A051866 -id:A051866 - cp's OEIS frontend

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

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

Offset: 0

Bruno Berselli, Dec 08 2012

Partial sums of A172073.

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

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.

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.

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

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

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 *)

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)

A275792 Decimal expansion of the sum of the reciprocals of the tetradecagonal numbers A051866.

Original entry on oeis.org

1, 1, 5, 0, 9, 8, 2, 3, 6, 8, 0, 9, 4, 6, 7, 6, 3, 8, 6, 3, 6, 3, 6, 8, 9, 8, 9, 6, 9, 5, 2, 6, 7, 5, 0, 5, 8, 3, 0, 9, 6, 6, 7, 0, 9, 5, 5, 1, 8, 7, 4, 9, 1, 0, 9, 8, 3, 9, 6, 4, 5, 7, 8, 4, 5, 0, 5, 0, 4, 2, 6, 9, 1, 0, 9, 1, 3, 6, 6, 7, 4, 1, 4, 0, 9, 6, 6, 7, 5, 5, 3, 7, 0, 6, 3, 0, 5, 1, 5

Offset: 1

Wolfdieter Lang, Sep 12 2016

See Table 1 of the Downey et al. link.
From Wolfdieter Lang, Nov 09 2017: (Start)
The general formula for S_{2*(k+1)} = Sum_{n>=0} 1/((n+1)*(k*n+1)) given in the Downey et al. link is a special case of the simpler formula for V(m,r) = Sum_{n>=0} 1/((n+1)*(m*n + r)), r = 1,2, ... ,m -1. V(m,r) = (m/(m-r))*v_m(r) in Koecher's notation. For this formula for m*v_m(r) see a comment in A294512.
The special case is m = k and r = 1, leading to S_{2*(k+1)} = V(k,1) = (log(k) + (Pi/2)*cot(Pi/k) - Sum_{j=1..k-1} cos(2*Pi*j/k)*log(2*sin(Pi*j/k)))/(k-1), for k >= 2.
S_14, for k=6, is then given by the formula below (also obtained from the more complicated formula of Downey et al.).
The partial sums are given in A294834/A294835.
(End)

			1.150982368094676386363689896952675058309...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193. See (6/5)*v_6(1) on p. 192.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.

Cf. A013661, A016627, A051866, A244641, A244645, A294512, A294834, A294835.

SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // G. C. Greubel, Mar 25 2024

RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)

2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ Michel Marcus, Nov 09 2017

numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # G. C. Greubel, Mar 25 2024

Sum_{n >= 1} 1/(n*(6*n - 5)) = 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10.

A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1).

Original entry on oeis.org

1, 15, 599, 23035, 2900123, 30112021, 1117973277, 96393597197, 6084978910411, 67042215785861, 4094947551504521, 274661892011507657, 20068897076286721961, 1586702257063428405419, 26992510145660626515763, 54017546409271099350401, 5242487768036648180534897, 180077149085745155963315797

Offset: 0

Wolfdieter Lang, Nov 20 2017

The corresponding denominators are given in A294835.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [6,1].
The limit of the series is V(6,1) = lim_{n -> oo} V(6,1;n) = (3/10)*log(3) + (2/5)*log(2) + (1/10)*Pi*sqrt(3). The value is 1.150982368094676386... given in A275792.

			The rationals V(6,1;n), n >= 0, begin: 1, 15/14, 599/546, 23035/20748, 2900123/2593500, 30112021/26799500, 1117973277/991581500, 96393597197/85276009000, 6084978910411/5372388567000, 67042215785861/59096274237000, 4094947551504521/3604872728457000, ...
V(6,1;10^6) = 1.150982200 (Maple, 10 digits) to be compared with the ten digits 1.150982368 obtained from V(6,1) given in A275792.

