A016970 -id:A016970 - cp's OEIS frontend

A016972 a(n) = (6*n + 5)^4.

625, 14641, 83521, 279841, 707281, 1500625, 2825761, 4879681, 7890481, 12117361, 17850625, 25411681, 35153041, 47458321, 62742241, 81450625, 104060401, 131079601, 163047361, 200533921, 244140625, 294499921, 352275361, 418161601, 492884401, 577200625, 671898241

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 (5,-10,10,-5,1).

Cf. A016969, A016970, A016971.

Subsequence of A000583.

[(6*n+5)^4: n in [0..40]]; // Vincenzo Librandi, May 07 2011

Table[(6 n + 5)^4, {n, 0, 20}] (* Michael De Vlieger, Mar 20 2017 *)

From Chai Wah Wu, Mar 20 2017: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-x^4 - 2396*x^3 - 16566*x^2 - 11516*x - 625)/(x - 1)^5. (End)
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^4 = A016970(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(3, 5/6)/7776. (End)

A016973 a(n) = (6*n + 5)^5.

Original entry on oeis.org

3125, 161051, 1419857, 6436343, 20511149, 52521875, 115856201, 229345007, 418195493, 714924299, 1160290625, 1804229351, 2706784157, 3939040643, 5584059449, 7737809375, 10510100501, 14025517307, 18424351793, 23863536599, 30517578125, 38579489651, 48261724457

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 (6,-15,20,-15,6,-1).

Cf. A016969, A016970, A016971, A016972.

Subsequence of A000584.

[(6*n+5)^5: n in [0..30]]; // Vincenzo Librandi, May 07 2011

(6*Range[0,20]+5)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3125,161051,1419857,6436343,20511149,52521875},20] (* Harvey P. Dale, Sep 24 2014 *)

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Sep 24 2014
G.f.: (3125 + 142301*x + 500426*x^2 + 270466*x^3 + 16801*x^4 + x^5)/(-1+x)^6. - Harvey P. Dale, Aug 13 2021
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^5.
Sum_{n>=0} 1/a(n) = 3751*zeta(5)/7776 - 11*Pi^5/(3888*sqrt(3)). (End)

A016974 a(n) = (6*n + 5)^6.

Original entry on oeis.org

15625, 1771561, 24137569, 148035889, 594823321, 1838265625, 4750104241, 10779215329, 22164361129, 42180533641, 75418890625, 128100283921, 208422380089, 326940373369, 496981290961, 735091890625, 1061520150601, 1500730351849, 2081951752609, 2839760855281, 3814697265625

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 (7,-21,35,-35,21,-7,1).

Cf. A016969 (6*n +5), A016970, A016971, A016972, A016973.

Subsequence of A001014 (n^6).

[(6*n+5)^6: n in [0..25]]; // Vincenzo Librandi, May 10 2011

(6Range[0,20]+5)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{15625,1771561,24137569,148035889,594823321,1838265625,4750104241},30] (* Harvey P. Dale, Apr 24 2025 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^6 = A016970(n)^3 = A016971(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(5, 5/6)/5598720. (End)

A016975 a(n) = (6*n + 5)^7.

Original entry on oeis.org

78125, 19487171, 410338673, 3404825447, 17249876309, 64339296875, 194754273881, 506623120463, 1174711139837, 2488651484819, 4902227890625, 9095120158391, 16048523266853, 27136050989627, 44231334895529, 69833729609375, 107213535210701, 160578147647843, 235260548044817

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 (8,-28,56,-70,56,-28,8,-1).

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

Subsequence of A001015 (n^7).

[(6*n+5)^7: n in [0..25]]; // Vincenzo Librandi, May 11 2011

(6Range[0,20]+5)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{78125,19487171,410338673,3404825447,17249876309,64339296875,194754273881,506623120463},20] (* Harvey P. Dale, Jan 30 2013 *)

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Jan 30 2013
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^7.
Sum_{n>=0} 1/a(n) = 138811*zeta(7)/279936 - 301*Pi^7/(1049760*sqrt(3)). (End)

A104777 Integer squares congruent to 1 mod 6.

