A085785 -id:A085785 - cp's OEIS frontend

A001082 Generalized octagonal numbers: k(3k-2), k=0, +- 1, +- 2, +-3, ...

0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85, 96, 120, 133, 161, 176, 208, 225, 261, 280, 320, 341, 385, 408, 456, 481, 533, 560, 616, 645, 705, 736, 800, 833, 901, 936, 1008, 1045, 1121, 1160, 1240, 1281, 1365, 1408, 1496, 1541, 1633, 1680, 1776, 1825, 1925, 1976

Offset: 1

N. J. A. Sloane and Tom Duff

Numbers of the form 3*m^2+2*m, m an integer.
3*a(n) + 1 is a perfect square.
a(n) mod 10 belongs to a periodic sequence: 0, 1, 5, 8, 6, 1, 3, 0, 6, 5, 5, 6, 0, 3, 1, 6, 8, 5, 1, 0. - Mohamed Bouhamida, Sep 04 2009
A089801 is the characteristic function. - R. J. Mathar, Oct 07 2011
Exponents of powers of q in one form of the quintuple product identity. (-x^-2 + 1) * q^0 + (x^-3 - x) * q^1 + (-x^-5 + x^3) * q^5 + (x^-6 - x^4) * q^8 + ... = Sum_{n>=0} q^(3*n^2 + 2*n) * (x^(3*n) - x^(-3*n - 2)) = Product_{k>0} (1 - x * q^(2*k - 1)) * (1 - x^-1 * q^(2*k - 1)) * (1 - q^(2*k)) * (1 - x^2 * q^(4*k)) * (1 - x^-2 * q^(4*k - 4)). - Michael Somos, Dec 21 2011
The offset 0 would also be valid here, all other entries of generalized k-gonal numbers have offset 0 (see cross references). - Omar E. Pol, Jan 12 2013
Also, x values of the Diophantine equation x(x+3)+(x+1)(x+2) = (x+y)^2+(x-y)^2. - Bruno Berselli, Mar 29 2013
Numbers n such that Sum_{i=1..n} 2*i*(n-i)/n is an integer (the addend is the harmonic mean of i and n-i). - Wesley Ivan Hurt, Sep 14 2014
Equivalently, integers of the form m*(m+2)/3 (nonnegative values of m are listed in A032766). - Bruno Berselli, Jul 18 2016
Exponents of q in the expansion of Sum_{n >= 0} ( q^n * Product_{k = 1..n} (1 - q^(2*k-1)) ) = 1 + q - q^5 - q^8 + q^16 + q^21 - - + + .... - Peter Bala, Dec 03 2020
Exponents of q in the expansion of Product_{n >= 1} (1 - q^(6*n))*(1 + q^(6*n-1))*(1 + q^(6*n-5)) = 1 + q + q^5 + q^8 + q^16 + q^21 + .... - Peter Bala, Dec 09 2020
Exponents of q in the expansion of Product_{n >= 1} (1 - q^n)^2*(1 - q^(4*n))^2 /(1 - q^(2*n)) = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - + ... (a consequence of the quintuple product identity). The series coefficients are a signed version of A001651. - Peter Bala, Feb 16 2021
From Peter Bala, Nov 26 2024: (Start)
Apart from the first two terms, the exponents of q in the expansion of Sum_{n >= 1} q^(3*n+2) * (Product_{k = 2..n} 1 - q^(2*k-1)) = q^5 + q^8 - q^16 - q^21 + + - - ... (in Andrews, equation 8, replace q with q^2 and set x = q).
Exponents of q^2 in the expansion of Sum_{n >= 0} q^n / (Product_{k = 1..n+1 } 1 + q^(2*k-1)) = 1 + (q^2)^1 - (q^2)^5 - (q^2)^8 + (q^2)^16 + (q^2)^21 - - + + ... (Chen, equation 22). (End)

			For the ninth comment: 65 is in the sequence because 65 = 13*(13+2)/3 or also 65 = -15*(-15+2)/3. - _Bruno Berselli_, Jul 18 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
George E. Andrews, Euler's Pentagonal Number Theorem, Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 279-284.
Sandy H. L. Chen, Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in q-Series, Open Journal of Discrete Mathematics, Vol.4, No.2, April 2014.
John Elias, Illustration of initial terms: Hourglass Octagonals.
John Elias, Illustration: Pentagonal-Octagonal Star Configurations
Ralf Stephan, On the solutions to 'px+1 is square'.
Zhi-Wei Sun, A result similar to Lagrange's theorem, arXiv preprint arXiv:1503.03743 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Partial sums of A022998.
Cf. A000567, A005563, A085785, A089801, A245031.
Column 4 of A195152. A045944.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), this sequence (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

a001082 n = a001082_list !! n
a001082_list = scanl (+) 0 $ tail a022998_list
-- Reinhard Zumkeller, Mar 31 2012

[n^2 - n - Floor(n/2)^2 : n in [1..50]]; // Wesley Ivan Hurt, Sep 14 2014

seq(n*(n-1)-floor(n/2)^2, n=1..51); # Gary Detlefs, Feb 23 2010

Table[If[EvenQ[n], n*(3*n-4)/4, (n-1) (3*n+1)/4], {n, 100}]
LinearRecurrence[{1,2,-2,-1,1},{0,1,5,8,16},60] (* Harvey P. Dale, Feb 03 2024 *)

