A232713 -id:A232713 - cp's OEIS frontend

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962

Offset: 0

Bruno Berselli, Jan 31 2014

After 0, first trisection of A011779 and right border of A177708.

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).

Partial sums of A004188.
Cf. A000217, A000332, A011779, A177708.
Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).
Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).
Cf. A232713, A260810.

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

Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,51,145},40] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A000326(A000217(n)).
a(n) = A000217(n) + 9*A000332(n+2).
Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

A063249 Doubly hexagonal numbers.

Original entry on oeis.org

0, 1, 66, 435, 1540, 4005, 8646, 16471, 28680, 46665, 72010, 106491, 152076, 210925, 285390, 378015, 491536, 628881, 793170, 987715, 1216020, 1481781, 1788886, 2141415, 2543640, 3000025, 3515226, 4094091, 4741660, 5463165

Offset: 0

Henry Bottomley, Jul 11 2001

			a(2)=66 since 6 is the 2nd hexagonal number and 66 is the 6th hexagonal number.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002817 for doubly triangular numbers, A000583 for doubly square numbers and A232713 for doubly pentagonal numbers.

LinearRecurrence[{5,-10,10,-5,1},{0,1,66,435,1540},30] (* Harvey P. Dale, Dec 02 2016 *)

concat(0, Vec(-x*(15*x^3+115*x^2+61*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Sep 14 2014

a(n) = n*(2*n-1)(4*n^2-2*n-1) = A000384(A000384(n)).

G.f.: -x*(15*x^3+115*x^2+61*x+1) / (x-1)^5. - Colin Barker, Sep 14 2014

A260810 a(n) = n^2(3n^2 - 1)/2.

Original entry on oeis.org

0, 1, 22, 117, 376, 925, 1926, 3577, 6112, 9801, 14950, 21901, 31032, 42757, 57526, 75825, 98176, 125137, 157302, 195301, 239800, 291501, 351142, 419497, 497376, 585625, 685126, 796797, 921592, 1060501, 1214550, 1384801, 1572352, 1778337, 2003926, 2250325, 2518776

Offset: 0

Bruno Berselli, Jul 31 2015

Pentagonal numbers with square indices.

After 0, a(k) is a square if k is in A072256.

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).

Subsequence of A001318 and A245288 (see Formula field).
Cf. A000326, A193218 (first differences).
Cf. A000583 (squares with square indices), A002593 (hexagonal numbers with square indices).
Cf. A232713 (pentagonal numbers with pentagonal indices), A236770 (pentagonal numbers with triangular indices).

Magma
```
[n^2*(3*n^2-1)/2: n in [0..40]];
```

I:=[0,1,22,117,376]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Aug 23 2015

A260810:=n->n^2*(3*n^2 - 1)/2: seq(A260810(n), n=0..50); # Wesley Ivan Hurt, Apr 25 2017

Table[n^2 (3 n^2 - 1)/2, {n, 0, 40}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 22, 117, 376}, 40] (* Vincenzo Librandi, Aug 23 2015 *)

PARI
```
vector(40, n, n--; n^2*(3*n^2-1)/2)
```
Sage
```
[n^2*(3*n^2-1)/2 for n in (0..40)]
```

G.f.: x*(1 + x)*(1 + 16*x + x^2)/(1 - x)^5.
a(n) = a(-n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A245288(2*n^2).
a(n) = A001318(2*n^2-1) with A001318(-1) = 0.
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi^2/3 - sqrt(3)*Pi*cot(Pi/sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi*cosec(Pi/sqrt(3)) - Pi^2/6 - 3. (End)

A329754 Doubly pentagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075, 83338750, 191592126, 410450976, 828497488, 1589341950, 2917620000, 5154021376, 8801526501, 14585352318, 23529456550, 37052820000, 57089119626, 86233820926, 127923156648, 186649920000, 268221484375, 380065968126

Offset: 0

Ilya Gutkovskiy, Nov 20 2019

Eric Weisstein's World of Mathematics, Pentagonal 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. A000326, A002411, A140236, A232713, A329753, A329755, A329756, A329757.

A002411[n_] := n^2 (n + 1)/2; a[n_] := A002411[A002411[n]]; Table[a[n], {n, 0, 26}]
Table[Sum[k (3 k - 1)/2, {k, 0, n^2 (n + 1)/2}], {n, 0, 26}]
nmax = 26; CoefficientList[Series[x (1 + 116 x + 1863 x^2 + 7570 x^3 + 9350 x^4 + 3474 x^5 + 304 x^6 + 2 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075}, 27]

G.f.: x*(1 + 116*x + 1863*x^2 + 7570*x^3 + 9350*x^4 + 3474*x^5 + 304*x^6 + 2*x^7)/(1 - x)^10.
a(n) = A002411(A002411(n)).
a(n) = Sum_{k=0..A002411(n)} A000326(k).
a(n) = n^4 *(n^3+n^2+2) *(n+1)^2 /16. - R. J. Mathar, Nov 28 2019

A047584 Numbers that are congruent to {1, 3, 5, 6, 7} mod 8.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 53, 54, 55, 57, 59, 61, 62, 63, 65, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99, 101, 102, 103, 105, 107

Offset: 1

N. J. A. Sloane

Numbers n such that A232713(n) is divisible by n. [Bruno Berselli, Dec 11 2013]

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A232713.

Select[Range@ 107, Or[OddQ@ Mod[#, 8], Mod[#, 8] == 6] &] (* Michael De Vlieger, Oct 23 2015 *)
#+{1,3,5,6,7}&/@(8*Range[0,20])//Flatten (* Harvey P. Dale, May 13 2019 *)

x='x+O('x^100); Vec((1+x)*(1+x+x^2+x^4)/((1-x)^2*(1+x+x^2+x^3+ x^4))) \\ Altug Alkan, Oct 22 2015

G.f.: (1 + x)*(1 + x + x^2 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)). [Bruno Berselli, Dec 11 2013]
From Wesley Ivan Hurt, Dec 30 2016: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
a(n) = (40n - 10 + 3*(n mod 5) + 3*((n+1) mod 5) - 2*((n+2) mod 5) - 2*((n+3) mod 5) - 2*((n+4) mod 5))/25. (End)

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

A236770 a(n) = n*(n + 1)*(3*n^2 + 3*n - 2)/8.

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A063249 Doubly hexagonal numbers.

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A260810 a(n) = n^2*(3*n^2 - 1)/2.

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A329754 Doubly pentagonal pyramidal numbers.

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A047584 Numbers that are congruent to {1, 3, 5, 6, 7} mod 8.

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

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Magma

Maple

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

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A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

A260810 a(n) = n^2(3n^2 - 1)/2.

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).