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

G. C. Greubel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A051866, A275792, A294512, A294835.

[Numerator((&+[1/((k + 1)*(6*k + 1)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018

Table[Numerator[Sum[1/((k + 1)*(6*k + 1)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 1)))); \\ Michel Marcus, Nov 21 2017

a(n) = numerator(V(6,1;n)) with V(6,1;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 1)) = Sum_{k=0..n} 1/A051866(k+1) = (1/5)*Sum_{k=0..n} (1/(k + 1/6) - 1/(k + 1)) = (-Psi(1/6) + Psi(n+7/6) - (gamma + Psi(n+2)))/5 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294835 Denominators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1), for k >= 0.

Original entry on oeis.org

1, 14, 546, 20748, 2593500, 26799500, 991581500, 85276009000, 5372388567000, 59096274237000, 3604872728457000, 241526472806619000, 17631432514883187000, 1392883168675771773000, 23679013867488120141000, 47358027734976240282000, 4593728690292695307354000, 157718018366715872219154000

Offset: 0

Wolfdieter Lang, Nov 20 2017

The corresponding numerators are given in A294834. Details are found there.

			See A294834 for the rationals.

Cf. A051866, A294834.

a(n) = denominator(sum(k=0, n, 1/((k + 1)*(6*k + 1)))); \\ Michel Marcus, Nov 21 2017

a(n) = denominator(V(6,1;n)) with V(6,1;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 1)) = Sum_{k=0..n} 1/A051866(k+1) = (1/5)*Sum_{k=0..n} (1/(k + 1/6) - 1/(k + 1)). For the formula in terms of the digamma function see A294834.

A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

Original entry on oeis.org

1, 17, 87, 287, 742, 1638, 3234, 5874, 9999, 16159, 25025, 37401, 54236, 76636, 105876, 143412, 190893, 250173, 323323, 412643, 520674, 650210, 804310, 986310, 1199835, 1448811, 1737477, 2070397, 2452472, 2888952, 3385448, 3947944, 4582809, 5296809, 6097119

Offset: 0

Ilya Gutkovskiy, Apr 12 2016

More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Triangular Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000217, A051866.

Cf. similar sequences of the convolution of triangular numbers and k-gonal numbers: A005585 (k=4), A051836 (k=5), A034263 (k=6), A027800 (k=7), A051843 (k=8), A051877 (k=9), A051878 (k=10), A051879 (k=11), A051880 (k=12), A056118 (k=13), this sequence (k=14).

/* From definition: */ P:=func; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 3)*P(i, 14): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 18 2016

[(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // Bruno Berselli, Apr 18 2016

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]
Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]

O.g.f.: (1 + 11*x)/(1 - x)^6.
E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.
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).
a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.
Sum_{n>=0} 1/a(n) = 20*((15552*(6*log(2) + 3*log(3) + 2*sqrt(3)*log(2 - sqrt(3)) + (2 - sqrt(3))*Pi) - 29449)/531867) = 1.07654258697...

Edited by Bruno Berselli, Apr 18 2016

A139600 Square array T(n,k) = n(k-1)k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.

The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011

			The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A   A   .   .   A    A    A    A
....................... 0   0   .   .   0    0    1    1
....................... 0   0   .   .   1    1    3    3
....................... 0   0   .   .   6    7    9    9
....................... 0   0   .   .   9    3    6    6
....................... 0   1   .   .   5    2    0    0
....................... 4   2   .   .   7    9    6    7
=========================================================
Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ...
Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ...
Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ...
Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ...
Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ...
Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ...
Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ...
9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ...
10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ...
11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ...
12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ...
...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.

Alois P. Heinz, Rows n = 0..140, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
Index to sequences related to polygonal numbers

Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
A formal extension negative n is in A326728.
Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).

