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k-1 + 2*x)/( x*(4*k+1 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
E.g.f.: (2*exp(-2*x) + exp(x))/3. - Ilya Gutkovskiy, Nov 12 2016
a(n) = A086893(n+2) - A061547(n+3), n >= 0. - Yosu Yurramendi, Jan 16 2017
a(n) = (-1)^n*A001045(n+1). - M. F. Hasler, Feb 13 2020
a(n) - a(n-1) = a(n-1) - a(n+1) = (-2)^n, a(n+1) = - a(n) + 2*a(n-1) = 1 - 2*a(n). - Michael Somos, Feb 22 2023

A082249 Reverse concatenation of 7 numbers that are multiples of 7.

Original entry on oeis.org

22212019181716, 29282726252423, 36353433323130, 43424140393837, 50494847464544, 57565554535251, 64636261605958, 71706968676665, 78777675747372, 85848382818079, 92919089888786, 99989796959493, 108107106105104103102
Offset: 1

Views

Author

Amarnath Murthy, Apr 11 2003

Keywords

Comments

It appears that if a(r) begins with k then a(r+1) ends in k+1. This is true up to the term 99989796959493.

Links

Crossrefs

Cf. A082242, A082243, A082247, A082248, A082250.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A082250 a(n) = A082249(n)/7.

Original entry on oeis.org

3173145597388, 4183246607489, 5193347617590, 6203448627691, 7213549637792, 8223650647893, 9233751657994, 10243852668095, 11253953678196, 12264054688297, 13274155698398, 14284256708499, 15443872300729157586
Offset: 1

Views

Author

Amarnath Murthy, Apr 11 2003

Keywords

Comments

Some pattern is visible which would change when numbers of three or more digits occur in A082249. Conjecture: If the concatenation of k consecutive r-digit numbers is divisible by n then the concatenation of the next k numbers would also be divisible by n provided it contains the same number of digits as the first number. This should be true for both forward and reverse concatenation.

Crossrefs

Cf. A082242, A082243, A082247, A082248, A082249.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
Showing 1-4 of 4 results.

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