A264850 -id:A264850

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A220212

Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

+10
13

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

LINKS

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).

Index to sequences related to pyramidal numbers.

FORMULA

G.f.: x*(1+11*x)/(1-x)^5.

a(n) = n*(n+1)*(n+2)*(3*n-2)/6.

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

MATHEMATICA

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]

CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

PROG

(Magma) A051866:=func<n | n*(6*n-5)>; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

(Magma) I:=[0, 1, 16, 70, 200]; [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..50]]; // Vincenzo Librandi, Aug 18 2013

CROSSREFS

Cf. A135021, A172073.

Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):

k= 2 (A000027 ): A000292;

k= 3 (A000217 ): A000332 (after the third term);

k= 4 (A000290 ): A002415 (after the first term);

k= 5 (A000326 ): A001296;

k= 6 (A000384*): A002417;

k= 7 (A000566 ): A002418;

k= 8 (A000567*): A002419;

k= 9 (A001106*): A051740;

k=10 (A001107*): A051797;

k=11 (A051682*): A051798;

k=12 (A051624*): A051799;

k=13 (A051865*): A055268.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Dec 08 2012

STATUS

approved

A264851

a(n) = n*(n + 1)*(n + 2)*(4*n - 3)/6.

+10
2

0, 1, 20, 90, 260, 595, 1176, 2100, 3480, 5445, 8140, 11726, 16380, 22295, 29680, 38760, 49776, 62985, 78660, 97090, 118580, 143451, 172040, 204700, 241800, 283725, 330876, 383670, 442540, 507935, 580320, 660176, 748000, 844305, 949620, 1064490, 1189476

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Partial sums of 18-gonal (or octadecagonal) pyramidal numbers. Therefore, this is the case k=8 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

LINKS

Table of n, a(n) for n=0..36.

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Pyramidal Number

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

G.f.: x*(1 + 15*x)/(1 - x)^5.

a(n) = Sum_{k = 0..n} A172078(k).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

MATHEMATICA

Table[n (n + 1) (n + 2) (4 n - 3)/6, {n, 0, 50}]

PROG

(Magma) [n*(n + 1)*(n + 2)*(4*n - 3)/6: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

(PARI) a(n)=n*(n+1)*(n+2)*(4*n-3)/6 \\ Charles R Greathouse IV, Jul 26 2016

CROSSREFS

Cf. A172078.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Nov 26 2015

STATUS

approved

A264852

a(n) = n*(n + 1)*(n + 2)*(9*n - 7)/12.

+10
1

0, 1, 22, 100, 290, 665, 1316, 2352, 3900, 6105, 9130, 13156, 18382, 25025, 33320, 43520, 55896, 70737, 88350, 109060, 133210, 161161, 193292, 230000, 271700, 318825, 371826, 431172, 497350, 570865, 652240, 742016, 840752, 949025, 1067430, 1196580, 1337106

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Partial sums of 20-gonal (or icosagonal) pyramidal numbers. Therefore, this is the case k=9 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

LINKS

Table of n, a(n) for n=0..36.

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Pyramidal Number

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

G.f.: x*(1 + 17*x)/(1 - x)^5.

a(n) = Sum_{k = 0..n} A172082(k).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

MATHEMATICA

Table[n (n + 1) (n + 2) (9 n - 7)/12, {n, 0, 50}]

PROG