T:= func< n,k | k*(n*(k-1)+2)/2 >;
A139600:= func< n,k | T(n-k, k) >;
[A139600(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T:= (n, k)-> n*(k-1)*k/2+k:
seq(seq(T(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2018

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def A139600Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield x
        x, y = x + y + n, y + n
for n in range(8):
    R = A139600Row(n)
    print([next(R) for  in range(11)]) # _Peter Luschny, Aug 04 2019

def T(n,k): return k*(n*(k-1)+2)/2
def A139600(n,k): return T(n-k, k)
flatten([[A139600(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = n*(k-1)*k/2+k.
T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016
From Stefano Spezia, Apr 12 2024: (Start)
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)

Edited by Omar E. Pol, Jan 05 2009

A195818 Generalized 14-gonal numbers: m(6m-5), m = 0,+1,-1,+2,-2,+3,-3,...

Original entry on oeis.org

0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175, 186, 246, 259, 329, 344, 424, 441, 531, 550, 650, 671, 781, 804, 924, 949, 1079, 1106, 1246, 1275, 1425, 1456, 1616, 1649, 1819, 1854, 2034, 2071, 2261, 2300, 2500, 2541, 2751, 2794, 3014, 3059, 3289

Offset: 0

Omar E. Pol, Sep 29 2011

Also generalized tetradecagonal numbers or generalized tetrakaidecagonal numbers.
Also A211014 and positive terms of A051866 interleaved. - Omar E. Pol, Aug 04 2012
Exponents in expansion of Product_{n >= 1} (1 + x^(12*n-11))*(1 + x^(12*n-1))*(1 - x^(12*n)) = 1 + x + x^11 + x^14 + x^34 + .... - Peter Bala, Dec 10 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Partial sums of A195817.
Column 10 of A195152.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), this sequence (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. A051866, A211014.

[(3*n*(n+1)+(2*n+1)*(-1)^n-1)/2: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

A195818:=func; [0] cat [A195818(n*m): m in [1,-1], n in [1..25]];

a:= n-> (m-> m*(6*m-5))(ceil(-(n+1)/2)*(-1)^n):
seq(a(n), n=0..46);  # Alois P. Heinz, Jun 08 2021

LinearRecurrence[{1,2,-2,-1,1},{0,1,11,14,34},50] (* Harvey P. Dale, Mar 13 2018 *)

Vec(-x*(x^2+10*x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 15 2013

a(n) = (3*n*(n+1) + (2*n+1)*(-1)^n - 1)/2. - Vincenzo Librandi, Sep 30 2011
G.f.: -x*(x^2+10*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 15 2013
Sum_{n>=1} 1/a(n) = 6/25 + sqrt(3)*Pi/5. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: (x*(3*x + 4)*cosh(x) + (3*x^2 + 8*x - 2)*sinh(x))/2. - Stefano Spezia, Jun 08 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (5*log(432)-6)/25. - Amiram Eldar, Feb 28 2022

A139273 a(n) = n(8n - 3).

Original entry on oeis.org

0, 5, 26, 63, 116, 185, 270, 371, 488, 621, 770, 935, 1116, 1313, 1526, 1755, 2000, 2261, 2538, 2831, 3140, 3465, 3806, 4163, 4536, 4925, 5330, 5751, 6188, 6641, 7110, 7595, 8096, 8613, 9146, 9695, 10260, 10841, 11438, 12051, 12680

Offset: 0

Omar E. Pol, Apr 26 2008

Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139277 in the same spiral.
Also, sequence of numbers of the form d*A000217(n-1) + 5*n with generating functions x*(5+(d-5)*x)/(1-x)^3; the inverse binomial transform is 0,5,d,0,0,.. (0 continued). See Crossrefs. - Bruno Berselli, Feb 11 2011
Even decagonal numbers divided by 2. - Omar E. Pol, Aug 19 2011

