A051868 -id:A051868

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A139600

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

+10
49

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

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

OFFSET

0,6

COMMENTS

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

LINKS

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

FORMULA

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)

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

PROG

(Python)

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

(Magma)

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

(SageMath)

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

CROSSREFS

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).

KEYWORD

nonn,tabl,easy

AUTHOR

Omar E. Pol, Apr 27 2008

EXTENSIONS

Edited by Omar E. Pol, Jan 05 2009

STATUS

approved

A274978

Integers of the form m*(m + 6)/7.

+10
32

0, 1, 13, 16, 40, 45, 81, 88, 136, 145, 205, 216, 288, 301, 385, 400, 496, 513, 621, 640, 760, 781, 913, 936, 1080, 1105, 1261, 1288, 1456, 1485, 1665, 1696, 1888, 1921, 2125, 2160, 2376, 2413, 2641, 2680, 2920, 2961, 3213, 3256, 3520, 3565, 3841, 3888, 4176, 4225, 4525, 4576

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

OFFSET

1,3

COMMENTS

Nonnegative values of m are listed in A047274.

Also, numbers h such that 7*h + 9 is a square.

Equivalently, numbers of the form i*(7*i - 6) with i = 0, 1, -1, 2, -2, 3, -3, ...

Infinitely many squares belong to this sequence.

Generalized 16-gonal (or hexadecagonal) numbers. See the third comment. - Omar E. Pol, Jun 06 2018

Partial sums of A317312. - Omar E. Pol, Jul 28 2018

Exponents in expansion of Product_{n >= 1} (1 + x^(14*n-13))*(1 + x^(14*n-1))*(1 - x^(14*n)) = 1 + x + x^13 + x^16+ x^40 + .... - Peter Bala, Dec 10 2020

LINKS

Bruno Berselli, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

FORMULA

O.g.f.: x^2*(1 + 12*x + x^2)/((1 + x)^2*(1 - x)^3).

E.g.f.: (5*(2*x + 1)*exp(-x) + (14*x^2 - 5)*exp(x))/8.

a(n) = (14*(n-1)*n - 5*(2*n-1)*(-1)^n - 5)/8.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 6. - Wesley Ivan Hurt, Dec 18 2020

Sum_{n>=2} 1/a(n) = (7 + 6*Pi*cot(Pi/7))/36. - Amiram Eldar, Feb 28 2022

EXAMPLE

88 is in the sequence because 88 = 22*(22+6)/7 or also 88 = 4*(7*4-6).

MATHEMATICA

Select[m = Range[0, 200]; m (m + 6)/7, IntegerQ] (* Jean-François Alcover, Jul 21 2016 *)

Select[Table[(n(n+6))/7, {n, 0, 200}], IntegerQ] (* Harvey P. Dale, Sep 20 2022 *)

PROG

(Sage)

def A274978_list(len):

h = lambda m: m*(m+6)/7

return [h(m) for m in (0..len) if h(m) in ZZ]

print(A274978_list(179)) # Peter Luschny, Jul 18 2016

(Magma) [t: m in [0..200] | IsIntegral(t) where t is m*(m+6)/7];

CROSSREFS

Supersequence of A051868.

Cf. A317312.

Cf. sequences of the form m*(m+k)/(k+1): A000290 (k=0), A000217 (k=1), A001082 (k=2), A074377 (k=3), A195162 (k=4), A144065 (k=5), A274978 (k=6), A274979 (k=7), A218864 (k=8).

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), A195818 (k=14), A277082 (k=15), this sequence (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).

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jul 15 2016

STATUS

approved

A139601

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

+10
16

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

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

OFFSET

0,6

COMMENTS

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

LINKS

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.

FORMULA

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)

EXAMPLE

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 ..............................................

========================================================

MATHEMATICA

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 *)

PROG

(Magma)

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

(SageMath)

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

CROSSREFS

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.

KEYWORD

nonn,tabl,easy

AUTHOR

Omar E. Pol, Apr 27 2008

STATUS

approved

A255184

25-gonal numbers: a(n) = n*(23*n-21)/2.

+10
12

0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980

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

OFFSET

0,3

COMMENTS

If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).

Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:

for k=4, A000290(n) = A000217(n) + A000217(n-1);

for k=5, A000326(n) = A000290(n) + A000217(n-1);

for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.

This is the case k=25.

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).

E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.

LINKS

Luciano Ancora, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: x*(-1 - 22*x)/(-1 + x)^3.

a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.

Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 23*x^2/2). - Nikolaos Pantelidis, Feb 05 2023

MATHEMATICA

Table[n (23 n - 21)/2, {n, 40}]

PROG

(Magma) k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015

(PARI) a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 03 2015

STATUS

approved

A317302

Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

+10
5

0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66

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

OFFSET

0,10

COMMENTS

Note that the formula gives several kinds of numbers, for example:

Row 0 gives 0 together with A258837.

Row 1 gives 0 together with A080956.

