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.

A168223 a(n) = A006369(n) - A006368(n).

Original entry on oeis.org

0, 0, 0, 0, -1, 3, -5, 4, -1, -1, -2, 7, -10, 7, -2, -1, -3, 10, -15, 11, -3, -2, -4, 14, -20, 14, -4, -2, -5, 17, -25, 18, -5, -3, -6, 21, -30, 21, -6, -3, -7, 24, -35, 25, -7, -4, -8, 28, -40, 28, -8, -4, -9, 31, -45, 32, -9, -5, -10, 35, -50, 35, -10, -5, -11, 38, -55, 39
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 20 2009

Keywords

Comments

A047342 and A168223 give record values and where they occur: a(A168224(n))=A047342(n) and a(m) < A047342(n) for m < A168224(n).

Links

Programs

  • Haskell
    a168223 n = a006369 n - a006368 n  -- Reinhard Zumkeller, Mar 15 2014
  • Mathematica
    LinearRecurrence[{-2,-2,0,3,4,3,0,-2,-2,-1},{0, 0, 0, 0, -1, 3, -5, 4, -1, -1},50] (* G. C. Greubel, Jul 16 2016 *)

Formula

a(12*n) = -10*n, a(12*n+1) = 7*n.
a(12*n+2) = -2*n, a(12*n+3) = -n.
a(12*n+4) = -2*n - 1, a(12*n+5) = 7*n + 3.
a(12*n+6) = -10*n - 5, a(12*n+7) = 7*n + 4.
a(12*n+8) = -2*n -1, a(12*n+9) = -n - 1.
a(12*n+10) = -2*n - 2, a(12*n+11) = 7*n + 7.
G.f.: -x^4*(x^2-x+1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^2+x+1)^2). - Colin Barker, Apr 04 2013

A082206 Digit sum of A082205(n).

Original entry on oeis.org

1, 4, 7, 10, 11, 14, 17, 18, 21, 24, 25, 28, 31, 32, 35, 38, 39, 42, 45, 46, 49, 52, 53, 56, 59, 60, 63, 66, 67, 70, 73, 74, 77, 80, 81, 84, 87, 88, 91, 94, 95, 98, 101, 102, 105, 108, 109, 112, 115, 116, 119, 122, 123, 126, 129, 130, 133, 136, 137, 140, 143, 144
Offset: 1

Views

Author

Amarnath Murthy, Apr 10 2003

Keywords

Examples

			The first six palindromes are 1, 22, 232, 3223, 22322, 232232.

Links

Crossrefs

Cf. A007953, A082204, A082205.
Essentially the same as A047342.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+3*x+3*x^2+2*x^3-2*x^4)/((1-x)*(1-x^3)) )); // G. C. Greubel, Jan 22 2024
  • Mathematica
    CoefficientList[Series[(1+3x+3x^2+2x^3-2x^4)/((1-x)*(1-x^3)),{x,0,70}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *)
Join[{1},LinearRecurrence[{1, 0, 1, -1},{4, 7, 10, 11},61]] (* Ray Chandler, Aug 25 2015 *)
  • SageMath
    def a(n): # a = A082206
    if n<5: return 3*n-2
    else: return a(n-3) + 7
[a(n) for n in range(1,71)] # G. C. Greubel, Jan 22 2024

Formula

For n>1, a(n+3) = a(n) + 7.
a(n) = A007953(A082205(n)).
G.f.: x*(1 + 3*x + 3*x^2 + 2*x^3 - 2*x^4)/((1-x)*(1-x^3)). - Vladimir Joseph Stephan Orlovsky, Jan 26 2012
a(n) = -a(n-1) - a(n-2) + 7*(n-1), for n >= 4, with a(n) = 3*n-2 for n < 4. - G. C. Greubel, Jan 22 2024

Extensions

Edited by Don Reble, Mar 13 2006
Offset corrected by Mohammed Yaseen, Aug 15 2023

A168224 Where record values occur in A168223.

Original entry on oeis.org

0, 5, 7, 11, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 55, 59, 65, 67, 71, 77, 79, 83, 89, 91, 95, 101, 103, 107, 113, 115, 119, 125, 127, 131, 137, 139, 143, 149, 151, 155, 161, 163, 167, 173, 175, 179, 185, 187, 191, 197, 199, 203, 209, 211, 215, 221, 223, 227, 233
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 20 2009

Keywords

Comments

A168223(a(n))=A047342(n) and A168223(m) < A047342(n) for m

Links

Programs

  • Mathematica
    With[{nn=300},DeleteDuplicates[Thread[{Range[0,nn-1],LinearRecurrence[{-2,-2,0,3,4,3,0,-2,-2,-1},{0,0,0,0,-1,3,-5,4,-1,-1},nn]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* Harvey P. Dale, Mar 01 2024 *)

A346798 Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 3, 6, 4, 4, 8, 9, 6, 10, 15, 12, 12, 21, 22, 18, 25, 36, 30, 32, 48, 52, 45, 60, 78, 72, 75, 105, 113, 105, 130, 166, 156, 166, 218, 236, 224, 274, 332, 325, 345, 436, 469, 462, 544, 649, 644, 688, 839, 907, 903, 1051
Offset: 0

Views

Author

Ludovic Schwob, Aug 04 2021

Keywords

Examples

			For n=19 the a(19)=6 solutions are 3+3+3+3+3+4, 3+3+3+3+7, 3+3+3+10, 3+4+4+4+4, 4+4+4+7, and 4+4+11.

Links

Crossrefs

Cf. A000041, A047342, A057570, A006950, A036820, A036822, A195848, A035363, A195849, A346797.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-3))(1 - x^(7*k-4))),{k,55}],{x,0,55}],x] (* Stefano Spezia, Aug 04 2021 *)

Formula

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-3))*(1 - x^(7*k-4))).
a(n) = a(n-3) + a(n-4) - a(n-13) - a(n-15) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 4, 13, 15, ... is the sequence A057570.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A315907 Coordination sequence Gal.3.40.3 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

Original entry on oeis.org

1, 6, 8, 14, 20, 22, 28, 34, 36, 42, 48, 50, 56, 62, 64, 70, 76, 78, 84, 90, 92, 98, 104, 106, 112, 118, 120, 126, 132, 134, 140, 146, 148, 154, 160, 162, 168, 174, 176, 182, 188, 190, 196, 202, 204, 210, 216, 218, 224, 230
Offset: 0

Views

Author

Brian Galebach and N. J. A. Sloane, Jun 18 2018

Keywords

Comments

Note that there may be other vertices in the Galebach list of u-uniform tilings with u <= 6 that have this same coordination sequence. See the Galebach link for the complete list of A-numbers for all these tilings.

Links

Crossrefs

Cf. A047342.

Formula

Conjectures from R. J. Mathar, May 02 2023: (Start)
G.f.: 1 + 2*x*(3+x+3*x^2) / ( (1+x+x^2)*(x-1)^2 ).
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
a(n) = 2*A047342(n+1) for n > 0. (End)
Showing 1-5 of 5 results.

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