A085601 -id:A085601 - cp's OEIS frontend

A102083 a(n) = 8n^2 + 4n + 1.

1, 13, 41, 85, 145, 221, 313, 421, 545, 685, 841, 1013, 1201, 1405, 1625, 1861, 2113, 2381, 2665, 2965, 3281, 3613, 3961, 4325, 4705, 5101, 5513, 5941, 6385, 6845, 7321, 7813, 8321, 8845, 9385, 9941, 10513, 11101, 11705, 12325, 12961, 13613, 14281, 14965, 15665

Offset: 0

N. J. A. Sloane, Feb 14 2005

If Y and Z are 2-blocks of a 2n-set X then, for n>=2, a(n-2) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Nov 18 2007
Equals binomial transform of [1, 12, 16, 0, 0, 0, ...]. - Gary W. Adamson, Jul 19 2008
Sequence found by reading the line from 1, in the direction 1, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 05 2011
First differences of A100157. - John Molokach, Jul 10 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A085601, A195241.

Table[8*n^2 + 4*n + 1, {n, 0, 300}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 13, 41}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)

a(n)=8*n^2+4*n+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+10*x+5*x^2)/(1-x)^3. - Paul Barry, Jun 04 2005
a(n) = 4*(4*n-1)+a(n-1) (with a(0)=1). - Vincenzo Librandi, Nov 16 2010
E.g.f.: (8*x^2 + 12*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017

A220978 a(n) = 3^(2n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6n+3) + 1.

Original entry on oeis.org

1, 19, 217, 2107, 19441, 176419, 1592137, 14342347, 129120481, 1162202419, 10460176057, 94142647387, 847287015121, 7625592702019, 68630363015977, 617673353237227, 5559060437415361, 50031544711579219, 450283904728735897, 4052555149532191867

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A198410(n+2): 3^(6*n+3) + 1 = (3^(2*n+1) + 1) * a(n) * A198410(n+2).

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (13, -39, 27).

Cf. A092440, A085601, A198410, A220979-A220990.

Table[3^(2n+1) - 3^(n+1) + 1, {n, 0, 30}]
LinearRecurrence[{13,-39,27},{1,19,217},30] (* Harvey P. Dale, Mar 17 2013 *)

Vec((1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)) + O(x^30)) \\ Michel Marcus, Feb 12 2015

a(n) = 13*a(n-1) - 39*a(n-2) + 27*a(n-3).

G.f.: (1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)).

A220979 a(n) = 5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1: the left Aurifeuillian factor of 5^(10n+5) - 1.

Original entry on oeis.org

11, 12851, 9384251, 6054921251, 3808599606251, 2383422998031251, 1490020755615156251, 931310653778075781251, 582075119020843503906251, 363797694444713592519531251, 227373652160169124603222656251, 142108544241637027263641113281251

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220980.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (781,-101530,2538250,-12203125,9765625).

Cf. A092440, A085601, A220978, A198410, A220980-A220990.

Table[5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1, {n, 0, 30}]

a(n)=5^(4*n+2)-5^(3*n+2)+3*5^(2*n+1)-5^(n+1)+1 \\ Charles R Greathouse IV, Sep 28 2015

Aurifeuillian factorization: 5^(10n+5) - 1 = (5^(2n+1) - 1) * a(n) * A220980(n).

G.f.: -(4296875*x^4+2662500*x^3+464450*x^2+4260*x+11) / ((x-1)*(5*x-1)*(25*x-1)*(125*x-1)*(625*x-1)). - Colin Barker, Jan 03 2013

A220990 a(n) = 12^(2n+1) + 6 * 12^n + 1: the right Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

19, 1801, 249697, 35842177, 5159904769, 743009863681, 106993223294977, 15407021789577217, 2218611109320327169, 319479999401581608961, 46005119909741205651457, 6624737266953695061344257, 953962166440743626203987969

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220989.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (157,-1884,1728).

Cf. A092440, A085601, A220978, A198410, A220979-A220989.

