A035608 -id:A035608 - cp's OEIS frontend

A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

0, -1, 10, -20, 38, -57, 84, -112, 148, -185, 230, -276, 330, -385, 448, -512, 584, -657, 738, -820, 910, -1001, 1100, -1200, 1308, -1417, 1534, -1652, 1778, -1905, 2040, -2176, 2320, -2465, 2618, -2772, 2934, -3097, 3268, -3440, 3620, -3801, 3990, -4180

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007586, A035608, A051682, A083392, A089594.

[(18*(-1)^n*n^2 + 4*(-1)^n*n - 7*(-1)^n + 7)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((18 n^2 + 4 n - 7) (-1)^n + 7)/8, {n, 0, 43}]
CoefficientList[Series[(x - 8 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
Accumulate[Times@@@Partition[Riffle[PolygonalNumber[11,Range[0,50]],{1,-1},{2,-1,2}],2]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{-2,0,2,1},{0,-1,10,-20},50] (* Harvey P. Dale, Aug 27 2019 *)

x='x+O('x^100); concat(0, Vec(-x*(1-8*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 8*x)/((1 - x)*(1 + x)^3).
a(n) = ((18*n^2 + 4*n - 7)*(-1)^n + 7)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A051682(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.
E.g.f.: (1/4)*(9*x^2 - 11*x)*cosh(x) - (1/4)*(9*x^2 - 11*x - 7)*sinh(x). - G. C. Greubel, Jan 27 2016

A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

Original entry on oeis.org

0, -1, 11, -22, 42, -63, 93, -124, 164, -205, 255, -306, 366, -427, 497, -568, 648, -729, 819, -910, 1010, -1111, 1221, -1332, 1452, -1573, 1703, -1834, 1974, -2115, 2265, -2416, 2576, -2737, 2907, -3078, 3258, -3439, 3629, -3820, 4020, -4221, 4431, -4642

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007587, A035608, A051624, A083392, A089594.

[1+(-1)^n*(5*n^2+n-2)/2: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[1 + (-1)^n (5 n^2 + n - 2)/2, {n, 0, 43}]
CoefficientList[Series[-x (1 - 9 x)/((1 - x) (1 + x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-9*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 9*x)/((1 - x)*(1 + x)^3).
a(n) = 1 + (-1)^n*(5*n^2 + n - 2)/2.
a(n) = Sum_{k = 0..n} (-1)^k*A051624(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

A337108 Square spiral constructed by greedy algorithm, so that each diagonal and antidiagonal contains distinct numbers.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Aug 16 2020

nonn

This sequence is a variant of A308896; here we walk a bishop, there a rook.

Visually, we have a superposition of two images that we can separate by considering the parity of the x and y coordinates (see illustrations in Links section).

			The spiral begins:
        7----7----6----7----1----7----4----7----7
        |                                       |
        6    5----5----4----1----5----5----5    6
        |    |                             |    |
        3    4    3----3----2----3----3    4    3
        |    |    |                   |    |    |
        6    5    2    1----1----1    2    4    6
        |    |    |    |         |    |    |    |
        0    0    0    0    0----0    0    0    0
        |    |    |    |              |    |    |
        6    3    2    2----1----2----2    3    6
        |    |    |                        |    |
        7    4    4----3----1----3----4----4    5
        |    |                                  |
        6    6----5----5----1----6----5----6----6
        |
        8----7----8----7----1----7----4----7----8

Rémy Sigrist, Table of n, a(n) for n = 0..10200
Rémy Sigrist, Colored representation of the spiral for -512 <= x, y <= 512
Rémy Sigrist, Colored representation of the spiral for -512 <= x, y <= 512 and x any y have the same parity
Rémy Sigrist, Colored representation of the spiral for -512 <= x, y <= 512 and x and y have different parity
Rémy Sigrist, PARI program for A337108

See A274641 and A308896 for similar sequences.

Cf. A035608.

PARI
```
\\ See Links section.
```

a(n) = 0 iff n belongs to A035608.

A104571 Triangle T(n,k) = A042948(n-k+1) read by rows, 0<=k<=n.

Original entry on oeis.org

1, 4, 1, 5, 4, 1, 8, 5, 4, 1, 9, 8, 5, 4, 1, 12, 9, 8, 5, 4, 1, 13, 12, 9, 8, 5, 4, 1, 16, 13, 12, 9, 8, 5, 4, 1

Offset: 0

Gary W. Adamson, Mar 16 2005

			The first few rows are:
1;
4, 1;
5, 4, 1;
8, 5, 4, 1;
9, 8, 5, 4, 1;
...

Cf. A042948, A035608 (row sums), A104570, A104569, A074377.

The triangle is extracted from the product of lower triangular matrices (with the rest of the terms all zeros): G * R (or R * G); G = [1; 3, 1; 1, 3, 1; 3, 1, 3, 1;...]; R = [1; 1, 1; 1, 1, 1;...].

A131307 (A127701 * A000012 + A000012(signed) * A127701) - A000012.

Original entry on oeis.org

1, 2, 3, 3, 2, 5, 4, 5, 2, 7, 5, 4, 7, 2, 9, 6, 7, 4, 9, 2, 11, 7, 6, 9, 4, 11, 2, 13, 8, 9, 6, 11, 4, 13, 2, 15, 9, 8, 11, 6, 13, 4, 15, 2, 17, 10, 11, 8, 13, 6, 15, 4, 17, 2, 19

Offset: 1

Gary W. Adamson, Sep 30 2007

Row sums = A035608: (1, 5, 10, 18, 27, 39, 52, ...).

			First few rows of the triangle:
   1;
   2,  3;
   3,  2,  5;
   4,  5,  2,  7;
   5,  4,  7,  2,  9;
   6,  7,  4,  9,  2, 11;
   7,  6,  9,  4, 11,  2, 13;
   8,  9,  6, 11,  4, 13,  2, 15;
   9,  8, 11,  6, 13,  4, 15,  2, 17;
  10, 11,  8, 13,  6, 15,  4, 17,  2, 19;
  ...

Cf. A000012, A127701, A035608.

(A127701 * A000012 + A000012(signed) * A127701) - A000012; as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; -1,1,-1,1; ...).

cp's OEIS Frontend

A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

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Magma

Mathematica

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A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

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Magma

Mathematica

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A337108 Square spiral constructed by greedy algorithm, so that each diagonal and antidiagonal contains distinct numbers.

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PARI

Formula

A104571 Triangle T(n,k) = A042948(n-k+1) read by rows, 0<=k<=n.

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Author

Keywords

Examples

Crossrefs

Formula

A131307 (A127701 * A000012 + A000012(signed) * A127701) - A000012.

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Author

Keywords

Comments

Examples

Crossrefs

Formula