A002065 -id:A002065 - cp's OEIS frontend

A055456 a(n) is the smallest number which is not the sum of exactly 1 or of n earlier terms.

1, 3, 2, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 183, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Henry Bottomley, May 19 2000

			a(3)=2 because 1 is already in the sequence, 2 has not yet appeared (i.e., is not the sum of 1 earlier term), and the sum of 3 earlier terms is 3, 5 or 7.
a(4)=13 because 1, 2, and 3 have already appeared and the sum of 4 earlier terms could be any integer from 4 through 12.

Cf. A035334. a(n) is not n-1 iff n-1 is in A024556 or equivalently in A002065.

If n-1 > 0 has not already appeared in sequence then a(n) = n-1, otherwise a(n) = n^2 - n + 1.

A065035 a(n+1) = a(n)^2 + 3*a(n) + 1.

Original entry on oeis.org

0, 1, 5, 41, 1805, 3263441, 10650056950805, 113423713055421844361000441, 12864938683278671740537145998360961546653259485195805, 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185441

Offset: 0

Henry Bottomley, Nov 03 2001

a(k) is the optimal competitive ratio of any memoryless algorithm for the weighted k-server problem (Chiplunkar and Vishwanathan). - David Eppstein, Dec 31 2013

This is a divisibility sequence, that is if n divides m then a(n) divides a(m). Cf. A002065. - Peter Bala, Mar 26 2018

			a(3) = a(2)^2 + 3*a(2) + 1 = 25 + 15 + 1 = 41.

Harry J. Smith, Table of n, a(n) for n = 0..12
Ashish Chiplunkar and Sundar Vishwanathan, On Randomized Memoryless Algorithms for the Weighted k-server Problem, 54th IEEE Symp. Foundations of Computer Science (FOCS 2013), pp. 11-19; arXiv:1301.0123
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000058, A002065, A007018, A028387, A060136.

PARI
```
a(n)=if(n<1,0,a(n-1)^2+3*a(n-1)+1);
```

{ for (n=0, 12, a=if(n, a^2 + 3*a + 1, 0); write("b065035.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009

a(n) = A007018(n)-1 = A000058(n)-2 = A060136(3, n) = A028387(a(n-1)). - Michael Somos, Feb 10 2002

A055776 a(n) = a(n-1)^3 + a(n-1)^2 + a(n-1) + 1.

Original entry on oeis.org

0, 1, 4, 85, 621436, 239988219843053389, 13821964488793901254190711941736196403535171578341580

Offset: 0

Henry Bottomley, Jul 12 2000

nonn

The next term has 157 digits. - Harvey P. Dale, Dec 08 2019

			a(3) = 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203

Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.

Cf. A002065.

[n le 1 select 0 else Self(n-1)^3 + Self(n-1)^2 + Self(n-1) + 1: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015

RecurrenceTable[{a[n] == a[n - 1]^3 + a[n - 1]^2 + a[n - 1] + 1, a[0] == 0}, a, {n, 0, 6}] (* Michael De Vlieger, Oct 05 2015 *)
NestList[#^3+#^2+#+1&,0,7] (* Harvey P. Dale, Dec 08 2019 *)

a=vector(6);a[1]=1;print1("0, 1, ");for(n=2,6,a[n]=a[n-1]^3+a[n-1]^2+a[n-1]+1;print1(a[n],", ")) \\ Gerald McGarvey, Dec 08 2007

a(n) = if(n==0, 0, a(n-1)^3 + a(n-1)^2 + a(n-1) + 1);
vector(10, n, a(n-1)) \\ Altug Alkan, Oct 06 2015

a(n) is asymptotic to c^(3^(n+1)) where c=1.056431004248312118265251254776175173104598976924006344252579493163876246969557582... - Gerald McGarvey, Dec 08 2007, corrected by Vaclav Kotesovec, Apr 03 2016

a(2n) mod 2 = 0 ; a(2n+1) mod 2 = 1. - Altug Alkan, Oct 04 2015

Next term is too big to include.

A081796 Continued cotangent for sin(Pi/3) = sqrt(3)/2.

Original entry on oeis.org

0, 1, 13, 196, 257087, 249639161983, 553029809670900697241813, 575598315149214535162520163688459972096324096213, 680813056961507163626080261194823226597566577785481001106845521689287461487322891517719568410606

Offset: 0

Benoit Cloitre, Apr 10 2003

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.

Steven R. Finch, Lehmer's Constant [broken link]
Steven R. Finch, Lehmer's Constant [From the Wayback machine]

Cf. A002065, A002666, A002667, A002668.

\p900
bn=vector(100);
bn[1]=sqrt(3)/2;
b(n)=if(n<0,0,bn[n]);
for(n=2,10,bn[n]=(b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))));
a(n)=floor(b(n+1));

sqrt(3)/2 = cot(Sum_{n>=0} (-1)^n*acot(a(n))).

Let b(0) = sqrt(3)/2, b(n) = (b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))) then a(n) = floor(b(n)).

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A055456 a(n) is the smallest number which is not the sum of exactly 1 or of n earlier terms.

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A065035 a(n+1) = a(n)^2 + 3*a(n) + 1.

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A055776 a(n) = a(n-1)^3 + a(n-1)^2 + a(n-1) + 1.

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A081796 Continued cotangent for sin(Pi/3) = sqrt(3)/2.

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