{a(n) = if( n%2, (n-1) * (3*n + 1) / 4, n * (3*n - 4) / 4)};

a(n) = n*(3*n-4)/4 if n even, (n-1)*(3*n+1)/4 if n odd.
a(n) = n^2 - n - floor(n/2)^2.
G.f.: Sum_{n>=0} (-1)^n*[x^(a(2n+1)) + x^(a(2n+2))] = 1/1 - (x-x^2)/1 - (x^2-x^4)/1 - (x^3-x^6)/1 - ... - (x^k - x^(2k))/1 - ... (continued fraction where k=1..inf). - Paul D. Hanna, Aug 16 2002
a(n+1) = ceiling(n/2)^2 + A046092(floor(n/2)).
a(2n) = n(3n-2) = A000567(n), a(2n+1) = n(3n+2) = A045944(n). - Mohamed Bouhamida, Nov 06 2007
O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - R. J. Mathar, Apr 15 2008
a(n) = n^2+n-ceiling(n/2)^2 with offset 0 and a(0)=0. - Gary Detlefs, Feb 23 2010
a(n) = (6*n^2-6*n-1-(2*n-1)*(-1)^n)/8. - Luce ETIENNE, Dec 11 2014
E.g.f.: (3*x^2*exp(x) + x*exp(-x) - sinh(x))/4. - Ilya Gutkovskiy, Jul 15 2016
Sum_{n>=2} 1/a(n) = (9 + 2*sqrt(3)*Pi)/12. - Vaclav Kotesovec, Oct 05 2016
Sum_{n>=2} (-1)^n/a(n) = 3*log(3)/2 - 3/4. - Amiram Eldar, Feb 28 2022

New sequence name from Matthew Vandermast, Apr 10 2003
Editorial changes by N. J. A. Sloane, Feb 03 2012
Edited by Omar E. Pol, Jun 09 2012

A073341 Number of steps to reach an integer starting with (2n+1)/4 and iterating the map x -> x*ceiling(x).

Original entry on oeis.org

3, 2, 3, 8, 1, 1, 3, 2, 2, 3, 2, 2, 1, 1, 7, 4, 4, 2, 4, 3, 1, 1, 2, 4, 2, 8, 4, 3, 1, 1, 6, 4, 3, 2, 5, 4, 1, 1, 5, 2, 2, 3, 2, 2, 1, 1, 4, 5, 6, 2, 3, 5, 1, 1, 2, 3, 2, 4, 3, 6, 1, 1, 7, 8, 3, 2, 4, 5, 1, 1, 3, 2, 2, 3, 2, 2, 1, 1, 7, 3, 4, 2, 7, 6, 1, 1, 2, 5, 2, 5, 5, 3, 1, 1, 3, 3, 3, 2, 10, 3, 1, 1, 4, 2, 2

Offset: 2

N. J. A. Sloane and J. C. Lagarias, Sep 04 2002

nonn

We conjecture that an integer is always reached.
Is S(n) = Sum_{k=2..n} a(k) asymptotic to 3*n? S(n) = 3n for n = 69, 127, 166, 169, 189, 197, 327, 328, 360, 389, 404, 405, 419, 428, 497, 519, 520, 540, 541, 544, 547, 652, 668, 669, 676, 682, 683...
The sign of 3n-S(n) seems to change often: 3n-S(n) = 3, 4, 4, -1, 1, 3, 3, 4, 5, 5, 6, 7, 9, 11, 7, 6, 5, 6, 5, 5, 7, 9, 10, 9, 10, 5, 4, 4, 6, 8, 5, 4, 4, 5, 3, 2, 4, 6, 4, 5, 6, 6, 7, 8, 10, 12, 11, 9, 6, 7, 7, 5, 7, 9, 10, 10, 11, 10, 10, 7, 9, 11, 7, 2, 2, 3, 2, 0, 2, 4, 4, 5, 6, 6, 7, 8, 10, 12, 8, 8, 7, 8, 4, 1, 3, 5, 6, 4, 5, 3, 1, 1, 3, 5, 5, 5, 5, 6, -1, ... Is 3n-S(n) bounded? - Benoit Cloitre, Sep 05 2002

Robert Israel, Table of n, a(n) for n = 2..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A073524, A074735, A085785, A085817, A085833.

g := proc(x) local M,t1,t2,t3; M := 4^100; t1 := ceil(x) mod M; t2 := x*t1; t3 := numer(t2) mod M; t3/denom(t2); end;
a := []; for n from 2 to 150 do t1 := (2*n+1)/4; for i from 1 to 100 do t1 := g(t1); if type(t1,`integer`) then break; fi; od: a := [op(a),i]; od: a;

a[n_] := Length @ NestWhileList[# Ceiling[#]&, (2n+1)/4, !IntegerQ[#]&] - 1;
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 31 2023 *)

a(n)=if(n<1,0,s=n/2+1/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c) \\ Benoit Cloitre, Sep 05 2002

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A001082 Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ...

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A073341 Number of steps to reach an integer starting with (2n+1)/4 and iterating the map x -> x*ceiling(x).

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A085817 Start at (2n+1)/4 and iterate the map x -> x*ceiling(x); sequence gives values of n for which the denominators in the orbit drop directly from 4 to 1, bypassing 2.

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A085833 Start at (2n+1)/4 and iterate the map x -> x*ceiling(x); sequence gives values of n for which the denominators in the orbit go from 4 to 2, instead of dropping directly to 1.

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A001082 Generalized octagonal numbers: k(3k-2), k=0, +- 1, +- 2, +-3, ...