A163761 -id:A163761 - cp's OEIS frontend

A262221 a(n) = 25n(n + 1)/2 + 1.

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

Offset: 0

Bruno Berselli, Sep 15 2015

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, ...

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

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).

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.

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

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 *)

PARI
```
vector(50, n, n--; 25*n*(n+1)/2+1)
```
Sage
```
[25*n*(n+1)/2+1 for n in (0..50)]
```

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

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 5, 4, 0, 0, 9, 12, 6, 0, 0, 14, 24, 21, 8, 0, 0, 20, 40, 45, 32, 10, 0, 0, 27, 60, 78, 72, 45, 12, 0, 0, 35, 84, 120, 128, 105, 60, 14, 0, 0, 44, 112, 171, 200, 190, 144, 77, 16, 0, 0, 54, 144, 231, 288, 300, 264, 189, 96, 18, 0

Offset: 0

Peter Luschny, Mar 22 2023

A detailed combinatorial interpretation can be found in A361682.

			[0] 0,  0,  0,   0,   0,   0,   0,    0, ...  A000004
[1] 0,  2,  5,   9,  14,  20,  27,   35, ...  A000096
[2] 0,  4, 12,  24,  40,  60,  84,  112, ...  A046092
[3] 0,  6, 21,  45,  78, 120, 171,  231, ...  A081266
[4] 0,  8, 32,  72, 128, 200, 288,  392, ...  A139098
[5] 0, 10, 45, 105, 190, 300, 435,  595, ...
[6] 0, 12, 60, 144, 264, 420, 612,  840, ...  A153792
[7] 0, 14, 77, 189, 350, 560, 819, 1127, ...
       | A028347 |     A163761
     A005843  A067725
.
[0] 0;
[1] 0,  0;
[2] 0,  2,   0;
[3] 0,  5,   4,   0;
[4] 0,  9,  12,   6,   0;
[5] 0, 14,  24,  21,   8,   0;
[6] 0, 20,  40,  45,  32,  10,   0;
[7] 0, 27,  60,  78,  72,  45,  12,  0;
[8] 0, 35,  84, 120, 128, 105,  60, 14,  0;
[9] 0, 44, 112, 171, 200, 190, 144, 77, 16, 0;

Rows: A000004, A000096, A046092, A081266, A139098, A153792.
Columns: A000004, A005843, A028347, A067725, A163761, A068380.
Cf. A361682.

A := (n, k) -> n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;

A(n, k) = n*k*(4 + n*(k - 1))/2.
T(n, k) = k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = A361682(n, k) - 1.

cp's OEIS Frontend

A262221 a(n) = 25n(n + 1)/2 + 1.

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Magma

Mathematica

PARI

Sage

Formula

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

cp's OEIS Frontend

A262221 a(n) = 25*n*(n + 1)/2 + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A262221 a(n) = 25n(n + 1)/2 + 1.