Original entry on oeis.org

1, 25, 49, 121, 169, 289, 361, 529, 625, 841, 961, 1225, 1369, 1681, 1849, 2209, 2401, 2809, 3025, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 8281, 9025, 9409, 10201, 10609, 11449, 11881, 12769, 13225, 14161, 14641, 15625, 16129

Offset: 1

Michael Somos, Mar 24 2005

Exponents of powers of q in expansion of eta(q^24).
Odd squares not divisible by 3. - Reinhard Zumkeller, Nov 14 2015
From Peter Bala, Jan 03 2025: (Start)
Exponents of q in the expansion of q*Product_{n >= 1} (1 - q^(24*n))^5/(1 - q^(48*n))^2 = q - 5*q^(5^2) + 7*q^(7^2) - 11*q^(11^2) + 13*q^(13^2) - 17*q^(17^2) + 19*q^(19)^2 - + ... (a consequence of the quintuple product identity).
Also, exponents in the expansion of q*Product_{n >= 1} (1 - q^(48*n))^13 / ( (1 - q^(24*n))*(1 - q^(96*n)) )^5 = q + 5*q^(5^2) + 7*q^(7^2) + 11*q^(11^2) - 13*q^(13^2) - 17*q^(17^2) - 19*q^(19^2) - 23*q^(23^2) + + + + - - - - ... (see Oliver, Theorem 1.1). (End)

			eta(q^24) = q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1)

Disjoint union of A016922 and A016970.

Cf. A007310, A001318, A033683, A100044.

a104777 = (^ 2) . a007310  -- Reinhard Zumkeller, Nov 14 2015

seq(9*(n-1/2)^2 + 1/4 + (-1)^n * (3*n - 3/2), n = 1 .. 100); # Robert Israel, Dec 12 2014

Select[Range[130]^2,Mod[#,6]==1&] (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,25,49,121,169},50] (* Harvey P. Dale, Mar 09 2017 *)

PARI
```
{a(n) = (3*n - 1 - n%2)^2};
```

A033683(a(n)) = 1.
G.f.: ( -1-24*x-22*x^2-24*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Feb 20 2011
a(n) = A007310(n)^2 = 1 + 24*A001318(n-1).
a(n) = 9*n^2 - 9*n + 5/2 + (-1)^n * (3*n - 3/2).  a(n+4) = 2*a(n+2) - a(n) + 72. - Robert Israel, Dec 12 2014
a(n) == 1 (mod 24). - Joerg Arndt, Jan 03 2017
Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - Amiram Eldar, Dec 19 2020

A016976 a(n) = (6*n + 5)^8.

Original entry on oeis.org

390625, 214358881, 6975757441, 78310985281, 500246412961, 2251875390625, 7984925229121, 23811286661761, 62259690411361, 146830437604321, 318644812890625, 645753531245761, 1235736291547681, 2252292232139041, 3936588805702081, 6634204312890625, 10828567056280801

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 (9,-36,84,-126,126,-84,36,-9,1).

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

Subsequence of A001016 (n^8).

[(6*n+5)^8: n in [0..20]]; // Vincenzo Librandi, May 12 2011

Table[(6*n + 5)^8, {n, 0, 20}] (* Amiram Eldar, Apr 01 2022 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^8 = A016970(n)^4 = A016972(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(7, 5/6)/8465264640. (End)

A017222 a(n) = (9*n + 5)^2.

Original entry on oeis.org

25, 196, 529, 1024, 1681, 2500, 3481, 4624, 5929, 7396, 9025, 10816, 12769, 14884, 17161, 19600, 22201, 24964, 27889, 30976, 34225, 37636, 41209, 44944, 48841, 52900, 57121, 61504, 66049, 70756

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences of the form (m*n+5)^2: A010864 (m=0), A000290 (m=1), A016754 (m=2), A016790 (m=3), A016814 (m=4), A016850 (m=5), A016970 (m=6), A017042 (m=7), A017126 (m=8), this sequence (m=9), A017330 (m=10), A017450 (m=11), A017582 (m=12).

Cf. A017221, A017246.

[(9*n+5)^2: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

(9Range[0,30]+5)^2 (* or *) LinearRecurrence[{3,-3,1},{25,196,529},30] (* Harvey P. Dale, May 22 2012 *)

a(n)=(9*n+5)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(9*n+5)^2 for n in range(41)] # G. C. Greubel, Dec 29 2022

a(n) = A017221(n)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 22 2012
G.f.: (25 + 121*x + 16*x^2)/(1-x)^3. - R. J. Mathar, Mar 20 2018
From G. C. Greubel, Dec 29 2022: (Start)
a(2*n+1) = 4*A017246(n).
a(n) = a(n-1) + 9*(18*n + 1).
E.g.f.: (25 + 171*x + 81*x^2)*exp(x). (End)

A016977 a(n) = (6*n + 5)^9.

Original entry on oeis.org

1953125, 2357947691, 118587876497, 1801152661463, 14507145975869, 78815638671875, 327381934393961, 1119130473102767, 3299763591802133, 8662995818654939, 20711912837890625, 45848500718449031, 95151694449171437, 186940255267540403, 350356403707485209, 630249409724609375

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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016969, A016970, A016971, A016972, A016973, A016974, A016975, A016976.

Subsequence of A001017.

[(6*n+5)^9: n in [0..35]]; // Vincenzo Librandi, May 13 2011

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^9 = A016971(n)^3.
Sum_{n>=0} 1/a(n) = 5028751*zeta(9)/10077696 - 15371*Pi^9/(529079040*sqrt(3)). (End)

A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n.