Table[12^(2n+1) + 6 * 12^n + 1, {n, 0, 10}]
LinearRecurrence[{157,-1884,1728},{19,1801,249697},20] (* Harvey P. Dale, Mar 26 2022 *)

a(n)=12^(2*n+1)+6*12^n+1 \\ Charles R Greathouse IV, Sep 28 2015

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * A220989(n) * a(n).

G.f.: -(2736*x^2-1182*x+19) / ((x-1)*(12*x-1)*(144*x-1)). - Colin Barker, Jan 03 2013

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

113, 34925927, 4651514210561, 556919483179733591, 65684998500756890925713, 7730533744900130305342957127, 909535949164303794596648514307361, 107006774488854204226839526889653524791, 12589253114717671385404089651370543317211313

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220984.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (137257, -2354918349, 5670354183893, -1944931485075299, 95029449572634843, -651636050170246351, 558545864083284007).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1, {n, 0, 20}]

a(n) = 7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * a(n) * A220984(n).
G.f.: -(184010736563880737*x^6 +268740854387875086*x^5 +14564007567924591*x^4 +73553506117028*x^3 +123792021759*x^2 +19415886*x +113) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A220984 The right Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

911, 46489241, 4845303761663, 560176314330212777, 65739735996793498937711, 7731453717973685046293120441, 909551411151743369070229385367263, 107007034358477098527617255914118283977, 12589257482346423369016062830670344414194511

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220983.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (137257, -2354918349, 5670354183893, -1944931485075299, 95029449572634843, -651636050170246351, 558545864083284007).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1, {n, 0, 20}]

a(n) = 7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * A220983(n) * a(n).
G.f.: -(1483484787696419039*x^6 -1087259214306211086*x^5 +71725962948861585*x^4 -562870083909028*x^3 +609660625665*x^2 -78551886*x +911) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A229768 Largest prime factor of 2^(2*n+1)+2^(n+1)+1.

Original entry on oeis.org

13, 41, 29, 109, 2113, 157, 1321, 26317, 525313, 1429, 1657, 268501, 279073, 536903681, 49477, 4327489, 7416361, 231769777, 21841, 43249589, 500177, 29247661, 7484047069, 19707683773, 1326700741, 586477649, 3630105520141, 275415303169, 104399276341

Offset: 1

Colin Barker, Sep 29 2013

nonn

2^(2*n+1)+2^(n+1)+1 is a factor of 4^(2*n+1)+1.

			For n=10, 2^(2*n+1)+2^(n+1)+1 = 2099201 = 13*113*1429, so a(10)=1429.

Daniel Suteu, Table of n, a(n) for n = 1..547

Cf. A085601, A207262, A229747, A229767.

Table[FactorInteger[2^(2n+1)+2^(n+1)+1][[-1,1]],{n,30}] (* Harvey P. Dale, Nov 03 2017 *)

a(n) = {f=factor(2^(2*n+1)+2^(n+1)+1); f[matsize(f)[1],1]}

A257418 Number of pieces after a sheet of paper is folded n times and cut diagonally.

Original entry on oeis.org

2, 3, 5, 8, 13, 23, 41, 77, 145, 281, 545, 1073, 2113, 4193, 8321, 16577, 33025, 65921, 131585, 262913, 525313, 1050113, 2099201, 4197377, 8392705, 16783361, 33562625, 67121153, 134234113, 268460033, 536903681, 1073790977, 2147549185, 4295065601, 8590065665

Offset: 0

Dirk Frettlöh, Apr 22 2015

Fold a rectangular sheet of paper in half (fold lower half up), and again into half (left half to the right), and again (lower half up), and again (left half to the right)... making n folds in all. Cut along the diagonal line from top left to bottom right of the resulting small rectangle. Sequence gives the number of pieces that are formed.

The even-numbered entries of this sequence are A343175 (essentially A085601). The odd numbered entries are A343176 (essentially A036562). [These bisections are easy to analyze and have simpler formulas. - N. J. A. Sloane, Apr 26 2021]

			n=1: Take a rectangular sheet of paper and fold it in half. Cutting along the diagonal of the resulting rectangle yields 3 smaller pieces of paper.
n=0: Cutting the sheet of paper (without any folding) along the diagonal yields two pieces.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A036562, A085601, A343175, A343176.

