A063249 -id:A063249 - cp's OEIS frontend

A232713 Doubly pentagonal numbers: a(n) = n(3n-2)(3n-1)(3n+1)/8.

0, 1, 35, 210, 715, 1820, 3876, 7315, 12650, 20475, 31465, 46376, 66045, 91390, 123410, 163185, 211876, 270725, 341055, 424270, 521855, 635376, 766480, 916895, 1088430, 1282975, 1502501, 1749060, 2024785, 2331890, 2672670, 3049501, 3464840, 3921225, 4421275

Offset: 0

Bruno Berselli, Nov 28 2013

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000326, A000332.
Cf. similar sequences: A000583 for A000290(A000290(n)); A002817 for A000217(A000217(n)); A063249 for A000384(A000384(n)).
Cf. A236770, A260810.

[n*(3*n-2)*(3*n-1)*(3*n+1)/8: n in [0..40]];

Table[n (3 n - 2) (3 n - 1) (3 n + 1)/8, {n, 0, 40}]

a(n)=n*(3*n-2)*(3*n-1)*(3*n+1)/8 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1 + 30*x + 45*x^2 + 5*x^3) / (1 - x)^5.
a(n) = A000326(A000326(n)) = A000332(3n+1).
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 4 + 2*Pi/sqrt(3) - 6*log(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 32*log(2)/3 - 4*Pi/(3*sqrt(3)) - 4. (End)

A329755 Doubly hexagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475, 243939410, 564840991, 1217275137, 2469392562, 4757404575, 8765621740, 15534503236, 26603512517, 44196596312, 71459197125, 112756874195, 174046844356, 263335062397, 391232840362, 571628456750, 822490729775

Offset: 0

Ilya Gutkovskiy, Nov 20 2019

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000384, A002412, A063249, A140236, A329753, A329754, A329756, A329757.

A002412[n_] := n (n + 1) (4 n - 1)/6; a[n_] := A002412[A002412[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (2 k - 1), {k, 0, n (n + 1) (4 n - 1)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 242 x + 4862 x^2 + 22425 x^3 + 30465 x^4 + 12424 x^5 + 1248 x^6 + 13 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475}, 26]

G.f.: x*(1 + 242*x + 4862*x^2 + 22425*x^3 + 30465*x^4 + 12424*x^5 + 1248*x^6 + 13*x^7)/(1 - x)^10.
a(n) = A002412(A002412(n)).
a(n) = Sum_{k=0..A002412(n)} A000384(k).
a(n) = n *(4*n-1) *(n+1) *(4*n^3+3*n^2-n+6) *(8*n^3+6*n^2-2*n-3) / 648 . - R. J. Mathar, Nov 28 2019

A264891 a(n) = n(5n - 3)(25n^2 - 15*n - 6)/8.

Original entry on oeis.org

0, 1, 112, 783, 2839, 7480, 16281, 31192, 54538, 89019, 137710, 204061, 291897, 405418, 549199, 728190, 947716, 1213477, 1531548, 1908379, 2350795, 2865996, 3461557, 4145428, 4925934, 5811775, 6812026, 7936137, 9193933, 10595614, 12151755, 13873306

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly heptagonal numbers.

G. C. Greubel, Table of n, a(n) for n = 0..1200
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000566, A002817, A000583, A232713, A063249.

[n*(5*n-3)*(25*n^2-15*n-6)/8: n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

seq(n*(5*n - 3)*(25*n^2 - 15*n - 6)/8, n=0..100); # Robert Israel, Dec 02 2015

Table[n (5 n - 3) (25 n^2 - 15 n - 6)/8, {n, 0, 35}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,112,783,2839},40] (* Harvey P. Dale, Nov 19 2019 *)

vector(100, n, n--; n*(5*n-3)*(25*n^2-15*n-6)/8) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 107*x + 233*x^2 + 34*x^3)/(1 - x)^5.
a(n) = A000566(A000566(n)).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 28 2015
Sum_{n>0} 1/a(n) = (4*(sqrt(33)*gamma + sqrt(33)*polygamma(0, 2/5) - 3*polygamma(0, (1/10)*(7 - sqrt(33))) + 3 polygamma(0, (1/10)* (7 + sqrt(33)))))/(9*sqrt(33)) = 1.0108420043...., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.
E.g.f.: x*(8 + 440*x + 600*x^2 + 125*x^3)*exp(x)/8, - Robert Israel, Dec 02 2015

A264892 a(n) = n(3n - 2)(9n^2 - 6*n - 2).

Original entry on oeis.org

0, 1, 176, 1281, 4720, 12545, 27456, 52801, 92576, 151425, 234640, 348161, 498576, 693121, 939680, 1246785, 1623616, 2080001, 2626416, 3273985, 4034480, 4920321, 5944576, 7120961, 8463840, 9988225, 11709776, 13644801, 15810256, 18223745, 20903520

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly octagonal numbers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Octagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000567, A002817, A000583, A232713, A063249.

