A153058 -id:A153058 - cp's OEIS frontend

A056520 a(n) = (n + 2)(2n^2 - n + 3)/6.

1, 2, 6, 15, 31, 56, 92, 141, 205, 286, 386, 507, 651, 820, 1016, 1241, 1497, 1786, 2110, 2471, 2871, 3312, 3796, 4325, 4901, 5526, 6202, 6931, 7715, 8556, 9456, 10417, 11441, 12530, 13686, 14911, 16207, 17576, 19020, 20541, 22141, 23822

Offset: 0

N. J. A. Sloane, Laura Kasavan (maui12129(AT)cswebmail.com), Aug 26 2000

Hankel transform of A030238. - Paul Barry, Oct 16 2007
Equals (1, 2, 3, 4, 5, ...) convolved with (1, 0, 3, 5, 7, 9, ...). - Gary W. Adamson, Jul 31 2010
a(n) equals n!^2 times the determinant of the n X n matrix whose (i,j)-entry is 1 + KroneckerDelta[i, j] (-1 + (1 + i^2)/i^2). - John M. Campbell, May 20 2011
Positions of ones in A253903 (with offset 1). - Harvey P. Dale, Mar 05 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 25 2012
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A153056, A153057, A153058, A179904.

Cf. A030238, A253903.

[(n+2)*(2*n^2-n+3)/6: n in [0..40]]; // Vincenzo Librandi, May 24 2011

a[n_] := (n+2)*(2*n^2-n+3)/6; Table[a[n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
s = 1; lst = {s}; Do[s += n^2; AppendTo[lst, s], {n, 1, 41, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
Table[n!^2*Det[Array[KroneckerDelta[#1,#2](((#1^2+1)/(#1^2))-1)+1&,{n,n}]],{n,1,20}] (* John M. Campbell, May 20 2011 *)
FoldList[#1 + #2^2 &, 1, Range@ 40] (* Robert G. Wilson v, Oct 28 2011 *)

a(n)=(n+2)*(2*n^2-n+3)/6 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = a(n-1) + n^2.
a(n) = A000330(n) + 1.
G.f.: (1 - 2*x + 4*x^2 - x^3)/(1 - x)^4. - Paul Barry, Apr 14 2010
Let b(0) = b(1) = 1, b(n) = max(b(n-1) + (n - 1)^2, b(n-2) + (n - 2)^2) for n >= 2; then a(n) = b(n+1). - Yalcin Aktar, Jul 28 2011

A028300 a(1) = 1; thereafter a(n+1) = a(n)^2 + n.

Original entry on oeis.org

1, 2, 6, 39, 1525, 2325630, 5408554896906, 29252466072845872288372843, 855706771342998810018458679815602502067088579902657

Offset: 1

A.R. Fink (fink(AT)cadvision.com)

			a(3) = a(2)^2+2 = 4+2 = 6.

Indranil Ghosh, Table of n, a(n) for n = 1..13
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A086851, A098152, A153058, A153059, A153060.

[n le 1 select 1 else Self(n-1)^2 +n-1: n in [1..14]]; // G. C. Greubel, Jan 03 2024

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

def a(n): return 1 if n==1 else a(n-1)^2 + n-1 # a = A028300
[a(n) for n in range(1,15)] # G. C. Greubel, Jan 03 2024

a(n) ~ c^(2^n), where c = 1.2574108318043003098123273077302829405940294\ 604970047023808427877694426442... . - Vaclav Kotesovec, Dec 18 2014

A098152 a(n) = a(n-1)^2 + n, with a(0)=0.

Original entry on oeis.org

0, 1, 3, 12, 148, 21909, 480004287, 230404115538378376, 53086056457022411804685755744397384, 2818129390158170901506703075470572449397357853477615482257305306043465

Offset: 0

Henry Bottomley, Oct 25 2004

nonn

			a(4) = a(3)^2 + 4 =12^2 + 4 = 148.

Indranil Ghosh, Table of n, a(n) for n = 0..12
Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Zbigniew Gołębiewski, On the number of unary-binary tree-like structures with restrictions on the unary height, arXiv:1510.01167 [math.CO], 2015, see Table 1, p. 19.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A153058, A153059, A153060, A086851.

[0] cat [n eq 1 select 1 else Self(n-1)^2+n: n in [1..10]]; // Vincenzo Librandi, Oct 06 2015

a=0; lst={}; Do[a=a^2+n; AppendTo[lst, a], {n, 0, 11}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
RecurrenceTable[{a[0]==0,a[n]==a[n-1]^2+n},a,{n,10}] (* Harvey P. Dale, Jul 28 2012 *)

For n>0, a(n) = floor(1.366609561487624975914833969579996...^(2^n)) = floor(A028300(n)^0.68178667449368682115305109818...) = ceiling(A003095(n)^1.53346965582393874689368175542252...).

A153056 a(0)=2, a(n) = n^2+a(n-1).