Original entry on oeis.org

30, 132, 306, 552, 870, 1260, 1722, 2256, 2862, 3540, 4290, 5112, 6006, 6972, 8010, 9120, 10302, 11556, 12882, 14280, 15750, 17292, 18906, 20592, 22350, 24180, 26082, 28056, 30102, 32220, 34410, 36672, 39006, 41412, 43890, 46440, 49062, 51756, 54522, 57360, 60270, 63252

Offset: 1

Craig Knecht, Aug 30 2017

The equilateral triangle composed of 144 smaller equilateral triangles is the smallest triangle that can be tiled with the sphinx. This triangle is used to form all orders of the hexagon.
Walter Trump enumerated all 830 sphinx tilings of this triangle and found six symmetrical examples one of which is used to produce this sequence.
Hyper-packing is a term that describes the ability of a shape to contain a greater area of subshapes than its own area by overlapping the subshapes. There are 864 unit triangles in the order 1 hexagon. 30 of the subshapes hyper-packed into this hexagon would contain 30x6x6 or 1080 unit triangles if summed individually.
The prime numbers cannot be described by a formula. Subsets of the primes such as the balanced primes are more formula friendly (see comments to puzzle 920 below). - Craig Knecht, Apr 19 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Craig Knecht, Eight sphinx tile frame tessellations.
Craig Knecht, Example for the sequence.
Craig Knecht, Order 4 Hexagon with 246 subshapes.
Craig Knecht, Sphinx Tile Component Triangles.
Craig Knecht, Sphinx tiling of the triangle used to make the hexagon.
Craig Knecht, T12 symmetric sphinx tilings.
Carlos Rivera, Puzzle 920. An enigma related to A291582, Primes puzzles and problems connections.
Wikipedia, Walter Trump.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016969, A049452.

Cf. A279887, A287999.

List([1..30], n -> 6*n*(6*n-1)); # G. C. Greubel, Dec 04 2018

[6*n*(6*n-1): n in [1..50]]; // Vincenzo Librandi, Sep 20 2017

seq(6*n*(6*n-1),n=1..100); # Robert Israel, Sep 19 2017

Array[6 # (6 # - 1) &, 42] (* Michael De Vlieger, Sep 19 2017 *)
CoefficientList[Series[2(15 + 21 x)/(1-x)^3,{x, 0, 50}], x] (* Vincenzo Librandi, Sep 20 2017 *)
CoefficientList[Series[6 E^x (5 + 17 x + 6 x^2), {x, 0, 50}], x]*
Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 07 2018 *)

a(n) = 6*n*(6*n-1); \\ Altug Alkan, Apr 08 2018

[6*n*(6*n-1) for n in (1..50)] # G. C. Greubel, Dec 04 2018

a(n) = 6*n*(6*n-1). - Walter Trump
G.f.: 2*x*(15+21*x)/(1-x)^3. - Vincenzo Librandi, Sep 20 2017
a(n) = 6*A049452(n) = 6*n*A016969(n-1). - Torlach Rush, Nov 28 2018
E.g.f.: 6*exp(x)*(5 + 17*x + 6*x^2). - Stefano Spezia, Dec 07 2018
a(n) = A016970(n-1) + A016969(n-1). - Torlach Rush, Dec 10 2018
From Amiram Eldar, Jul 30 2024: (Start)
Sum_{n>=1} 1/a(n) = log(2)/3 + log(3)/4 - sqrt(3)*Pi/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/6 - arccoth(sqrt(3))/sqrt(3). (End)

A016978 a(n) = (6*n + 5)^10.

Original entry on oeis.org

9765625, 25937424601, 2015993900449, 41426511213649, 420707233300201, 2758547353515625, 13422659310152401, 52599132235830049, 174887470365513049, 511116753300641401, 1346274334462890625, 3255243551009881201, 7326680472586200649, 15516041187205853449

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

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

Subsequence of A008454 (n^10).

[(6*n+5)^10: n in [0..30]]; // Vincenzo Librandi, May 14 2011

(6*Range[0,20]+5)^10 (* Harvey P. Dale, Aug 26 2012 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^10 = A016970(n)^5 = A016973(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(9, 5/6)/21941965946880. (End)

cp's OEIS Frontend

A016972 a(n) = (6*n + 5)^4.

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A016973 a(n) = (6*n + 5)^5.

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A016974 a(n) = (6*n + 5)^6.

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A016975 a(n) = (6*n + 5)^7.

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A104777 Integer squares congruent to 1 mod 6.

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A016976 a(n) = (6*n + 5)^8.

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Magma

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A017222 a(n) = (9*n + 5)^2.

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A016977 a(n) = (6*n + 5)^9.

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