G. C. Greubel, Table of n, a(n) for n = 0..5000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734.

[ n*(8*n-3) : n in [0..40] ];  // Bruno Berselli, Feb 11 2011

Table[n (8 n - 3), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 26}, 40] (* Harvey P. Dale, Feb 02 2012 *)

a(n)=n*(8*n-3) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 8*n^2 - 3*n.
Sequences of the form a(n) = 8*n^2 + c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - R. J. Mathar, May 12 2008
a(n) = 16*n + a(n-1) - 11 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: x*(5 + 11*x)/(1 - x)^3.
a(n) = 4*A000217(n) + A051866(n). (End)
a(n) = A028994(n)/2. - Omar E. Pol, Aug 19 2011
a(0)=0, a(1)=5, a(2)=26; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 02 2012
E.g.f.: (8*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 18 2017
Sum_{n>=1} 1/a(n) = 4*log(2)/3 - (sqrt(2)-1)*Pi/6 - sqrt(2)*arccoth(sqrt(2))/3. - Amiram Eldar, Jul 03 2020

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1,  3,  6, 10,  15,  21,  28,
Squares ...... A000290: 0, 1,  4,  9, 16,  25,  36,  49,
Pentagonals .. A000326: 0, 1,  5, 12, 22,  35,  51,  70,
Hexagonals ... A000384: 0, 1,  6, 15, 28,  45,  66,  91,
Heptagonals .. A000566: 0, 1,  7, 18, 34,  55,  81, 112,
Octagonals ... A000567: 0, 1,  8, 21, 40,  65,  96, 133,
9-gonals ..... A001106: 0, 1,  9, 24, 46,  75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52,  85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58,  95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A033568 Second pentagonal numbers with odd index: a(n) = (2n-1)(3*n-1).

Original entry on oeis.org

1, 2, 15, 40, 77, 126, 187, 260, 345, 442, 551, 672, 805, 950, 1107, 1276, 1457, 1650, 1855, 2072, 2301, 2542, 2795, 3060, 3337, 3626, 3927, 4240, 4565, 4902, 5251, 5612, 5985, 6370, 6767, 7176, 7597, 8030, 8475, 8932, 9401, 9882, 10375, 10880, 11397, 11926

Offset: 0

N. J. A. Sloane

Sequence found by reading the segment (1, 2) together with the line (one of the diagonal axes) from 2, in the direction 2, 15, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001318, A005449, A033570, A049452, A049453, A051866.

List([0..50], n-> (2*n-1)*(3*n-1)); # G. C. Greubel, Oct 12 2019

[(2*n-1)*(3*n-1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq((2*n-1)*(3*n-1), n=0..50); # G. C. Greubel, Oct 12 2019

Table[(2*n-1)*(3*n-1),{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{3,-3,1},{1,2,15},50] (* Ray Chandler, Dec 08 2011 *)

a(n)=(2*n-1)*(3*n-1) \\ Charles R Greathouse IV, Sep 24 2015

[(2*n-1)*(3*n-1) for n in range(50)] # G. C. Greubel, Oct 12 2019

G.f.: (1-x+12*x^2)/(1-x)^3.
a(n) = a(n-1) + 12*n - 11 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
a(n) = 6*n^2 - 5*n + 1 = A051866(n) + 1. - Omar E. Pol, Jul 18 2012
E.g.f.: (1 + x + 6*x^2)*exp(x). - G. C. Greubel, Oct 12 2019
From Amiram Eldar, Feb 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + Pi/(2*sqrt(3)) + 2*log(2) - 3*log(3)/2.
Sum_{n>=0} (-1)^n/a(n) = 1 + (1/sqrt(3) - 1/2)*Pi - log(2). (End)

More terms from Ray Chandler, Dec 08 2011

cp's OEIS Frontend

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

A275792 Decimal expansion of the sum of the reciprocals of the tetradecagonal numbers A051866.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)*(6*k + 1) = A051866(k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A294835 Denominators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)*(6*k + 1) = A051866(k+1), for k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

Extensions

A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

SageMath

Formula

A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1).

A294835 Denominators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1), for k >= 0.

A139600 Square array T(n,k) = n(k-1)k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

A195818 Generalized 14-gonal numbers: m(6m-5), m = 0,+1,-1,+2,-2,+3,-3,...

A139273 a(n) = n(8n - 3).

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

A033568 Second pentagonal numbers with odd index: a(n) = (2n-1)(3*n-1).