A255184 -id:A255184

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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

A303304

Generalized 25-gonal (or icosipentagonal) numbers: m*(23*m - 21)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

+10
29

0, 1, 22, 25, 67, 72, 135, 142, 226, 235, 340, 351, 477, 490, 637, 652, 820, 837, 1026, 1045, 1255, 1276, 1507, 1530, 1782, 1807, 2080, 2107, 2401, 2430, 2745, 2776, 3112, 3145, 3502, 3537, 3915, 3952, 4351, 4390, 4810, 4851, 5292, 5335, 5797, 5842, 6325, 6372, 6876, 6925

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

OFFSET

0,3

COMMENTS

Numbers k for which 184*k + 441 is a square. - Bruno Berselli, Jul 10 2018

Partial sums of A317321. - Omar E. Pol, Jul 28 2018

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

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

FORMULA

From Colin Barker, Jul 10 2018: (Start)

G.f.: x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2).

a(n) = n*(23*n + 42)/8 for n even.

a(n) = (23*n - 19)*(n + 1)/8 for n odd.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

(End)

Sum_{n>=1} 1/a(n) = 46/441 + 2*Pi*cot(2*Pi/23)/21. - Amiram Eldar, Mar 01 2022

MAPLE

seq(coeff(series(x*(x^2+21*x+1)/((1-x)^3*(1+x)^2), x, n+1), x, n), n=0..50); # Muniru A Asiru, Jul 10 2018

MATHEMATICA

CoefficientList[Series[x (1 + 21 x + x^2)/((1 - x)^3*(1 + x)^2), {x, 0, 49}], x] (* or *)

Array[PolygonalNumber[25, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 50, 0] (* Michael De Vlieger, Jul 10 2018 *)

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 22, 25, 67}, 50] (* Robert G. Wilson v, Jul 15 2018 *)

PROG

(PARI) concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 10 2018

(GAP) a:=[0, 1, 22, 25, 67];; for n in [6..50] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 10 2018

CROSSREFS

Cf. A255184, A317321.

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), 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), this sequence (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

KEYWORD

nonn,easy

AUTHOR

Omar E. Pol, Jul 10 2018

STATUS

approved

A139601

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

+10
17

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

A254474

30-gonal numbers: a(n) = n*(14*n-13).

+10
15

0, 1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860, 2197, 2562, 2955, 3376, 3825, 4302, 4807, 5340, 5901, 6490, 7107, 7752, 8425, 9126, 9855, 10612, 11397, 12210, 13051, 13920, 14817, 15742, 16695, 17676, 18685, 19722, 20787, 21880

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

OFFSET

0,3

COMMENTS

See comments in A255184.

Also star 15-gonal numbers.

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (28th row of the table).

LINKS

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

Luciano Ancora, Polygonal and Pyramidal numbers, Section 1.

Index to sequences related to polygonal numbers

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

FORMULA

G.f.: x*(-1 - 27*x)/(-1 + x)^3.

a(n) = A000217(n) + 27*A000217(n-1).

a(n) = A051867(n) + 15*A000217(n-1).

Product_{n>=2} (1 - 1/a(n)) = 14/15. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 14*x^2). - Nikolaos Pantelidis, Feb 05 2023

MATHEMATICA

Table[n (14 n - 13), {n, 40}]

PROG

(PARI) a(n)=n*(14*n-13) \\ Charles R Greathouse IV, Jun 17 2017

CROSSREFS

Cf. similar sequences listed in A255184.

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 04 2015

STATUS

approved

A255187

29-gonal numbers: a(n) = n*(27*n-25)/2.

+10
11

0, 1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794, 2119, 2471, 2850, 3256, 3689, 4149, 4636, 5150, 5691, 6259, 6854, 7476, 8125, 8801, 9504, 10234, 10991, 11775, 12586, 13424, 14289, 15181, 16100, 17046, 18019, 19019, 20046, 21100

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

OFFSET

0,3

COMMENTS

See comments in A255184.

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (27th row of the table).

LINKS

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

Index to sequences related to polygonal numbers

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

FORMULA

G.f.: x*(-1 - 26*x)/(-1 + x)^3.

a(n) = A000217(n) + 26*A000217(n-1).

Product_{n>=2} (1 - 1/a(n)) = 27/29. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 27*x^2/2). - Nikolaos Pantelidis, Feb 06 2023

MATHEMATICA

Table[n (27 n - 25)/2, {n, 40}]

PolygonalNumber[29, Range[0, 40]] (* Harvey P. Dale, Jul 25 2021 *)

PROG

(PARI) a(n)=n*(27*n-25)/2 \\ Charles R Greathouse IV, Jun 17 2017

CROSSREFS

Cf. similar sequences listed in A255184.

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 04 2015

STATUS

approved

A255185

26-gonal numbers: a(n) = n*(12*n-11).

+10
10

0, 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, 1596, 1885, 2198, 2535, 2896, 3281, 3690, 4123, 4580, 5061, 5566, 6095, 6648, 7225, 7826, 8451, 9100, 9773, 10470, 11191, 11936, 12705, 13498, 14315, 15156, 16021, 16910, 17823, 18760

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

OFFSET

0,3

COMMENTS

See comments in A255184.

Also star 13-gonal number: a(n) = A051865(n) + 13*A000217(n-1).

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (24th row of the table).

LINKS

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

Luciano Ancora, Polygonal and Pyramidal numbers, Section 1.

