A060985 -id:A060985 - cp's OEIS frontend

A237889 Decimal expansion of a constant related to A060985.

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

Offset: 0

Vaclav Kotesovec, Feb 15 2014

			0.2827628505953449...

Cf. A060985.

Equals lim n->infinity A060985(n)/2^n.

A061885 n + largest triangular number less than or equal to n.

Original entry on oeis.org

0, 2, 3, 6, 7, 8, 12, 13, 14, 15, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 42, 43, 44, 45, 46, 47, 48, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 74, 75, 76, 77, 78, 79, 80, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 132

Offset: 0

Henry Bottomley, May 12 2001

nonn

A253607(a(n)) > 0. - Reinhard Zumkeller, Jan 05 2015

Also possible values of floor(x*floor(x)) for real x < 1. - Jianing Song, Feb 16 2021

			a(9) = 9+6 = 15;
a(10) = 10+10 = 20;
a(11) = 11+10 = 21.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A060985.

Cf. A064801 (complement), A253607.

a061885 n = n + a057944 n  -- Reinhard Zumkeller, Feb 03 2012

from math import comb, isqrt
def A061885(n): return n+comb((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),2) # Chai Wah Wu, Nov 09 2024

a(n) = n+A057944(n) = 2n-A002262(n) = n+[(sqrt(1+8*n)-1)/2]*[(sqrt(1+8*n)+1)/2]/2.

a(n) = A004201(n+1) - 1. - Franklin T. Adams-Watters, Jul 05 2009

A060984 a(1) = 1; a(n+1) = a(n) + (largest square <= a(n)).

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 21, 37, 73, 137, 258, 514, 998, 1959, 3895, 7739, 15308, 30437, 60713, 121229, 242333, 484397, 967422, 1933711, 3865811, 7730967, 15459367, 30912128, 61814609, 123625653, 247235577, 494448306, 988888002, 1977738918, 3955408759, 7910812423

Offset: 1

R. K. Guy, May 11 2001

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.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E26.

Henry J. Smith and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 500 terms from Henry J. Smith)

Cf. A060985, A237888.

a060984 n = a060984_list !! (n-1)
a060984_list = iterate (\x -> x + a048760 x) 1
-- Reinhard Zumkeller, Dec 24 2013

a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[ Sqrt[ a[n - 1] ] ]^2; Table[ a[n], {n, 1, 40} ]
RecurrenceTable[{a[1]==1,a[n]==a[n-1]+Floor[Sqrt[a[n-1]]]^2},a,{n,40}] (* Harvey P. Dale, Nov 19 2011 *)
NestList[#+Floor[Sqrt[#]]^2&,1,40] (* Harvey P. Dale, Jan 22 2013 *)

{ default(realprecision, 100); for (n=1, 500, if (n==1, a=1, a+=floor(sqrt(a))^2); write("b060984.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 15 2009

from sympy import integer_nthroot
A060984_list = [1]
for i in range(10**3): A060984_list.append(A060984_list[-1]+integer_nthroot(A060984_list[-1],2)[0]**2) # Chai Wah Wu, Apr 02 2021

from math import isqrt
from itertools import accumulate
def f(an, _): return an + isqrt(an)**2
print(list(accumulate([1]*36, f))) # Michael S. Branicky, Apr 02 2021

a(n+1) = a(n)+[sqrt(a(n))]^2 = a(n)+A061886(n) = a(n)+A048760(a(n)) = A061887(a(n)). - Henry Bottomley, May 12 2001

a(n) ~ c * 2^n, where c = 0.11511532187216693... (see A237888). - Vaclav Kotesovec, Feb 15 2014

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

A061883 Largest triangular number less than or equal to sum of previous terms with a(0)=1.

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 21, 36, 78, 153, 300, 595, 1176, 2346, 4656, 9316, 18528, 37128, 74305, 148240, 296835, 593505, 1186570, 2372931, 4744740, 9489546, 18975880, 37953828, 75909681, 151806600, 303626403, 607243825, 1214480970, 2428940451

Offset: 0

Henry Bottomley, May 12 2001

nonn

a(5)=10 since 1+1+1+3+6=12 and 10 is the largest triangular number less than or equal to this.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

a061883 n = a061883_list !! n
a061883_list = 1 : zipWith (-) (tail a060985_list) a060985_list
-- Reinhard Zumkeller, Feb 03 2012

ltn[n_]:=Module[{c=Floor[(Sqrt[8*n+1]-1)/2]},(c(c+1))/2]; nxt[{t_, a_}] := Module[ {m = ltn[t]}, {t + m, m}]; Transpose[NestList[nxt,{1,1},40]] [[2]] (* Harvey P. Dale, Feb 12 2016 *)

a(n) = A060985(n+1)-A060985(n) = A057944(A060985(n)) = A000217(A003056(A060985(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

Jonathan Vos Post, Mar 22 2008

			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.|
==================================================================

Cf. A060984, A060985.

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

More terms from R. J. Mathar, Jun 12 2008

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A237889 Decimal expansion of a constant related to A060985.

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A061885 n + largest triangular number less than or equal to n.

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A060984 a(1) = 1; a(n+1) = a(n) + (largest square <= a(n)).

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A061883 Largest triangular number less than or equal to sum of previous terms with a(0)=1.

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

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