[n*(3*n-2)*(9*n^2-6*n-2): n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Table[n (3 n - 2) (9 n^2 - 6 n - 2), {n, 0, 30}]

concat(0, Vec(x*(1+171*x+411*x^2+65*x^3)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 171*x + 411*x^2 + 65*x^3)/(1 - x)^5.
a(n) = A000567(A000567(n)).
Sum_{n>0} 1/a(n) = (sqrt(3)*gamma + sqrt(3)*polygamma(0, 1/3) - polygamma(0, (1/3)*(2 - sqrt(3))) + polygamma(0, (1/3)*(2 + sqrt(3))))/(4*sqrt(3)) = 1.006842786293...,where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

A264894 a(n) = n(7n - 5)(49n^2 - 35*n - 10)/8.

Original entry on oeis.org

0, 1, 261, 1956, 7291, 19500, 42846, 82621, 145146, 237771, 368875, 547866, 785181, 1092286, 1481676, 1966875, 2562436, 3283941, 4148001, 5172256, 6375375, 7777056, 9398026, 11260041, 13385886, 15799375, 18525351, 21589686, 25019281, 28842066, 33087000

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly 9-gonal (or nonagonal) numbers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Nonagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A001106, A002817, A000583, A232713, A063249.

[n*(7*n-5)*(49*n^2-35*n-10)/8: n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Table[n (7 n - 5) (49 n^2 - 35 n - 10)/8, {n, 0, 30}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,261,1956,7291},40] (* Harvey P. Dale, Apr 29 2017 *)

vector(100, n, n--; n*(7*n-5)*(49*n^2-35*n-10)/8) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 256*x + 661*x^2 + 111*x^3)/(1 - x)^5.
a(n) = A001106(A001106(n)).
Sum_{n>0} 1/a(n) = (4*(sqrt(65)*gamma + sqrt(65)*polygamma(0, 2/7) - 5*polygamma(0, (1/14)*(9 - sqrt(65))) + 5*polygamma(0, (1/14)*(9 + sqrt(65)))))/(25*sqrt(65)) = 1.0045877861645573..., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

A264895 a(n) = n(4n - 3)(16n^2 - 12*n - 3).

Original entry on oeis.org

0, 1, 370, 2835, 10660, 28645, 63126, 121975, 214600, 351945, 546490, 812251, 1164780, 1621165, 2200030, 2921535, 3807376, 4880785, 6166530, 7690915, 9481780, 11568501, 13981990, 16754695, 19920600, 23515225, 27575626, 32140395, 37249660, 42945085, 49269870

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly 10-gonal (or decagonal) numbers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Decagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A001107, A002817, A000583, A232713, A063249.

[n*(4*n - 3)*(16*n^2 - 12*n - 3): n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Table[n (4 n - 3) (16 n^2 - 12 n - 3), {n, 0, 30}]
LinearRecurrence[{5,-10,10,-5,1}, {0, 1, 370, 2835, 10660}, 50] (* G. C. Greubel, Sep 07 2018 *)

vector(100, n, n--; n*(4*n-3)*(16*n^2-12*n-3)) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 365*x + 995*x^2 + 175*x^3)/(1 - x)^5.
a(n) = A001107(A001107(n)).
Sum_{n>0} 1/a(n) = (sqrt(21)*gamma + sqrt(21)*polygamma(0, 1/4)  - 3*polygamma(0, (1/8)*(5 - sqrt(21))) + 3*polygamma(0, (1/8)*(5 + sqrt(21))))/(9*sqrt(21))= 1.00322253307732984...., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

cp's OEIS Frontend

A232713 Doubly pentagonal numbers: a(n) = n*(3*n-2)*(3*n-1)*(3*n+1)/8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A329755 Doubly hexagonal pyramidal numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A264891 a(n) = n*(5*n - 3)*(25*n^2 - 15*n - 6)/8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A264892 a(n) = n*(3*n - 2)*(9*n^2 - 6*n - 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A264894 a(n) = n*(7*n - 5)*(49*n^2 - 35*n - 10)/8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A264895 a(n) = n*(4*n - 3)*(16*n^2 - 12*n - 3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A232713 Doubly pentagonal numbers: a(n) = n(3n-2)(3n-1)(3n+1)/8.

A264891 a(n) = n(5n - 3)(25n^2 - 15*n - 6)/8.

A264892 a(n) = n(3n - 2)(9n^2 - 6*n - 2).

A264894 a(n) = n(7n - 5)(49n^2 - 35*n - 10)/8.

A264895 a(n) = n(4n - 3)(16n^2 - 12*n - 3).