[2,3,5,8] cat [Floor((2^n+2^(n/2)*(1+(-1)^n+3*Sqrt(2)*(1-(-1)^n)/4)+2)/2):n in [4..40]]; // Vincenzo Librandi, May 05 2015

2, seq(floor((2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2),n=1..25);

Table[Floor[(2^n + 2^(n/2)*(1 + (-1)^n + 3 Sqrt[2]*(1 - (-1)^n)/4) + 2)/2], {n, 0, 25}] (* Michael De Vlieger, Apr 24 2015 *)

concat(2,vector(30,n,round((2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2))) \\ Derek Orr, Apr 27 2015

Vec((2 - 3*x - 4*x^2 + 5*x^3 - x^4 + 2*x^5) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)) + O(x^35)) \\ Colin Barker, Feb 05 2020

a(n) = (2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2 for n>1. (Johan Nilsson)
a(0) = 2, a(1) = 3, a(n+1) = 2*a(n)-2^(floor((n-1)/2))-1.
G.f.: -(2*x^5-x^4+5*x^3-4*x^2-3*x+2)/((x-1)*(2*x-1)*(2*x^2-1)). - Alois P. Heinz, Apr 23 2015
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n>5. - Colin Barker, Feb 05 2020
E.g.f.: (1/4)*(-2 - 2*x + 2*cosh(2*x) + 4*cosh(sqrt(2)*x) + 4*sinh(x) + 4*cosh(x)*(1 + sinh(x)) + 3*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Feb 05 2020

A281482 a(n) = 2^(n + 1) * (2^n + 1) - 1.

Original entry on oeis.org

3, 11, 39, 143, 543, 2111, 8319, 33023, 131583, 525311, 2099199, 8392703, 33562623, 134234111, 536903679, 2147549183, 8590065663, 34360000511, 137439477759, 549756862463, 2199025352703, 8796097216511, 35184380477439, 140737505132543, 562949986975743

Offset: 0

Jaroslav Krizek, Jan 22 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281481 (2^(n - 1) * (2^n + 1) + 1).

[2^(n + 1) * (2^n + 1) - 1: n in [0..200]];

Vec((3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

A343175 a(0)=2; for n > 0, a(n) = 2^(2*n-1) + 2^n + 1.

Original entry on oeis.org

2, 5, 13, 41, 145, 545, 2113, 8321, 33025, 131585, 525313, 2099201, 8392705, 33562625, 134234113, 536903681, 2147549185, 8590065665, 34360000513, 137439477761, 549756862465, 2199025352705, 8796097216513, 35184380477441, 140737505132545, 562949986975745, 2251799880794113

Offset: 0

N. J. A. Sloane, Apr 26 2021

A bisection of A257418. Apart from first term, same as A085601.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A257418, A085601, A343176.

LinearRecurrence[{7,-14,8},{2,5,13,41},30] (* Harvey P. Dale, Aug 04 2024 *)

From Chai Wah Wu, Apr 26 2021: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n > 3.
G.f.: (-4*x^3 - 6*x^2 + 9*x - 2)/((x - 1)*(2*x - 1)*(4*x - 1)). (End)

a(18)-a(26) from Martin Ehrenstein, Apr 26 2021

cp's OEIS Frontend

A102083 a(n) = 8*n^2 + 4*n + 1.

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A220978 a(n) = 3^(2*n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6*n+3) + 1.

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A220979 a(n) = 5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1: the left Aurifeuillian factor of 5^(10n+5) - 1.

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A220990 a(n) = 12^(2n+1) + 6 * 12^n + 1: the right Aurifeuillian factor of 12^(6n+3) + 1.

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A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

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A220984 The right Aurifeuillian factor of 7^(14n+7) + 1.

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A229768 Largest prime factor of 2^(2*n+1)+2^(n+1)+1.

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A257418 Number of pieces after a sheet of paper is folded n times and cut diagonally.

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A102083 a(n) = 8n^2 + 4n + 1.

A220978 a(n) = 3^(2n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6n+3) + 1.