A016922 -id:A016922 - cp's OEIS frontend

A134934 a(n) = (14*n+1)^2.

1, 225, 841, 1849, 3249, 5041, 7225, 9801, 12769, 16129, 19881, 24025, 28561, 33489, 38809, 44521, 50625, 57121, 64009, 71289, 78961, 87025, 95481, 104329, 113569, 123201, 133225, 143641, 154449, 165649, 177241, 189225, 201601, 214369, 227529, 241081

Offset: 0

Hans Isdahl, Jan 26 2008

Number of rats in population after n years, starting with one rat at year 0 (see A016754 for more details).

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+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), this sequence (m=14).

Cf. A016754.

[(14*n+1)^2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

(14*Range[0,50]+1)^2 (* G. C. Greubel, Dec 24 2022 *)

a(n)=(14*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(14*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

O.g.f.: (1+222*x+169*x^2)/(1-x)^3 = 169/(1-x) - 560/(1-x)^2 + 392/(1-x)^3. - R. J. Mathar, Jan 31 2008
a(n) = A016754(7*n).
E.g.f.: (1 + 224*x + 196*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

A016924 a(n) = (6*n + 1)^4.

Original entry on oeis.org

1, 2401, 28561, 130321, 390625, 923521, 1874161, 3418801, 5764801, 9150625, 13845841, 20151121, 28398241, 38950081, 52200625, 68574961, 88529281, 112550881, 141158161, 174900625, 214358881, 260144641, 312900721, 373301041, 442050625, 519885601, 607573201, 705911761

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016921, A016922, A016923, A016925, A016926.

[(6*n+1)^4: n in [0..50]]; // Vincenzo Librandi, May 04 2011

(6Range[0,30]+1)^4 (* Harvey P. Dale, May 18 2011 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^4 = A016922(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(3, 1/6)/7776. (End)

A016925 a(n) = (6*n + 1)^5.

Original entry on oeis.org

1, 16807, 371293, 2476099, 9765625, 28629151, 69343957, 147008443, 282475249, 503284375, 844596301, 1350125107, 2073071593, 3077056399, 4437053125, 6240321451, 8587340257, 11592740743, 15386239549, 20113571875, 25937424601, 33038369407, 41615795893, 51888844699

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Marián Štofka, Problem 11715, The American Mathematical Monthly, Vol. 120, No. 6 (2013), p. 569; An Infinite Sum Introduces a Zeta, Solution to Problem 11715 by Michael Hoffman, ibid., Vol. 122, No. 6 (2015), pp. 608-609.
Roberto Tauraso, Problem 11715.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A016921, A016922, A016923, A016924, A016926.

[(6*n+1)^5: n in [0..50]]; // Vincenzo Librandi, May 04 2011

Table[(6*n + 1)^5, {n, 0, 40}] (* Amiram Eldar, Mar 28 2022 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^5.
Sum_{n>=0} 1/a(n) = ((1-1/2^5)*(1-1/3^5)*zeta(5) + 11*(Pi/3)^5/(8*sqrt(3)))/2 (Štofka, 2013). (End)

A016926 a(n) = (6*n + 1)^6.

Original entry on oeis.org

1, 117649, 4826809, 47045881, 244140625, 887503681, 2565726409, 6321363049, 13841287201, 27680640625, 51520374361, 90458382169, 151334226289, 243087455521, 377149515625, 567869252041, 832972004929, 1194052296529, 1677100110841, 2313060765625, 3138428376721

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016921, A016922, A016923, A016924, A016925.

[(6*n+1)^6: n in [0..40]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+1)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,117649,4826809,47045881,244140625,887503681,2565726409},20] (* Harvey P. Dale, Aug 19 2019 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^6 = A016922(n)^3 = A016923(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(5, 1/6)/5598720. (End)

A016927 a(n) = (6*n + 1)^7.

Original entry on oeis.org

1, 823543, 62748517, 893871739, 6103515625, 27512614111, 94931877133, 271818611107, 678223072849, 1522435234375, 3142742836021, 6060711605323, 11047398519097, 19203908986159, 32057708828125, 51676101935731, 80798284478113, 122987386542487, 182803912081669

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926.

