A143760 -id:A143760 - cp's OEIS frontend

A143761 a(n+1) = a(n)^2 - n*a(n) + n^2, a(1) = 1.

1, 1, 3, 9, 61, 3441, 11819871, 139709267717593, 19518679486184955909459972969, 380978848884417414427615903969045678210740619589070918321

Offset: 1

Reinhard Zumkeller, Sep 01 2008

nonn

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=a(n), f(n,+1)=A143762(n),
f(n,-2)=A143763(n), f(n,+2)=A143764(n), f(n,-3)=A143765(n),
f(n,+3)=A143766(n).

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From _Reinhard Zumkeller_, Sep 11 2008]

RecurrenceTable[{a[n+1] == a[n]^2 - n*a[n] + n^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
nxt[{n_,a_}]:={n+1,a^2-a(n+1)+(n+1)^2}; Join[{1},NestList[nxt,{1,1},10][[All,2]]] (* Harvey P. Dale, Oct 15 2017 *)

a(n) ~ c^(2^n), where c = 1.1356768837971801294990742257077941319336784846128736022759470343698650014... . - Vaclav Kotesovec, Dec 18 2014

A143762 a(n+1) = a(n)^2 + n*a(n) + n^2, a(1) = 1.

Original entry on oeis.org

1, 3, 19, 427, 184053, 33876427099, 1147612313197120118431, 1317014021401644919149757309088631306730827, 1734525932568532421128602190712731410907662021613907396581559184320372648524685950609

Offset: 1

Reinhard Zumkeller, Sep 01 2008

nonn

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=a(n),
f(n,-2)=A143763(n), f(n,+2)=A143764(n), f(n,-3)=A143765(n),
f(n,+3)=A143766(n).

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From _Reinhard Zumkeller_, Sep 11 2008]

RecurrenceTable[{a[n+1] == a[n]^2 + n*a[n] + n^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
nxt[{n_, a_}] := {n + 1, a^2 + n*a + n^2}; NestList[nxt,{1,1},10][[All,2]] (* Harvey P. Dale, Sep 12 2018 *)

a(n) ~ c^(2^n), where c = 1.460594463210996899745335217057197049886968082959330102210304982704405107261... . - Vaclav Kotesovec, Dec 18 2014

A143763 a(n+1) = (a(n)-n)^2, a(1) = 1.

Original entry on oeis.org

1, 0, 4, 1, 9, 16, 100, 8649, 74666881, 5575141774264384, 31082205803147712138788845611876, 966103517589229313003894215813508352493573272034098666228778225

Offset: 1

Reinhard Zumkeller, Sep 01 2008

nonn

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=A143762(n),
f(n,-2)=a(n), f(n,+2)=A143764(n), f(n,-3)=A143765(n), f(n,+3)=A143766(n).

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From _Reinhard Zumkeller_, Sep 11 2008]

Cf. A000290.

a=1;Table[a=(n-a)^2,{n,0,16}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)
RecurrenceTable[{a[n+1] == (a[n]-n)^2, a[1] == 1}, a, {n, 1, 15}] (* Vaclav Kotesovec, Dec 18 2014 *)
nxt[{n_,a_}]:={n+1,(a-n)^2}; NestList[nxt,{1,1},12][[All,2]] (* Harvey P. Dale, Feb 02 2022 *)

a(n) ~ c^(2^n), where c = 1.0360416217849060104872070636721393146594472258894124389077919684140867897... . - Vaclav Kotesovec, Dec 18 2014

A143764 a(n+1) = (a(n)+n)^2, a(1) = 1.

Original entry on oeis.org

1, 4, 36, 1521, 2325625, 5408554896900, 29252466072845872288372836, 855706771342998810018458679815602502067088579902649, 732234078522259249473123638277942895348884466303559008943210424176413224873174808848983192723595659649

Offset: 1

Reinhard Zumkeller, Sep 01 2008

nonn

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=A143762(n),
f(n,-2)=A143763(n), f(n,+2)=a(n), f(n,-3)=A143765(n), f(n,+3)=A143766(n).

