A016970 -id:A016970 - cp's OEIS frontend

A174371 a(n) = (6*n-1)^2.

1, 25, 121, 289, 529, 841, 1225, 1681, 2209, 2809, 3481, 4225, 5041, 5929, 6889, 7921, 9025, 10201, 11449, 12769, 14161, 15625, 17161, 18769, 20449, 22201, 24025, 25921, 27889, 29929, 32041, 34225, 36481, 38809, 41209, 43681, 46225, 48841, 51529

Offset: 0

Juri-Stepan Gerasimov, Mar 17 2010

Unit together with numbers of form (6*n+5)^2.

Sequence may be obtained by starting with the segment (1, 25) followed by the line from 25 in the direction 25, 121,... in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, Jul 29 2016

			a(0)=1 because (6*0-1)^2=1, a(1)=25 because (6*1-1)^2=25.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016970, A033579.

[(6*n-1)^2: n in [0..50]]; // Vincenzo Librandi, May 07 2011

CoefficientList[Series[(49*x^2 + 22*x + 1)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 27 2013 *)

a(n)=(6*n-1)^2 \\ Charles R Greathouse IV, Jul 28 2016

a(n) = A016970(n-1), n >= 1.
G.f.: (49*x^2 + 22*x + 1)/(1 - x)^3. - Vincenzo Librandi, Jan 27 2013
a(n) = 6*A033579(n) + 1. - Miquel Cerda, Jul 28 2016
a(n) = 36n^2 - 12n + 1. - Omar E. Pol, Jul 28 2016
E.g.f.: exp(x)*(1 + 24*x + 36*x^2). - Stefano Spezia, Aug 19 2023

Offset and formula corrected by R. J. Mathar, Apr 16 2010

A016979 a(n) = (6*n + 5)^11.

Original entry on oeis.org

48828125, 285311670611, 34271896307633, 952809757913927, 12200509765705829, 96549157373046875, 550329031716248441, 2472159215084012303, 9269035929372191597, 30155888444737842659, 87507831740087890625, 231122292121701565271, 564154396389137449973

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974, A016975, A016976, A016977, A016978.

Subsequence of A008455 (n^11).

[(6*n+5)^11 : n in [0..20]]; // Vincenzo Librandi, May 17 2011

(6Range[0,20]+5)^11 (* or *) LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{48828125,285311670611,34271896307633,952809757913927,12200509765705829,96549157373046875,550329031716248441,2472159215084012303,9269035929372191597,30155888444737842659,87507831740087890625,231122292121701565271},20] (* Harvey P. Dale, Dec 17 2024 *)

a(n) = (6*n + 5)^11.
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^11.
Sum_{n>=0} 1/a(n) = 181308931*zeta(11)/362797056 - 1261501*Pi^11/(428554022400*sqrt(3)). (End)

A017642 a(n) = (12*n+10)^2.

Original entry on oeis.org

100, 484, 1156, 2116, 3364, 4900, 6724, 8836, 11236, 13924, 16900, 20164, 23716, 27556, 31684, 36100, 40804, 45796, 51076, 56644, 62500, 68644, 75076, 81796, 88804, 96100, 103684, 111556, 119716

Offset: 0

N. J. A. Sloane

			a(5) = (12*5+10)^2 = 70^2 = 4900.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016969, A016970, A017641, A017643.

[(12*n+10)^2 : n in [0..40]]; // Wesley Ivan Hurt, Dec 22 2020

(12*Range[0,30]+10)^2 (* or *) LinearRecurrence[{3,-3,1},{100,484,1156},30] (* Harvey P. Dale, May 08 2017 *)
CoefficientList[Series[4*(25 + 46 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* Wesley Ivan Hurt, Dec 22 2020 *)

a(n)=(12*n+10)^2 \\ Charles R Greathouse IV, Jun 17 2017

From Wesley Ivan Hurt, Dec 22 2020: (Start)
G.f.: 4*(25 + 46*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = 144*n^2 + 240*n + 100.
a(n) = A017641(n)^2. (End)

A016980 a(n) = (6*n + 5)^12.

Original entry on oeis.org

244140625, 3138428376721, 582622237229761, 21914624432020321, 353814783205469041, 3379220508056640625, 22563490300366186081, 116191483108948578241, 491258904256726154641, 1779197418239532716881, 5688009063105712890625, 16409682740640811134241, 43439888521963583647921

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974, A016975, A016976, A016977, A016978, A016979.

Subsequence of A008456 (n^12).

[(6*n+5)^12: n in [0..40]]; // Vincenzo Librandi, May 19 2011

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^12 = A016970(n)^6 = A016971(n)^4 = A016972(n)^3 = A016974(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(11, 5/6)/86890185149644800. (End)

cp's OEIS Frontend

A174371 a(n) = (6*n-1)^2.

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A016979 a(n) = (6*n + 5)^11.

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A017642 a(n) = (12*n+10)^2.

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A016980 a(n) = (6*n + 5)^12.

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