[(6*n+1)^7: n in [0..40]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+1)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,823543,62748517,893871739,6103515625,27512614111,94931877133,271818611107},20] (* Harvey P. Dale, May 12 2015 *)

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, May 12 2015
From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^7.
Sum_{n>=0} 1/a(n) = 301*Pi^7/(1049760*sqrt(3)) + 138811*zeta(7)/279936. (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

A016928 a(n) = (6*n + 1)^8.

Original entry on oeis.org

1, 5764801, 815730721, 16983563041, 152587890625, 852891037441, 3512479453921, 11688200277601, 33232930569601, 83733937890625, 191707312997281, 406067677556641, 806460091894081, 1517108809906561, 2724905250390625, 4702525276151521, 7837433594376961

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927.

[(6*n+1)^8: n in [0..30]]; // Vincenzo Librandi, May 04 2011

Table[(6*n + 1)^8, {n, 0, 20}] (* Amiram Eldar, Mar 28 2022 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^8 = A016922(n)^4 = A016924(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(7, 1/6)/8465264640. (End)

A016929 a(n) = (6*n + 1)^9.

Original entry on oeis.org

1, 40353607, 10604499373, 322687697779, 3814697265625, 26439622160671, 129961739795077, 502592611936843, 1628413597910449, 4605366583984375, 11694146092834141, 27206534396294947, 58871586708267913, 119851595982618319, 231616946283203125, 427929800129788411

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928.

[(6*n+1)^9: n in [0..25]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+1)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,40353607,10604499373,322687697779,3814697265625,26439622160671,129961739795077,502592611936843,1628413597910449,4605366583984375},20] (* Harvey P. Dale, Mar 22 2015 *)

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Mar 22 2015
From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^9 = A016923(n)^3.
Sum_{n>=0} 1/a(n) = 15371*Pi^9/(529079040*sqrt(3)) + 5028751*zeta(9)/10077696. (End)

A016930 a(n) = (6*n + 1)^10.

Original entry on oeis.org

1, 282475249, 137858491849, 6131066257801, 95367431640625, 819628286980801, 4808584372417849, 21611482313284249, 79792266297612001, 253295162119140625, 713342911662882601, 1822837804551761449, 4297625829703557649, 9468276082626847201, 19687440434072265625

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. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928, A016929.

[(6*n+1)^10: n in [0..25]]; // Vincenzo Librandi, May 04 2011

(6 Range[0, 15] + 1)^10 (* Wesley Ivan Hurt, Jan 15 2022 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,282475249,137858491849,6131066257801,95367431640625,819628286980801,4808584372417849,21611482313284249,79792266297612001,253295162119140625,713342911662882601},20] (* Harvey P. Dale, Sep 05 2023 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^10 = A016922(n)^5 = A016925(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(9, 1/6)/21941965946880. (End)

A016931 a(n) = (6*n + 1)^11.

Original entry on oeis.org

1, 1977326743, 1792160394037, 116490258898219, 2384185791015625, 25408476896404831, 177917621779460413, 929293739471222707, 3909821048582988049, 13931233916552734375, 43513917611435838661, 122130132904968017083, 313726685568359708377, 747993810527520928879

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. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928, A016929, A016930.

[(6*n+1)^11: n in [0..25]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+1)^11 (* Harvey P. Dale, Aug 23 2017 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^11.
Sum_{n>=0} 1/a(n) = 1261501*Pi^11/(428554022400*sqrt(3)) + 181308931*zeta(11)/362797056. (End)

cp's OEIS Frontend

A134934 a(n) = (14*n+1)^2.

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A016924 a(n) = (6*n + 1)^4.

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A016925 a(n) = (6*n + 1)^5.

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A016926 a(n) = (6*n + 1)^6.

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Magma

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A016927 a(n) = (6*n + 1)^7.

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Magma

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A104777 Integer squares congruent to 1 mod 6.

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A016928 a(n) = (6*n + 1)^8.

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A016929 a(n) = (6*n + 1)^9.

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