Row 2 gives A001477, the nonnegative numbers.

For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).

LINKS

Table of n, a(n) for n=0..84.

Omar E. Pol, Polygonal numbers.

The OEIS, Polygonal numbers.

University of Surrey, Dept. of Mathematics, Polygonal Numbers - or Numbers as Shapes.

Eric Weisstein's World of Mathematics, Figurate Number.

Eric Weisstein's World of Mathematics, Polygonal Number.

Wikipedia, Polygonal number.

Index to sequences related to polygonal numbers

FORMULA

T(n,k) = A139600(n-2,k) if n >= 2.

T(n,k) = A139601(n-3,k) if n >= 3.

EXAMPLE

Array begins:

------------------------------------------------------------------------

n\k Numbers Seq. No. 0 1 2 3 4 5 6 7 8

------------------------------------------------------------------------

0 ............ (A258837): 0, 1, 0, -3, -8, -15, -24, -35, -48, ...

1 ............ (A080956): 0, 1, 1, 0, -2, -5, -9, -14, -20, ...

2 Nonnegatives A001477: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...

3 Triangulars A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, ...

4 Squares A000290: 0, 1, 4, 9, 16, 25, 36, 49, 64, ...

5 Pentagonals A000326: 0, 1, 5, 12, 22, 35, 51, 70, 92, ...

6 Hexagonals A000384: 0, 1, 6, 15, 28, 45, 66, 91, 120, ...

7 Heptagonals A000566: 0, 1, 7, 18, 34, 55, 81, 112, 148, ...

8 Octagonals A000567: 0, 1, 8, 21, 40, 65, 96, 133, 176, ...

9 9-gonals A001106: 0, 1, 9, 24, 46, 75, 111, 154, 204, ...

10 10-gonals A001107: 0, 1, 10, 27, 52, 85, 126, 175, 232, ...

11 11-gonals A051682: 0, 1, 11, 30, 58, 95, 141, 196, 260, ...

12 12-gonals A051624: 0, 1, 12, 33, 64, 105, 156, 217, 288, ...

13 13-gonals A051865: 0, 1, 13, 36, 70, 115, 171, 238, 316, ...

14 14-gonals A051866: 0, 1, 14, 39, 76, 125, 186, 259, 344, ...

15 15-gonals A051867: 0, 1, 15, 42, 82, 135, 201, 280, 372, ...

...

CROSSREFS

Column 0 gives A000004.

Column 1 gives A000012.

Column 2 gives A001477, which coincides with the row numbers.

Main diagonal gives A060354.

Row 0 gives 0 together with A258837.

Row 1 gives 0 together with A080956.

Row 2 gives A001477, the same as column 2.

For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).

Cf. A139600, A139601.

Cf. A303301 (similar table but with generalized polygonal numbers).

KEYWORD

sign,tabl,easy

AUTHOR

Omar E. Pol, Aug 09 2018

STATUS

approved

A131877

a(n) = 14*n + 1.

+10
4

1, 15, 29, 43, 57, 71, 85, 99, 113, 127, 141, 155, 169, 183, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 393, 407, 421, 435, 449, 463, 477, 491, 505, 519, 533, 547, 561, 575, 589, 603, 617, 631, 645, 659, 673, 687, 701, 715, 729

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

OFFSET

0,2

COMMENTS

Left column of triangle A131876.

Binomial transform of (1, 14, 0, 0, 0, ...).

Partial sums give A051868. - Leo Tavares, Mar 19 2023

LINKS

Table of n, a(n) for n=0..52.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

a(n) = 14*n + 1.

From Elmo R. Oliveira, Apr 03 2024: (Start)

G.f.: (1+13*x)/(1-x)^2.

E.g.f.: exp(x)*(1 + 14*x).

a(n) = A051868(n+1) - A051868(n).

a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

EXAMPLE

a(2) = 29 = 2*14 + 1.

a(2) = 29 = (1, 2, 1) dot (1, 14, 0) = (1 + 28 + 0).

MATHEMATICA

Range[1, 1000, 14] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

CROSSREFS

Cf. A008596, A131876.

Cf. A051868.

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jul 22 2007

STATUS

approved

A172076

a(n) = n*(n+1)*(14*n-11)/6.

+10
4

0, 1, 17, 62, 150, 295, 511, 812, 1212, 1725, 2365, 3146, 4082, 5187, 6475, 7960, 9656, 11577, 13737, 16150, 18830, 21791, 25047, 28612, 32500, 36725, 41301, 46242, 51562, 57275, 63395, 69936, 76912, 84337, 92225, 100590, 109446, 118807, 128687

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

OFFSET

0,3

COMMENTS

Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=7.

From Bruno Berselli, Dec 14 2010: (Start)

In fact, the sequence is related to A001106 by a(n) = n*A001106(n) - Sum_{k=0..n-1} A001106(k) and this is the case d=7 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6.

Also 16-gonal (or hexadecagonal) pyramidal numbers.