Index entries for sequences of form a(n+1)=a(n)^2 + ... [_Reinhard Zumkeller_, Sep 11 2008]

Subsequence of A000290.

s=1;lst={};Do[AppendTo[lst,s*=s+=n],{n,0,9}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 20 2009 *)
RecurrenceTable[{a[n+1] == (a[n]+n)^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
FoldList[(#1+#2)^2&,1,Range@8] (* Ivan N. Ianakiev, May 08 2015 *)
nxt[{n_,a_}]:={n+1,(a+n)^2}; NestList[nxt,{1,1},10][[;;,2]] (* Harvey P. Dale, Aug 16 2024 *)

a(n) ~ c^(2^n), where c = 1.581081999938782403516250856478461283288009405562783587423785805068844988... . - Vaclav Kotesovec, Dec 18 2014

A143765 a(n+1) = a(n)^2 - 3na(n) + n^2, a(1) = 1.

Original entry on oeis.org

1, -1, 11, 31, 605, 356975, 127424725111, 16237060566937994735039, 263642135854412795003324875413502371940690649, 69507175797876652622009028770643203522181284529919017784559264153993383198392412717393759

Offset: 1

Reinhard Zumkeller, Sep 01 2008

sign

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=A143762(n),
f(n,-2)=A143763(n), f(n,+2)=A143764(n), f(n,-3)=a(n), f(n,+3)=A143766(n).

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From _Reinhard Zumkeller_, Sep 11 2008]

RecurrenceTable[{a[n+1] == a[n]^2 - 3*n*a[n] + n^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(2^n), where c = 1.22112407764058519321076802441458002508833724488233983960016657090909521541... . - Vaclav Kotesovec, Dec 18 2014

A143766 a(n+1) = a(n)^2 + 3na(n) + n^2, a(1) = 1.

Original entry on oeis.org

1, 5, 59, 4021, 16216709, 262981894041341, 69159476593575838635509822455, 4783033202697364284917104840982811414253511628131328498629, 22877406618105405861781317490149379589769149890660405723416585348109182559037843469373513563751798569651299138846801

Offset: 1

Reinhard Zumkeller, Sep 01 2008

nonn

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=A143762(n),
f(n,-2)=A143763(n), f(n,+2)=A143764(n), f(n,-3)=A143765(n), f(n,+3)=a(n).

			Contribution from _Reinhard Zumkeller_, Sep 11 2008: (Start)
a(4)=A000040(556);
a(5)=19*199*4289;
a(6)=3686299*71340359;
a(7)=5*89*23581*36190079671*182112572569;
A055642(a(8))=58; A001221(a(8))=A001222(a(8))=3;
A055642(A020639(a(8)))=4, A020639(a(8))=2459;
A055642(A006530(a(8)))=43, A006530(a(8))=1145781805709434583439407716589323093429591;
A055642(a(9))=116; A001221(a(9))=A001222(a(9))=3;
A055642(A020639(a(9)))=5, A020639(a(9))=52291;
A055642(A087039(a(9)))=39, A087039(a(9))=823717865733493312451872329574156137131;
A055642(A006530(a(9)))=72, A006530(a(9))=531130643259166452223939782963931943654770628199012648274446497807560081;
factorizations made with Dario Alpern's ECM applet. (End)

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From _Reinhard Zumkeller_, Sep 11 2008]
Dario A. Alpern, Factorization using the Elliptic Curve Method [From _Reinhard Zumkeller_, Sep 11 2008]

RecurrenceTable[{a[n+1] == a[n]^2 + 3*n*a[n] + n^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(2^n), where c = 1.68000796750332615134775696497253700657744224375254906378714756508286... . - Vaclav Kotesovec, Dec 18 2014

cp's OEIS Frontend

A143761 a(n+1) = a(n)^2 - n*a(n) + n^2, a(1) = 1.

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A143762 a(n+1) = a(n)^2 + n*a(n) + n^2, a(1) = 1.

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A143763 a(n+1) = (a(n)-n)^2, a(1) = 1.

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A143764 a(n+1) = (a(n)+n)^2, a(1) = 1.

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Mathematica

Formula

A143765 a(n+1) = a(n)^2 - 3*n*a(n) + n^2, a(1) = 1.

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Programs

Mathematica

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A143766 a(n+1) = a(n)^2 + 3*n*a(n) + n^2, a(1) = 1.

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A143765 a(n+1) = a(n)^2 - 3na(n) + n^2, a(1) = 1.

A143766 a(n+1) = a(n)^2 + 3na(n) + n^2, a(1) = 1.