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.

A237888 Decimal expansion of a constant related to A060984.

Original entry on oeis.org

1, 1, 5, 1, 1, 5, 3, 2, 1, 8, 7, 2, 1, 6, 6, 9, 3, 0, 8, 1, 8, 1, 1, 6, 6, 5, 4, 9, 8, 0, 8, 5, 4, 7, 2, 8, 5, 7, 1, 0, 6, 6, 7, 6, 4, 3, 9, 9, 4, 5, 6, 1, 8, 2, 1, 3, 0, 5, 6, 5, 9, 8, 5, 8, 7, 8, 7, 4, 8, 2, 6, 0, 3, 2, 6, 5, 4, 4, 6, 3, 8, 9, 1, 7, 9, 9, 5, 4, 1, 1, 7, 7, 9, 0, 8, 0, 0, 3, 7, 7, 5, 2, 0, 7, 6, 4
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 15 2014

Keywords

Examples

			0.11511532187216693...

Crossrefs

Cf. A060984.

Formula

Equals lim n->infinity A060984(n)/2^n.

A060985 a(1) = 1; a(n+1) = a(n) + (largest triangular number <= a(n)).

Original entry on oeis.org

1, 2, 3, 6, 12, 22, 43, 79, 157, 310, 610, 1205, 2381, 4727, 9383, 18699, 37227, 74355, 148660, 296900, 593735, 1187240, 2373810, 4746741, 9491481, 18981027, 37956907, 75910735, 151820416, 303627016, 607253419, 1214497244, 2428978214, 4857918665
Offset: 1

Views

Author

R. K. Guy, May 11 2001

Keywords

Comments

Arises in analyzing 'put-or-take' games (see Winning Ways, 484-486, 501-503), the prototype being Epstein's Put-or-Take-a-Square game.

References

  • E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982.

Links

Crossrefs

Cf. A060984, A237889.

Programs

  • Haskell
    a060985 n = a060985_list !! (n-1)
a060985_list = iterate a061885 1  -- Reinhard Zumkeller, Feb 03 2012
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Block[ {k = 1}, While[ k*(k + 1)/2 <= a[n - 1], k++ ]; a[n - 1] + k*(k - 1)/2]; Table[ a[n], {n, 1, 40} ]
f[n_]:=Module[{c=Floor[(Sqrt[1+8n]-1)/2]},(c(c+1))/2]; NestList[#+f[#]&, 1, 40] (* Harvey P. Dale, Jun 19 2011 *)
  • PARI
    { default(realprecision, 1000); for (n=1, 1000, if (n<2, a=1, k=(sqrt(1 + 8*a) - 1)\2; a+=k*(k + 1)/2 ); write("b060985.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 16 2009

Formula

a(n+1) = a(n) + A061883(n) = a(n) + A057944(a(n)) = A061885(a(n)). - Henry Bottomley, May 12 2001
a(n) ~ 0.28276... * 2^n. - Charles R Greathouse IV, Jun 19 2011

Extensions

More terms from David W. Wilson, Henry Bottomley and Robert G. Wilson v, May 12 2001

A061886 Largest square less than or equal to sum of previous terms.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 9, 16, 36, 64, 121, 256, 484, 961, 1936, 3844, 7569, 15129, 30276, 60516, 121104, 242064, 483025, 966289, 1932100, 3865156, 7728400, 15452761, 30902481, 61811044, 123609924, 247212729, 494439696, 988850916, 1977669841
Offset: 0

Views

Author

Henry Bottomley, May 12 2001

Keywords

Examples

			a(6) = 9 since 1+1+1+1+4+4 = 12 and 9 is the largest square less than or equal to this.

Links

Crossrefs

Cf. A061883.

Programs

  • Haskell
    a061886 n = a061886_list !! n
a061886_list = 1 : zipWith (-) (tail a060984_list) a060984_list
-- Reinhard Zumkeller, Dec 24 2013

Formula

For n > 0: a(n) = A060984(n+1)-A060984(n) = A048760(A060984(n)).

Extensions

Formula corrected by Reinhard Zumkeller, Dec 24 2013

A061887 n + largest square less than or equal to n; numbers in the range [2k^2,2k^2+2k] for some k.

Original entry on oeis.org

0, 2, 3, 4, 8, 9, 10, 11, 12, 18, 19, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 38, 39, 40, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 128, 129
Offset: 0

Views

Author

Henry Bottomley, May 12 2001

Keywords

Examples

			a(15)=15+9=24; a(16)=16+16=32; a(17)=17+16=33.

Crossrefs

Cf. A060984, A061885.

Programs

  • Mathematica
    Table[n+Floor[Sqrt[n]]^2,{n,0,70}] (* Harvey P. Dale, Aug 23 2012 *)

Formula

a(n) = n+[sqrt(n)]^2 = n+A048760(n) = 2n-A053186(n).

A136311 Array read by antidiagonals: a(1) = 1; a(n+1) = a(n) + (largest k-gonal number <= a(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 6, 1, 2, 3, 4, 12, 1, 2, 3, 4, 8, 22, 1, 2, 3, 4, 5, 12, 43, 1, 2, 3, 4, 5, 10, 21, 79, 1, 2, 3, 4, 5, 6, 15, 37, 157, 1, 2, 3, 4, 5, 6, 12, 27, 73, 310, 1, 2, 3, 4, 5, 6, 7, 18, 49, 137, 610, 1, 2, 3, 4, 5, 6, 7, 14, 33, 84, 258, 1205, 1, 2, 3, 4, 5, 6, 7, 8, 21, 61
Offset: 1

Views

Author

Jonathan Vos Post, Mar 22 2008

Keywords

Examples

			The array begins:
==================================================================
n=..|.1.|.2.|.3.|.4.|..5.|..6.|..7.|..8.|...9.|..10.|..11.|...12.|
==================================================================
k=3.|.1.|.2.|.3.|.6.|.12.|.22.|.43.|.79.|.157.|.310.|.610.|.1205.|.A060985
k=4.|.1.|.2.|.3.|.4.|..8.|.12.|.21.|.37.|..73.|.137.|.258.|..514.|.A060984
k=5.|.1.|.2.|.3.|.4.|..5.|.10.|.15.|.27.|..49.|..84.|.154.|..299.|
k=6.|.1.|.2.|.3.|.4.|..5.|..6.|.12.|.18.|..33.|..61.|.106.|..197.|
k=7.|.1.|.2.|.3.|.4.|..5.|..6.|..7.|.14.|..21.|..39.|..73.|..128.|
k=8.|.1.|.2.|.3.|.4.|..5.|..6.|..7.|..8.|..16.|..24.|..45.|...85.|
k=9.|.1.|.2.|.3.|.4.|..5.|..6.|..7.|..8.|...9.|..18.|..27.|...51.|
==================================================================

Crossrefs

Cf. A060984, A060985.

Programs

  • Maple
    A081422 := proc(k,n) n/2*((k-2)*n-k+4) ; end: A136311 := proc(k,n) option remember ; local aprev,n2 ; if n = 1 then RETURN(1) ; else aprev := A136311(k,n-1) ; for n2 from 0 do if A081422(k,n2) > aprev then RETURN( aprev+A081422(k,n2-1)); fi; od: fi ; end: for d from 4 to 20 do for n from 1 to d-3 do printf("%d,", A136311(d-n,n)) ; od: od: # R. J. Mathar, Jun 12 2008

Extensions

More terms from R. J. Mathar, Jun 12 2008
Showing 1-5 of 5 results.

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