Inverse binomial transform of this sequence: 0, 1, 15, 14, 0, 0 (0 continued). (End)

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [From Bruno Berselli, Feb 13 2014]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.

Index to sequences related to pyramidal numbers.

Index to sequences related to pyramidal numbers

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: x*(1+13*x)/(1-x)^4. - Bruno Berselli, Dec 15 2010

a(n) = Sum_{i=0..n} A051868(i). - Bruno Berselli, Dec 15 2010

a(n) = Sum_{i=0..n-1} (n-i)*(14*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

E.g.f.: x*(6 + 45*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

MAPLE

A172076:=n->n*(n+1)*(14*n-11)/6; seq(A172076(n), n=0..50); # Wesley Ivan Hurt, Feb 26 2014

MATHEMATICA

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 17, 62}, 50] (* Vincenzo Librandi, Mar 01 2012 *)

PROG

(PARI) vector(40, n, n*(n-1)*(14*n-25)/6) \\ G. C. Greubel, Aug 30 2019

(Magma) [n*(n+1)*(14*n-11)/6: n in [0..40]] // G. C. Greubel, Aug 30 2019

(Sage) [n*(n+1)*(14*n-11)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

(GAP) List([0..40], n-> n*(n+1)*(14*n-11)/6); # G. C. Greubel, Aug 30 2019

CROSSREFS

Cf. similar sequences listed in A237616.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Jan 25 2010

STATUS

approved

A256650

30-gonal pyramidal numbers: a(n) = n*(n+1)*(28*n-25)/6.

+10
2

0, 1, 31, 118, 290, 575, 1001, 1596, 2388, 3405, 4675, 6226, 8086, 10283, 12845, 15800, 19176, 23001, 27303, 32110, 37450, 43351, 49841, 56948, 64700, 73125, 82251, 92106, 102718, 114115, 126325, 139376, 153296, 168113, 183855, 200550, 218226, 236911, 256633

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

OFFSET

0,3

COMMENTS

See comments in A256645.

This sequence is related to A051868 by a(n) = n*A051868(n) - Sum_{i=0..n-1} A051868(i). [Bruno Berselli, Apr 09 2015]

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (28th row of the table).

LINKS

Luciano Ancora, Table of n, a(n) for n = 0..1000

Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: x*(1 + 27*x)/(1 - x)^4.

a(n) = A000292(n) + 27*A000292(n-1).

MATHEMATICA

Table[n (n + 1) (28 n - 25)/6, {n, 0, 40}]

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 31, 118}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

PROG

(Magma) [n*(n+1)*(28*n-25)/6: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

CROSSREFS

Partial sums of A254474.

Cf. similar sequences listed in A237616.

Cf. A000292, A051868.

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 07 2015

STATUS

approved

A360436

32-gonal numbers: a(n) = n*(15*n-14).

+10
1

0, 1, 32, 93, 184, 305, 456, 637, 848, 1089, 1360, 1661, 1992, 2353, 2744, 3165, 3616, 4097, 4608, 5149, 5720, 6321, 6952, 7613, 8304, 9025, 9776, 10557, 11368, 12209, 13080, 13981, 14912, 15873, 16864, 17885, 18936, 20017, 21128, 22269, 23440, 24641, 25872

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..42.

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: x*(1 + 29*x)/(1 - x)^3.

E.g.f.: exp(x)*(x + 15*x^2).

MATHEMATICA

Table[n (15 n - 14), {n, 30}]

PROG

(PARI) a(n)=n*(15*n-14) \\ Charles R Greathouse IV, Feb 07 2023

CROSSREFS

Cf. A051868, A254474, A161935, A282853.

KEYWORD

nonn,easy

AUTHOR

Nikolaos Pantelidis, Feb 07 2023

STATUS

approved

A373665

Positive integers that cannot be written as a sum of a practical number and a 16-gonal number.

+10
1

10, 11, 14, 15, 23, 26, 27, 35, 38, 39, 45, 50, 59, 62, 68, 71, 74, 83, 86, 95, 98, 102, 103, 107, 110, 114, 115, 119, 122, 131, 134, 137, 138, 139, 143, 155, 158, 159, 164, 167, 170, 174, 179, 182, 183, 186, 190, 191, 194, 202, 203, 206, 215, 219, 227, 230

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

OFFSET

1,1

COMMENTS

Somu and Tran (2024) proved that there are infinitely many such integers. More generally, infinitely many positive integers cannot be written as a sum of a practical number and an s-gonal number if s is congruent to 4 modulo 12.

LINKS

Duc Van Khanh Tran, Table of n, a(n) for n = 1..10000

Sai Teja Somu and Duc Van Khanh Tran, On sums of practical numbers and polygonal numbers, Journal of Integer Sequences, 27(5), 2024.

CROSSREFS

Cf. A005153, A051868.

KEYWORD

nonn

AUTHOR

Duc Van Khanh Tran, Jun 12 2024

STATUS

approved

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