Index to sequences related to polygonal numbers

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

FORMULA

G.f.: x*(1 + 23*x)/(1 - x)^3.

a(n) = A000217(n) + 23*A000217(n-1).

Product_{n>=2} (1 - 1/a(n)) = 12/13. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 12*x^2). - Nikolaos Pantelidis, Feb 05 2023

MATHEMATICA

Table[n (12 n - 11), {n, 50}]

PolygonalNumber[26, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 26}, 50] (* Harvey P. Dale, Feb 02 2017 *)

PROG

(PARI) a(n)=n*(12*n-11) \\ Charles R Greathouse IV, Jun 17 2017

(Magma) [n*(12*n-11): n in [0..50]]; // G. C. Greubel, Jul 12 2024

(SageMath) [n*(12*n-11) for n in range(51)] # G. C. Greubel, Jul 12 2024

CROSSREFS

Cf. similar sequences listed in A255184.

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 04 2015

STATUS

approved

A256645

25-gonal pyramidal numbers: a(n) = n*(n+1)*(23*n-20)/6.

+10
9

0, 1, 26, 98, 240, 475, 826, 1316, 1968, 2805, 3850, 5126, 6656, 8463, 10570, 13000, 15776, 18921, 22458, 26410, 30800, 35651, 40986, 46828, 53200, 60125, 67626, 75726, 84448, 93815, 103850, 114576, 126016, 138193, 151130, 164850, 179376, 194731, 210938

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

OFFSET

0,3

COMMENTS

If b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6 is n-th k-gonal pyramidal number, then b(n,k) = A000292(n) + (k-3)*A000292(n-1) (see Deza in References section, p. 96).

Also, b(n,k) = b(n,k-1) + A000292(n-1) (see Deza in References section, p. 95). Some examples:

for k=4, A000330(n) = A000292(n) + A000292(n-1);

for k=5, A002411(n) = A000330(n) + A000292(n-1);

for k=6, A002412(n) = A002411(n) + A000292(n-1), etc.

This is the case k=25.

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (23rd row of the table).

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index to sequences related to polygonal numbers

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

FORMULA

G.f.: x*(1 + 22*x)/(1 - x)^4.

a(n) = A000292(n) + 22*A000292(n-1) = A256716(n) + A000292(n-1), see comments.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Colin Barker, Apr 07 2015

MATHEMATICA

Table[n (n + 1) (23 n - 20)/6, {n, 0, 40}]

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 26, 98}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

PROG

(PARI) concat(0, Vec(x*(1 + 22*x)/(1 - x)^4 + O(x^100))) \\ Colin Barker, Apr 07 2015

(Magma) k:=25; [n*(n+1)*((k-2)*n-(k-5))/6: n in [0..40]]; // Vincenzo Librandi, Apr 08 2015

CROSSREFS

Partial sums of A255184.

Cf. similar sequences listed in A237616.

Cf. A000292, A256716.

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 07 2015

STATUS

approved

A255186

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

+10
8

0, 1, 27, 78, 154, 255, 381, 532, 708, 909, 1135, 1386, 1662, 1963, 2289, 2640, 3016, 3417, 3843, 4294, 4770, 5271, 5797, 6348, 6924, 7525, 8151, 8802, 9478, 10179, 10905, 11656, 12432, 13233, 14059, 14910, 15786, 16687, 17613, 18564, 19540

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

OFFSET

0,3

COMMENTS

See comments in A255184.

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (25th row of the table).

LINKS

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

Index to sequences related to polygonal numbers

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

FORMULA

G.f.: x*(-1 - 24*x)/(-1 + x)^3.

a(n) = A000217(n) + 24*A000217(n-1).

Product_{n>=2} (1 - 1/a(n)) = 25/27. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 25*x^2/2). - Nikolaos Pantelidis, Feb 06 2023

MATHEMATICA

Table[n (25 n - 23)/2, {n, 40}]

PROG

(PARI) a(n)=n*(25*n-23)/2 \\ Charles R Greathouse IV, Jun 17 2017

CROSSREFS

Cf. similar sequences listed in A255184.

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Apr 04 2015

STATUS

approved

A262221

a(n) = 25*n*(n + 1)/2 + 1.

+10
8

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651

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

OFFSET

0,2

COMMENTS

Also centered 25-gonal (or icosipentagonal) numbers.

This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.

For k=2*n, the formula shown above gives A011379.

Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.

Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.

Index entries for sequences related to centered polygonal numbers.

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

FORMULA

G.f.: (1 + 23*x + x^2)/(1 - x)^3.

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

a(n) = A123296(n) + 1.

a(n) = A000217(5*n+2) - 2.

a(n) = A034856(5*n+1).

a(n) = A186349(10*n+1).

a(n) = A054254(5*n+2) with n>0, a(0)=1.

a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.

Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=0} a(n)/n! = 77*e/2.

Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)

E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

MATHEMATICA

Table[25 n (n + 1)/2 + 1, {n, 0, 50}]

25*Accumulate[Range[0, 50]]+1 (* or *) LinearRecurrence[{3, -3, 1}, {1, 26, 76}, 50] (* Harvey P. Dale, Jan 29 2023 *)

PROG

(PARI) vector(50, n, n--; 25*n*(n+1)/2+1)

(Sage) [25*n*(n+1)/2+1 for n in (0..50)]

(Magma) [25*n*(n+1)/2+1: n in [0..50]];

CROSSREFS

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.

Cf. centered polygonal numbers listed in A069190.

Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Sep 15 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

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