Original entry on oeis.org

2, 3, 7, 16, 32, 57, 93, 142, 206, 287, 387, 508, 652, 821, 1017, 1242, 1498, 1787, 2111, 2472, 2872, 3313, 3797, 4326, 4902, 5527, 6203, 6932, 7716, 8557, 9457, 10418, 11442, 12531, 13687, 14912, 16208, 17577, 19021, 20542, 22142, 23823, 25587

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A056520, A153057, A153058, A179904.

a=2;lst={};Do[a=n^2+a;AppendTo[lst,a],{n,0,5!}];lst
nxt[{n_,a_}]:={n+1,(n+1)^2+a}; NestList[nxt,{0,2},50][[;;,2]] (* or *) LinearRecurrence[{4,-6,4,-1},{2,3,7,16},50] (* Harvey P. Dale, Sep 05 2023 *)

a(n) = n*(n+1)*(2*n+1)/6 + 2; \\ Altug Alkan, Apr 30 2018

G.f.: (2-5x+7x^2-2x^3)/(1-x)^4. a(n)=2+n(1+2n^2+3n)/6 = 2+A000330(n). - R. J. Mathar, Jan 08 2009

A153057 a(0)=3; a(n) = n^2 + a(n-1) for n>0.

Original entry on oeis.org

3, 4, 8, 17, 33, 58, 94, 143, 207, 288, 388, 509, 653, 822, 1018, 1243, 1499, 1788, 2112, 2473, 2873, 3314, 3798, 4327, 4903, 5528, 6204, 6933, 7717, 8558, 9458, 10419, 11443, 12532, 13688, 14913, 16209, 17578, 19022, 20543, 22143, 23824, 25588

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A056520, A153056, A153058, A179904.

I:=[3,4,8,17]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, May 09 2017

a=3;lst={};Do[a=n^2+a;AppendTo[lst,a],{n,0,5!}];lst
CoefficientList[Series[(3 - 8 x + 10 x^2 - 3 x^3) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 09 2017 *)

From R. J. Mathar, Jan 17 2009: (Start)
G.f.: (3-8*x + 10*x^2 - 3*x^3)/(1 - x)^4.
a(n) = 3+A000330(n). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 09 2017

Added indices to definition. Corrected offset R. J. Mathar, Jan 17 2009

A179904 a(n) = A056520(n)+1 for n>0, a(0)=1.

Original entry on oeis.org

1, 3, 7, 16, 32, 57, 93, 142, 206, 287, 387, 508, 652, 821, 1017, 1242, 1498, 1787, 2111, 2472, 2872, 3313, 3797, 4326, 4902, 5527, 6203, 6932, 7716, 8557, 9457, 10418, 11442, 12531, 13687, 14912, 16208, 17577, 19021, 20542, 22142, 23823, 25587

Offset: 0

Gary W. Adamson, Jul 31 2010

Original name: (1,3,5,7,9,..) = A005408 convolved with (1,0,2,3,4,..) = 1 followed by A087156.

			a(3) = 16 = 1 + A056520(3) = (1 + 15).
a(4) = 32 = (9, 7, 5, 3, 1) dot (1, 0, 2, 3, 4) = (9 + 0 + 10 + 9 + 4).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A056520, A153056, A153057, A153058.

LinearRecurrence[{4,-6,4,-1},{1,3,7,16,32},50] (* Harvey P. Dale, Apr 25 2020 *)

From Bruno Berselli, Aug 26 2011: (Start)
G.f.: (1 + x)*(1 - 2*x + 3*x^2 - x^3)/(1 - x)^4.
a(n) = (1/6)*(2*n^3 + 3*n^2 + n + 12) for n>0, a(0)=1. (End)
a(n) = A153056(n) for n > 0. - Georg Fischer, Oct 24 2018

More terms and a(20) added by Bruno Berselli, Aug 26 2011

A153062 a(0)=1, a(n)=a(n-1)^2-n^2.

Original entry on oeis.org

1, 0, -4, 7, 33, 1064, 1132060, 1281559843551, 1642395632602463596289537, 2697463413991646582332856337058890220473935674288, 7276308869823469318922107258255539214217660183827400648026742290333726278585952350082821224306844

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 17 2008

sign

Indranil Ghosh, Table of n, a(n) for n = 0..13
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A153058, A153059, A086851, A153060, A098152, A028300, A153061

a=1;lst={};Do[a=a^2-n^2;AppendTo[lst,a],{n,0,12}];lst
RecurrenceTable[{a[0]==1, a[n]==a[n-1]^2-n^2}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(2^n), where c = 1.24334516566159378905145559484975509773481252881469517052010008004602693351819... . - Vaclav Kotesovec, Dec 18 2014

cp's OEIS Frontend

A056520 a(n) = (n + 2)*(2*n^2 - n + 3)/6.

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A028300 a(1) = 1; thereafter a(n+1) = a(n)^2 + n.

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A098152 a(n) = a(n-1)^2 + n, with a(0)=0.

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A153056 a(0)=2, a(n) = n^2+a(n-1).

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A153057 a(0)=3; a(n) = n^2 + a(n-1) for n>0.

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A179904 a(n) = A056520(n)+1 for n>0, a(0)=1.

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A153062 a(0)=1, a(n)=a(n-1)^2-n^2.

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A056520 a(n) = (n + 2)(2n^2 - n + 3)/6.